AI For Students Gemini

AI For Students Gemini — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  IRows

  iRows was a web-based spreadsheet in beta with a GUI similar to the traditional desktop-based spreadsheet applications, such as Microsoft Excel and OpenOffice.org. It was shut down on December 31, 2006, after it was announced that its two founders had been hired by Google. iRows used Ajax and XML. It was described as an example of a Web 2.0 system. iRows supported conventional spreadsheet features functions, value formatting and charts and added web oriented spreadsheet capabilities like collaboration (multiple people using a shared spreadsheet, sending a spreadsheet as a link instead of an attachment and ability to publish spreadsheets on other web pages (e.g. blogs).

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  Quadratic unconstrained binary optimization

  Quadratic unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP), is a combinatorial optimization problem with a wide range of applications from finance and economics to machine learning. QUBO is an NP hard problem, and for many classical problems from theoretical computer science, like maximum cut, graph coloring and the partition problem, embeddings into QUBO have been formulated. Embeddings for machine learning models include support-vector machines, clustering and probabilistic graphical models. Moreover, due to its close connection to Ising models, QUBO constitutes a central problem class for adiabatic quantum computation, where it is solved through a physical process called quantum annealing. == Definition == Let B = { 0 , 1 } {\displaystyle \mathbb {B} =\lbrace 0,1\rbrace } the set of binary digits (or bits), then B n {\displaystyle \mathbb {B} ^{n}} is the set of binary vectors of fixed length n ∈ N {\displaystyle n\in \mathbb {N} } . Given a symmetric or upper triangular matrix Q ∈ R n × n {\displaystyle {\boldsymbol {Q}}\in \mathbb {R} ^{n\times n}} , whose entries Q i j {\displaystyle Q_{ij}} define a weight for each pair of indices i , j ∈ { 1 , … , n } {\displaystyle i,j\in \lbrace 1,\dots ,n\rbrace } , we can define the function f Q : B n → R {\displaystyle f_{\boldsymbol {Q}}:\mathbb {B} ^{n}\rightarrow \mathbb {R} } that assigns a value to each binary vector x {\displaystyle {\boldsymbol {x}}} through f Q ( x ) = x ⊺ Q x = ∑ i = 1 n ∑ j = 1 n Q i j x i x j . {\displaystyle f_{\boldsymbol {Q}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }{\boldsymbol {Qx}}=\sum _{i=1}^{n}\sum _{j=1}^{n}Q_{ij}x_{i}x_{j}.} Alternatively, the linear and quadratic parts can be separated as f Q ′ , q ( x ) = x ⊺ Q ′ x + q ⊺ x , {\displaystyle f_{{\boldsymbol {Q}}',{\boldsymbol {q}}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }{\boldsymbol {Q}}'{\boldsymbol {x}}+{\boldsymbol {q}}^{\intercal }{\boldsymbol {x}},} where Q ′ ∈ R n × n {\displaystyle {\boldsymbol {Q}}'\in \mathbb {R} ^{n\times n}} and q ∈ R n {\displaystyle {\boldsymbol {q}}\in \mathbb {R} ^{n}} . This is equivalent to the previous definition through Q = Q ′ + diag ⁡ [ q ] {\displaystyle {\boldsymbol {Q}}={\boldsymbol {Q}}'+\operatorname {diag} [{\boldsymbol {q}}]} using the diag operator, exploiting that x = x ⋅ x {\displaystyle x=x\cdot x} for all binary values x {\displaystyle x} . Intuitively, the weight Q i j {\displaystyle Q_{ij}} is added if both x i = 1 {\displaystyle x_{i}=1} and x j = 1 {\displaystyle x_{j}=1} . The QUBO problem consists of finding a binary vector x ∗ {\displaystyle {\boldsymbol {x}}^{}} that minimizes f Q {\displaystyle f_{\boldsymbol {Q}}} , i.e., ∀ x ∈ B n : f Q ( x ∗ ) ≤ f Q ( x ) {\displaystyle \forall {\boldsymbol {x}}\in \mathbb {B} ^{n}:~f_{\boldsymbol {Q}}({\boldsymbol {x}}^{})\leq f_{\boldsymbol {Q}}({\boldsymbol {x}})} . In general, x ∗ {\displaystyle {\boldsymbol {x}}^{}} is not unique, meaning there may be a set of minimizing vectors with equal value w.r.t. f Q {\displaystyle f_{\boldsymbol {Q}}} . The complexity of QUBO arises from the number of candidate binary vectors to be evaluated, as | B n | = 2 n {\displaystyle \left|\mathbb {B} ^{n}\right|=2^{n}} grows exponentially in n {\displaystyle n} . Sometimes, QUBO is defined as the problem of maximizing f Q {\displaystyle f_{\boldsymbol {Q}}} , which is equivalent to minimizing f − Q = − f Q {\displaystyle f_{-{\boldsymbol {Q}}}=-f_{\boldsymbol {Q}}} . == Properties == QUBO is scale invariant for positive factors α > 0 {\displaystyle \alpha >0} , which leave the optimum x ∗ {\displaystyle {\boldsymbol {x}}^{}} unchanged: f α Q ( x ) = x ⊺ ( α Q ) x = α ( x ⊺ Q x ) = α f Q ( x ) {\displaystyle f_{\alpha {\boldsymbol {Q}}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }(\alpha {\boldsymbol {Q}}){\boldsymbol {x}}=\alpha ({\boldsymbol {x}}^{\intercal }{\boldsymbol {Qx}})=\alpha f_{\boldsymbol {Q}}({\boldsymbol {x}})} . In its general form, QUBO is NP-hard and cannot be solved efficiently by any known polynomial-time algorithm. However, there are polynomially-solvable special cases, where Q {\displaystyle {\boldsymbol {Q}}} has certain properties, for example: If all coefficients are positive, the optimum is trivially x ∗ = ( 0 , … , 0 ) ⊺ {\displaystyle {\boldsymbol {x}}^{}=(0,\dots ,0)^{\intercal }} . Similarly, if all coefficients are negative, the optimum is x ∗ = ( 1 , … , 1 ) ⊺ {\displaystyle {\boldsymbol {x}}^{}=(1,\dots ,1)^{\intercal }} . If Q {\displaystyle {\boldsymbol {Q}}} is diagonal, the bits can be optimized independently, and the problem is solvable in O ( n ) {\displaystyle {\mathcal {O}}(n)} . The optimal variable assignments are simply x i ∗ = 1 {\displaystyle x_{i}^{}=1} if Q i i < 0 {\displaystyle Q_{ii}<0} , and x i ∗ = 0 {\displaystyle x_{i}^{}=0} otherwise. If all off-diagonal elements of Q {\displaystyle {\boldsymbol {Q}}} are non-positive, the corresponding QUBO problem is solvable in polynomial time. QUBO can be solved using integer linear programming solvers like CPLEX or Gurobi Optimizer. This is possible since QUBO can be reformulated as a linear constrained binary optimization problem. To achieve this, substitute the product x i x j {\displaystyle x_{i}x_{j}} by an additional binary variable z i j ∈ B {\displaystyle z_{ij}\in \mathbb {B} } and add the constraints x i ≥ z i j {\displaystyle x_{i}\geq z_{ij}} , x j ≥ z i j {\displaystyle x_{j}\geq z_{ij}} and x i + x j − 1 ≤ z i j {\displaystyle x_{i}+x_{j}-1\leq z_{ij}} . Note that z i j {\displaystyle z_{ij}} can also be relaxed to continuous variables within the bounds zero and one. == Applications == QUBO is a structurally simple, yet computationally hard optimization problem. It can be used to encode a wide range of optimization problems from various scientific areas. === Maximum Cut === Given a graph G = ( V , E ) {\displaystyle G=(V,E)} with vertex set V = { 1 , … , n } {\displaystyle V=\lbrace 1,\dots ,n\rbrace } and edges E ⊆ V × V {\displaystyle E\subseteq V\times V} , the maximum cut (max-cut) problem consists of finding two subsets S , T ⊆ V {\displaystyle S,T\subseteq V} with T = V ∖ S {\displaystyle T=V\setminus S} , such that the number of edges between S {\displaystyle S} and T {\displaystyle T} is maximized. The more general weighted max-cut problem assumes edge weights w i j ≥ 0 ∀ i , j ∈ V {\displaystyle w_{ij}\geq 0~\forall i,j\in V} , with ( i , j ) ∉ E ⇒ w i j = 0 {\displaystyle (i,j)\notin E\Rightarrow w_{ij}=0} , and asks for a partition S , T ⊆ V {\displaystyle S,T\subseteq V} that maximizes the sum of edge weights between S {\displaystyle S} and T {\displaystyle T} , i.e., max S ⊆ V ∑ i ∈ S , j ∉ S w i j . {\displaystyle \max _{S\subseteq V}\sum _{i\in S,j\notin S}w_{ij}.} By setting w i j = 1 {\displaystyle w_{ij}=1} for all ( i , j ) ∈ E {\displaystyle (i,j)\in E} this becomes equivalent to the original max-cut problem above, which is why we focus on this more general form in the following. For every vertex in i ∈ V {\displaystyle i\in V} we introduce a binary variable x i {\displaystyle x_{i}} with the interpretation x i = 0 {\displaystyle x_{i}=0} if i ∈ S {\displaystyle i\in S} and x i = 1 {\displaystyle x_{i}=1} if i ∈ T {\displaystyle i\in T} . As T = V ∖ S {\displaystyle T=V\setminus S} , every i {\displaystyle i} is in exactly one set, meaning there is a 1:1 correspondence between binary vectors x ∈ B n {\displaystyle {\boldsymbol {x}}\in \mathbb {B} ^{n}} and partitions of V {\displaystyle V} into two subsets. We observe that, for any i , j ∈ V {\displaystyle i,j\in V} , the expression x i ( 1 − x j ) + ( 1 − x i ) x j {\displaystyle x_{i}(1-x_{j})+(1-x_{i})x_{j}} evaluates to 1 if and only if i {\displaystyle i} and j {\displaystyle j} are in different subsets, equivalent to logical XOR. Let W ∈ R + n × n {\displaystyle {\boldsymbol {W}}\in \mathbb {R} _{+}^{n\times n}} with W i j = w i j ∀ i , j ∈ V {\displaystyle W_{ij}=w_{ij}~\forall i,j\in V} . By extending above expression to matrix-vector form we find that x ⊺ W ( 1 − x ) + ( 1 − x ) ⊺ W x = − 2 x ⊺ W x + ( W 1 + W ⊺ 1 ) ⊺ x {\displaystyle {\boldsymbol {x}}^{\intercal }{\boldsymbol {W}}({\boldsymbol {1}}-{\boldsymbol {x}})+({\boldsymbol {1}}-{\boldsymbol {x}})^{\intercal }{\boldsymbol {Wx}}=-2{\boldsymbol {x}}^{\intercal }{\boldsymbol {Wx}}+({\boldsymbol {W1}}+{\boldsymbol {W}}^{\intercal }{\boldsymbol {1}})^{\intercal }{\boldsymbol {x}}} is the sum of weights of all edges between S {\displaystyle S} and T {\displaystyle T} , where 1 = ( 1 , 1 , … , 1 ) ⊺ ∈ R n {\displaystyle {\boldsymbol {1}}=(1,1,\dots ,1)^{\intercal }\in \mathbb {R} ^{n}} . As this is a quadratic function over x {\displaystyle {\boldsymbol {x}}} , it is a QUBO problem whose parameter matrix we can read from above expression as Q = 2 W − diag ⁡ [ W 1 + W ⊺ 1 ] , {\displaystyle {\boldsymbol {Q}}=2{\boldsymbol {W}}-\operatorname {diag} [{\boldsymbol {W1}}+{\boldsymbol {W}}^{\intercal }{\bol

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  Margin-infused relaxed algorithm

  Margin-infused relaxed algorithm (MIRA) is a machine learning and online algorithm for multiclass classification problems. It is designed to learn a set of parameters (vector or matrix) by processing all the given training examples one-by-one and updating the parameters according to each training example, so that the current training example is classified correctly with a margin against incorrect classifications at least as large as their loss. The change of the parameters is kept as small as possible. A two-class version called binary MIRA simplifies the algorithm by not requiring the solution of a quadratic programming problem (see below). When used in a one-vs-all configuration, binary MIRA can be extended to a multiclass learner that approximates full MIRA, but may be faster to train. The flow of the algorithm looks as follows: The update step is then formalized as a quadratic programming problem: Find m i n ‖ w ( i + 1 ) − w ( i ) ‖ {\displaystyle min\|w^{(i+1)}-w^{(i)}\|} , so that s c o r e ( x t , y t ) − s c o r e ( x t , y ′ ) ≥ L ( y t , y ′ ) ∀ y ′ {\displaystyle score(x_{t},y_{t})-score(x_{t},y')\geq L(y_{t},y')\ \forall y'} , i.e. the score of the current correct training y {\displaystyle y} must be greater than the score of any other possible y ′ {\displaystyle y'} by at least the loss (number of errors) of that y ′ {\displaystyle y'} in comparison to y {\displaystyle y} .

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  Fitness approximation

  Fitness approximation aims to approximate the objective or fitness functions in evolutionary optimization by building up machine learning models based on data collected from numerical simulations or physical experiments. The machine learning models for fitness approximation are also known as meta-models or surrogates, and evolutionary optimization based on approximated fitness evaluations are also known as surrogate-assisted evolutionary approximation. Fitness approximation in evolutionary optimization can be seen as a sub-area of data-driven evolutionary optimization. == Approximate models in function optimization == === Motivation === In many real-world optimization problems including engineering problems, the number of fitness function evaluations needed to obtain a good solution dominates the optimization cost. In order to obtain efficient optimization algorithms, it is crucial to use prior information gained during the optimization process. Conceptually, a natural approach to utilizing the known prior information is building a model of the fitness function to assist in the selection of candidate solutions for evaluation. A variety of techniques for constructing such a model, often also referred to as surrogates, metamodels or approximation models – for computationally expensive optimization problems have been considered. === Approaches === Common approaches to constructing approximate models based on learning and interpolation from known fitness values of a small population include: Low-degree polynomials and regression models Fourier surrogate modeling Artificial neural networks including Multilayer perceptrons Radial basis function network Support vector machines Due to the limited number of training samples and high dimensionality encountered in engineering design optimization, constructing a globally valid approximate model remains difficult. As a result, evolutionary algorithms using such approximate fitness functions may converge to local optima. Therefore, it can be beneficial to selectively use the original fitness function together with the approximate model.

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  GPT-5

  GPT-5 is a multimodal large language model developed by OpenAI and the fifth in its series of generative pre-trained transformer (GPT) foundation models. Preceded in the series by GPT-4, it was launched on August 7, 2025. It is publicly accessible to users of the chatbot products ChatGPT and Microsoft Copilot as well as to developers through the OpenAI API. == Background == On April 14, 2023, Sam Altman, the chief executive officer of OpenAI, spoke at an event at the Massachusetts Institute of Technology and said that the company was not training GPT-5 at that time. He stated that OpenAI was "prioritizing GPT-4 development" and that "we are not and won't for some time" release GPT-5. On July 18, OpenAI filed for a "GPT-5" trademark in the United States. On November 13, Altman confirmed to the Financial Times that the company was working to develop GPT-5. According to The Information, "[f]or much of the second half of 2024, OpenAI was developing a model known internally as Orion and intended to become GPT-5", "[b]ut the Orion effort failed to produce a better model, and the company instead released it as GPT-4.5 in February [2025]." By late July 2025, OpenAI was widely anticipated as planning to release GPT-5 in early August. On July 30, The Verge reported that "Microsoft is getting ready for GPT-5" as "sources familiar with Microsoft's AI plans" told an editor that the company was testing a new mode for its Copilot chatbot that would offer a model that "thinks deeply or quickly based on the task". On August 5, in the leadup to the release of GPT-5, OpenAI released GPT-OSS, a set of two open-weight models that have reasoning capabilities. GPT-5 was then unveiled during a livestream event on August 7. == Capabilities == At the time of its release, GPT-5 had state-of-the-art performance on benchmarks that test mathematics, programming, finance, and multimodal understanding. According to OpenAI, improvements over its predecessor models include faster response times, better coding and writing skills, more accurate answers to health questions, and lower levels of hallucination. Also, compared to previous models, GPT-5 aims to give safe, high-level responses to potentially harmful queries rather than outright declining them, an approach that OpenAI refers to as "safe completions", aiming to result "in GPT-5 being able to refuse more unsafe questions, while offering fewer rejections to users seeking harmless information." In addition, GPT-5 was trained to give more critical, "less effusively agreeable" answers compared to its predecessor models. Days before the launch of GPT-5, two early testers of the model stated that they were "impressed" by its ability to code and to solve mathematical and scientific problems. They suggested that the model shows great improvement from GPT-4, but not as large of a gain as from GPT-3 to GPT-4. A day prior to the release of GPT-5, during a press briefing, Sam Altman, the chief executive officer of OpenAI, called GPT-5 "a significant step along the path to AGI", referring to artificial general intelligence, the hypothetical level of intelligence that OpenAI defines as the ability to perform any economically valuable task that a human can. According to Altman, GPT-5 is "significantly better" than its predecessors, offering "PhD-level" abilities across a wide range of tasks. The exact energy consumption of GPT-5 use has not been disclosed by OpenAI. Researchers at the University of Rhode Island estimated that a medium-length response consumes slightly over 18 watt-hours, equivalent to using an incandescent bulb for 18 minutes. === Architecture === GPT-5 is a system that contains a fast, high-throughput model, a deeper reasoning model, and a real-time router that decides which model to use based on conversation type, complexity, tool needs, and explicit user intent. Altman had previously criticized the manual model picker for being overly complex, suggesting a need for unification. GPT-5 also includes agentic functionality through which it can set up its own desktop and can use its browser to search autonomously for sources that relate to its task. The GPT-5 system card defines two fast, high-throughput models – gpt-5-main and gpt-5-main-mini – and two thinking models – gpt-5-thinking and gpt-5-thinking-mini. In the OpenAI API, developers can access the thinking model, its mini version, and gpt-5-thinking-nano, an even smaller and faster nano version of the thinking model. The version of GPT-5 that is accessible via the API has adjustable reasoning effort (low, medium, high, or minimal) and verbosity (low, medium, or high). Additionally, ChatGPT provides access to gpt-5-thinking with a setting that makes use of parallel test-time compute, referred to as gpt-5-thinking-pro. == Limitations == === Safety === Neuraltrust, a security research company, claimed to have successfully compromised GPT-5 within its first day of testing the model. According to its report, it enabled GPT-5 to generate detailed instructions for manufacturing explosive devices. SPLX, another company, conducted similar tests and came to similar conclusions about GPT-5's security. Their assessments suggest that GPT-5 has significant security gaps, potentially rendering it as being unsafe for use in a corporate environment. == Training == According to AIMultiple, GPT-5 is natively multimodal, meaning that it was trained from scratch on multiple modalities (like text and images) at once without relying on already-trained language or vision models. Its training process involved three stages: unsupervised pretraining, supervised fine-tuning, and reinforcement learning from human feedback. Pretraining used a large-scale multilingual dataset of books, articles, web pages, academic papers, and licensed sources. GPT-5's visual and text capabilities were described as having been developed alongside each other throughout training, unlike with GPT-4. == Use == GPT-5 is used in ChatGPT. Although GPT-5 is free for all ChatGPT users, Plus users get higher use limits while Pro users get unlimited access to GPT-5 as well as limited access to GPT-5 Pro. Standard limits for lower-tier users on responses per hour still apply. Additionally, with the introduction of GPT-5, ChatGPT's "Advanced Voice Mode" was replaced by "ChatGPT Voice", which is supposed to enable more natural-sounding conversations. OpenAI stated that "Standard Voice Mode retires on September 9, 2025, unifying all users on ChatGPT Voice". On November 24, 2025, the feature of shopping research was added to ChatGPT, claimed to be a mini model post-trained on gpt-5-thinking-mini. GPT-5 is also available in Microsoft Copilot, and Microsoft stated that it will incorporate GPT-5 into a wide variety of its products. According to 9to5Mac, Apple Inc. is planning to integrate the model into the Apple Intelligence feature in its iOS 26, iPadOS 26, and macOS Tahoe operating systems. It is also accessible via the OpenAI API. A number of American companies were reported as having received access to GPT-5 ahead of its launch. OpenAI stated that the private health insurance company Oscar Health was checking applications from its policyholders with the model. In addition, Uber was using GPT-5 for its customer support system; GitLab, Windsurf, and Cursor were using the model for software development; and the Spanish bank BBVA was using it for financial analysis. Other companies that OpenAI listed as having used GPT-5 pre-release include Amgen, Lowe's, and Notion. == Reception == === Critical reviews === Grace Huckins in MIT Technology Review found that, "[w]hereas o1 was a major technological advancement, GPT-5 is, above all else, a refined product." In response to claims that Sam Altman, the chief executive officer of OpenAI, had made about the model, she stated that "GPT-5 will furnish a more pleasant and seamless user experience. That's not nothing, but it falls far short of the transformative AI future that Altman has spent much of the past year hyping." In response to Altman's claim that GPT-5 is "a significant step along the path" to artificial general intelligence, she noted: "[M]aybe he's right—but if so, it's a very small step." In The Information, Stephanie Palazzolo praised GPT-5's coding capabilities. According to Matteo Wong in The Atlantic, GPT-5 "is intuitive, fast, and efficient; adapts to human preferences and intentions; and is easy to personalize." He stated: "At this stage of the AI boom, when every major chatbot is legitimately helpful in numerous ways, benchmarks, science, and rigor feel almost insignificant. What matters is how the chatbot feels [...]". John Herrman from the New York magazine wrote: "Casual users who encounter GPT-5 through ChatGPT aren't likely to feel like they're using a completely different product [...] while people who use it for software development or in a corporate context are more likely to notice a major change." Mashable's Christian de Looper found that "GPT-5

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  Softmax function

  The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution over K possible outcomes. It is a generalization of the logistic function to multiple dimensions, and is used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes. == Definition == The softmax function takes as input a tuple z of K real numbers, and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some tuple components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval ( 0 , 1 ) {\displaystyle (0,1)} , and the components will add up to 1, so that they can be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities. Formally, the standard (unit) softmax function σ : R K → ( 0 , 1 ) K {\displaystyle \sigma :\mathbb {R} ^{K}\to (0,1)^{K}} , where ⁠ K > 1 {\displaystyle K>1} ⁠, takes a tuple z = ( z 1 , … , z K ) ∈ R K {\displaystyle \mathbf {z} =(z_{1},\dotsc ,z_{K})\in \mathbb {R} ^{K}} and computes each component of vector σ ( z ) ∈ ( 0 , 1 ) K {\displaystyle \sigma (\mathbf {z} )\in (0,1)^{K}} with σ ( z ) i = e z i ∑ j = 1 K e z j . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{z_{i}}}{\sum _{j=1}^{K}e^{z_{j}}}}\,.} In words, the softmax applies the standard exponential function to each element z i {\displaystyle z_{i}} of the input tuple z {\displaystyle \mathbf {z} } (consisting of K {\displaystyle K} real numbers), and normalizes these values by dividing by the sum of all these exponentials. The normalization ensures that the sum of the components of the output vector σ ( z ) {\displaystyle \sigma (\mathbf {z} )} is 1. The term "softmax" derives from the amplifying effects of the exponential on any maxima in the input tuple. For example, the standard softmax of ( 1 , 2 , 8 ) {\displaystyle (1,2,8)} is approximately ( 0.001 , 0.002 , 0.997 ) {\displaystyle (0.001,0.002,0.997)} , which amounts to assigning almost all of the total unit weight in the result to the position of the tuple's maximal element (of 8). In general, instead of e a different base b > 0 can be used. As above, if b > 1 then larger input components will result in larger output probabilities, and increasing the value of b will create probability distributions that are more concentrated around the positions of the largest input values. Conversely, if 0 < b < 1 then smaller input components will result in larger output probabilities, and decreasing the value of b will create probability distributions that are more concentrated around the positions of the smallest input values. Writing b = e β {\displaystyle b=e^{\beta }} or b = e − β {\displaystyle b=e^{-\beta }} (for real β) yields the expressions: σ ( z ) i = e β z i ∑ j = 1 K e β z j or σ ( z ) i = e − β z i ∑ j = 1 K e − β z j for i = 1 , … , K . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{\beta z_{i}}}{\sum _{j=1}^{K}e^{\beta z_{j}}}}{\text{ or }}\sigma (\mathbf {z} )_{i}={\frac {e^{-\beta z_{i}}}{\sum _{j=1}^{K}e^{-\beta z_{j}}}}{\text{ for }}i=1,\dotsc ,K.} A value proportional to the reciprocal of β is sometimes referred to as the temperature: β = 1 / k T {\textstyle \beta =1/kT} , where k is typically 1 or the Boltzmann constant and T is the temperature. A higher temperature results in a more uniform output distribution (i.e. with higher entropy; it is "more random"), while a lower temperature results in a sharper output distribution, with one value dominating. In some fields, the base is fixed, corresponding to a fixed scale, while in others the parameter β (or T) is varied. The softmax function is a multiple-variable generalization of the logistic function. == Interpretations == === Smooth arg max === The Softmax function is a smooth approximation to the arg max function: the function whose value is the index of a tuple's largest element. The name "softmax" may be misleading. Softmax is not a smooth maximum (that is, a smooth approximation to the maximum function). The term "softmax" is also used for the closely related LogSumExp function, which is a smooth maximum. For this reason, some prefer the more accurate term "softargmax", though the term "softmax" is conventional in machine learning. This section uses the term "softargmax" for clarity. Formally, instead of considering the arg max as a function with categorical output 1 , … , n {\displaystyle 1,\dots ,n} (corresponding to the index), consider the arg max function with one-hot representation of the output (assuming there is a unique maximum arg): a r g m a x ⁡ ( z 1 , … , z n ) = ( y 1 , … , y n ) = ( 0 , … , 0 , 1 , 0 , … , 0 ) , {\displaystyle \operatorname {arg\,max} (z_{1},\,\dots ,\,z_{n})=(y_{1},\,\dots ,\,y_{n})=(0,\,\dots ,\,0,\,1,\,0,\,\dots ,\,0),} where the output coordinate y i = 1 {\displaystyle y_{i}=1} if and only if i {\displaystyle i} is the arg max of ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} , meaning z i {\displaystyle z_{i}} is the unique maximum value of ( z 1 , … , z n ) {\displaystyle (z_{1},\,\dots ,\,z_{n})} . For example, in this encoding a r g m a x ⁡ ( 1 , 5 , 10 ) = ( 0 , 0 , 1 ) , {\displaystyle \operatorname {arg\,max} (1,5,10)=(0,0,1),} since the third argument is the maximum. This can be generalized to multiple arg max values (multiple equal z i {\displaystyle z_{i}} being the maximum) by dividing the 1 between all max args; formally 1/k where k is the number of arguments assuming the maximum. For example, a r g m a x ⁡ ( 1 , 5 , 5 ) = ( 0 , 1 / 2 , 1 / 2 ) , {\displaystyle \operatorname {arg\,max} (1,\,5,\,5)=(0,\,1/2,\,1/2),} since the second and third argument are both the maximum. In case all arguments are equal, this is simply a r g m a x ⁡ ( z , … , z ) = ( 1 / n , … , 1 / n ) . {\displaystyle \operatorname {arg\,max} (z,\dots ,z)=(1/n,\dots ,1/n).} Points z with multiple arg max values are singular points (or singularities, and form the singular set) – these are the points where arg max is discontinuous (with a jump discontinuity) – while points with a single arg max are known as non-singular or regular points. With the last expression given in the introduction, softargmax is now a smooth approximation of arg max: as ⁠ β → ∞ {\displaystyle \beta \to \infty } ⁠, softargmax converges to arg max. There are various notions of convergence of a function; softargmax converges to arg max pointwise, meaning for each fixed input z as ⁠ β → ∞ {\displaystyle \beta \to \infty } ⁠, σ β ( z ) → a r g m a x ⁡ ( z ) . {\displaystyle \sigma _{\beta }(\mathbf {z} )\to \operatorname {arg\,max} (\mathbf {z} ).} However, softargmax does not converge uniformly to arg max, meaning intuitively that different points converge at different rates, and may converge arbitrarily slowly. In fact, softargmax is continuous, but arg max is not continuous at the singular set where two coordinates are equal, while the uniform limit of continuous functions is continuous. The reason it fails to converge uniformly is that for inputs where two coordinates are almost equal (and one is the maximum), the arg max is the index of one or the other, so a small change in input yields a large change in output. For example, σ β ( 1 , 1.0001 ) → ( 0 , 1 ) , {\displaystyle \sigma _{\beta }(1,\,1.0001)\to (0,1),} but σ β ( 1 , 0.9999 ) → ( 1 , 0 ) , {\displaystyle \sigma _{\beta }(1,\,0.9999)\to (1,\,0),} and σ β ( 1 , 1 ) = 1 / 2 {\displaystyle \sigma _{\beta }(1,\,1)=1/2} for all inputs: the closer the points are to the singular set ( x , x ) {\displaystyle (x,x)} , the slower they converge. However, softargmax does converge compactly on the non-singular set. Conversely, as ⁠ β → − ∞ {\displaystyle \beta \to -\infty } ⁠, softargmax converges to arg min in the same way, where here the singular set is points with two arg min values. In the language of tropical analysis, the softmax is a deformation or "quantization" of arg max and arg min, corresponding to using the log semiring instead of the max-plus semiring (respectively min-plus semiring), and recovering the arg max or arg min by taking the limit is called "tropicalization" or "dequantization". It is also the case that, for any fixed β, if one input ⁠ z i {\displaystyle z_{i}} ⁠ is much larger than the others relative to the temperature, T = 1 / β {\displaystyle T=1/\beta } , the output is approximately the arg max. For example, a difference of 10 is large relative to a temperature of 1: σ ( 0 , 10 ) := σ 1 ( 0 , 10 ) = ( 1 / ( 1 + e 10 ) , e 10 / ( 1 + e 10 ) ) ≈ ( 0.00005 , 0.99995 ) {\displaystyle \sigma (0,\,10):=\sigma _{1}(0,\,10)=\left(1/\left(1+e^{10}\right),\,e^{10}/\left(1+e^{10}\right)\right)\approx (0.00005

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  Ground truth

  Ground truth is information that is known to be real or true, provided by direct observation and measurement (i.e. empirical evidence) as opposed to information provided by inference. The term ground truth appeared in remote sensing literature as early as 1972, when NASA described it as essential "data about ... materials on the earth's surface" used to calibrate measurements. It was later adopted by the statistical modeling and machine learning communities. == Etymology == The Oxford English Dictionary (s.v. ground truth) records the use of the word Groundtruth in the sense of 'fundamental truth' from Henry Ellison's poem "The Siberian Exile's Tale", published in 1833. == Usage == The term "ground truth" can be used as a noun, adjective, and verb. Noun: "ground truth" (no hyphen). Example: "The ground truth is essential for training accurate models." Adjective: "ground-truth" (hyphenated compound adjective). Example: "We need to use ground-truth data to validate the model." Verb: "to ground-truth" or "to groundtruth" (compound verb,). Example: "We need to ground-truth the results to ensure their accuracy." == Statistics and machine learning == In statistics and machine learning, ground truth is the ideal expected result, used in statistical models to prove or disprove research hypotheses. "Ground truthing" is the process of gathering the good data for this test. Ground truth is typically included in labeled data. In machine learning, "ground truth" is not necessarily objectively correct or true. For example, in training AI models or relevance rankers, it may be a set of judgments made by people or inferred from user behavior, which may depend on context. For example, in Bayesian spam filtering, a supervised learning system is typically trained by examples labeled as spam and non-spam. Although these labels may be subjective or inaccurate, they are considered ground truth. True ground truth in machine learning is objective data. For example, suppose we are testing a stereo vision system to see how well it can estimate 3D positions. A calibrated laser rangefinder may provide accurate distances as ground truth. == Remote sensing == In remote sensing, "ground truth" refers to information collected at the imaged location. Ground truth allows image data to be related to real features and materials on the ground. The collection of ground truth data enables calibration of remote-sensing data, and aids in the interpretation and analysis of what is being sensed. Examples include cartography, meteorology, analysis of aerial photographs, satellite imagery and other techniques in which data are gathered at a distance. More specifically, ground truth may refer to a process in which "pixels" on a satellite image are compared to what is imaged (at the time of capture) in order to verify the contents of the "pixels" in the image (noting that the concept of "pixel" is imaging-system-dependent). In the case of a classified image, supervised classification can help to determine the accuracy of the classification by the remote sensing system which can minimize error in the classification. Ground truth is usually done on site, correlating what is known with surface observations and measurements of various properties of the features of the ground resolution cells under study in the remotely sensed digital image. The process also involves taking geographic coordinates of the ground resolution cell with GPS technology and comparing those with the coordinates of the "pixel" being studied provided by the remote sensing software to understand and analyze the location errors and how it may affect a particular study. Ground truth is important in the initial supervised classification of an image. When the identity and location of land cover types are known through a combination of field work, maps, and personal experience these areas are known as training sites. The spectral characteristics of these areas are used to train the remote sensing software using decision rules for classifying the rest of the image. These decision rules such as Maximum Likelihood Classification, Parallelopiped Classification, and Minimum Distance Classification offer different techniques to classify an image. Additional ground truth sites allow the remote sensor to establish an error matrix that validates the accuracy of the classification method used. Different classification methods may have different percentages of error for a given classification project. It is important that the remote sensor chooses a classification method that works best with the number of classifications used while providing the least amount of error. Ground truth also helps with atmospheric correction. Since images from satellites have to pass through the atmosphere, they can get distorted because of absorption in the atmosphere. So ground truth can help fully identify objects in satellite photos. === Errors of commission === An example of an error of commission is when a pixel reports the presence of a feature (such a tree) that, in reality, is absent (no tree is actually present). Ground truthing ensures that the error matrices have a higher accuracy percentage than would be the case if no pixels were ground-truthed. This value is the complement of the user's accuracy, i.e. Commission Error = 1 - user's accuracy. === Errors of omission === An example of an error of omission is when pixels of a certain type, for example, maple trees, are not classified as maple trees. The process of ground-truthing helps to ensure that the pixel is classified correctly and the error matrices are more accurate. This value is the complement of the producer's accuracy, i.e. Omission Error = 1 - producer's accuracy == Geographical information systems == In GIS the spatial data is modeled as field (like in remote sensing raster images) or as object (like in vectorial map representation). They are modeled from the real world (also named geographical reality), typically by a cartographic process (illustrated). Geographic information systems such as GIS, GPS, and GNSS, have become so widespread that the term "ground truth" has taken on special meaning in that context. If the location coordinates returned by a location method such as GPS are an estimate of a location, then the "ground truth" is the actual location on Earth. A smart phone might return a set of estimated location coordinates such as 43.87870, −103.45901. The ground truth being estimated by those coordinates is the tip of George Washington's nose on Mount Rushmore. The accuracy of the estimate is the maximum distance between the location coordinates and the ground truth. We could say in this case that the estimate accuracy is 10 meters, meaning that the point on Earth represented by the location coordinates is thought to be within 10 meters of George's nose—the ground truth. In slang, the coordinates indicate where we think George Washington's nose is located, and the ground truth is where it really is. In practice a smart phone or hand-held GPS unit is routinely able to estimate the ground truth within 6–10 meters. Specialized instruments can reduce GPS measurement error to under a centimeter. == Military usage == US military slang uses "ground truth" to refer to the facts comprising a tactical situation—as opposed to intelligence reports, mission plans, and other descriptions reflecting the conative or policy-based projections of the industrial·military complex. The term appears in the title of the Iraq War documentary film The Ground Truth (2006), and also in military publications, for example Stars and Stripes saying: "Stripes decided to figure out what the ground truth was in Iraq."

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  Sufficient dimension reduction

  In statistics, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of dimension reduction with the concept of sufficiency. Dimension reduction has long been a primary goal of regression analysis. Given a response variable y and a p-dimensional predictor vector x {\displaystyle {\textbf {x}}} , regression analysis aims to study the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , the conditional distribution of y {\displaystyle y} given x {\displaystyle {\textbf {x}}} . A dimension reduction is a function R ( x ) {\displaystyle R({\textbf {x}})} that maps x {\displaystyle {\textbf {x}}} to a subset of R k {\displaystyle \mathbb {R} ^{k}} , k < p, thereby reducing the dimension of x {\displaystyle {\textbf {x}}} . For example, R ( x ) {\displaystyle R({\textbf {x}})} may be one or more linear combinations of x {\displaystyle {\textbf {x}}} . A dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} is said to be sufficient if the distribution of y ∣ R ( x ) {\displaystyle y\mid R({\textbf {x}})} is the same as that of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, no information about the regression is lost in reducing the dimension of x {\displaystyle {\textbf {x}}} if the reduction is sufficient. == Graphical motivation == In a regression setting, it is often useful to summarize the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} graphically. For instance, one may consider a scatterplot of y {\displaystyle y} versus one or more of the predictors or a linear combination of the predictors. A scatterplot that contains all available regression information is called a sufficient summary plot. When x {\displaystyle {\textbf {x}}} is high-dimensional, particularly when p ≥ 3 {\displaystyle p\geq 3} , it becomes increasingly challenging to construct and visually interpret sufficiency summary plots without reducing the data. Even three-dimensional scatter plots must be viewed via a computer program, and the third dimension can only be visualized by rotating the coordinate axes. However, if there exists a sufficient dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} with small enough dimension, a sufficient summary plot of y {\displaystyle y} versus R ( x ) {\displaystyle R({\textbf {x}})} may be constructed and visually interpreted with relative ease. Hence sufficient dimension reduction allows for graphical intuition about the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , which might not have otherwise been available for high-dimensional data. Most graphical methodology focuses primarily on dimension reduction involving linear combinations of x {\displaystyle {\textbf {x}}} . The rest of this article deals only with such reductions. == Dimension reduction subspace == Suppose R ( x ) = A T x {\displaystyle R({\textbf {x}})=A^{T}{\textbf {x}}} is a sufficient dimension reduction, where A {\displaystyle A} is a p × k {\displaystyle p\times k} matrix with rank k ≤ p {\displaystyle k\leq p} . Then the regression information for y ∣ x {\displaystyle y\mid {\textbf {x}}} can be inferred by studying the distribution of y ∣ A T x {\displaystyle y\mid A^{T}{\textbf {x}}} , and the plot of y {\displaystyle y} versus A T x {\displaystyle A^{T}{\textbf {x}}} is a sufficient summary plot. Without loss of generality, only the space spanned by the columns of A {\displaystyle A} need be considered. Let η {\displaystyle \eta } be a basis for the column space of A {\displaystyle A} , and let the space spanned by η {\displaystyle \eta } be denoted by S ( η ) {\displaystyle {\mathcal {S}}(\eta )} . It follows from the definition of a sufficient dimension reduction that F y ∣ x = F y ∣ η T x , {\displaystyle F_{y\mid x}=F_{y\mid \eta ^{T}x},} where F {\displaystyle F} denotes the appropriate distribution function. Another way to express this property is y ⊥ ⊥ x ∣ η T x , {\displaystyle y\perp \!\!\!\perp {\textbf {x}}\mid \eta ^{T}{\textbf {x}},} or y {\displaystyle y} is conditionally independent of x {\displaystyle {\textbf {x}}} , given η T x {\displaystyle \eta ^{T}{\textbf {x}}} . Then the subspace S ( η ) {\displaystyle {\mathcal {S}}(\eta )} is defined to be a dimension reduction subspace (DRS). === Structural dimensionality === For a regression y ∣ x {\displaystyle y\mid {\textbf {x}}} , the structural dimension, d {\displaystyle d} , is the smallest number of distinct linear combinations of x {\displaystyle {\textbf {x}}} necessary to preserve the conditional distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, the smallest dimension reduction that is still sufficient maps x {\displaystyle {\textbf {x}}} to a subset of R d {\displaystyle \mathbb {R} ^{d}} . The corresponding DRS will be d-dimensional. === Minimum dimension reduction subspace === A subspace S {\displaystyle {\mathcal {S}}} is said to be a minimum DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} if it is a DRS and its dimension is less than or equal to that of all other DRSs for y ∣ x {\displaystyle y\mid {\textbf {x}}} . A minimum DRS S {\displaystyle {\mathcal {S}}} is not necessarily unique, but its dimension is equal to the structural dimension d {\displaystyle d} of y ∣ x {\displaystyle y\mid {\textbf {x}}} , by definition. If S {\displaystyle {\mathcal {S}}} has basis η {\displaystyle \eta } and is a minimum DRS, then a plot of y versus η T x {\displaystyle \eta ^{T}{\textbf {x}}} is a minimal sufficient summary plot, and it is (d + 1)-dimensional. == Central subspace == If a subspace S {\displaystyle {\mathcal {S}}} is a DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} , and if S ⊂ S drs {\displaystyle {\mathcal {S}}\subset {\mathcal {S}}_{\text{drs}}} for all other DRSs S drs {\displaystyle {\mathcal {S}}_{\text{drs}}} , then it is a central dimension reduction subspace, or simply a central subspace, and it is denoted by S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . In other words, a central subspace for y ∣ x {\displaystyle y\mid {\textbf {x}}} exists if and only if the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} of all dimension reduction subspaces is also a dimension reduction subspace, and that intersection is the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . The central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does not necessarily exist because the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} is not necessarily a DRS. However, if S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does exist, then it is also the unique minimum dimension reduction subspace. === Existence of the central subspace === While the existence of the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} is not guaranteed in every regression situation, there are some rather broad conditions under which its existence follows directly. For example, consider the following proposition from Cook (1998): Let S 1 {\displaystyle {\mathcal {S}}_{1}} and S 2 {\displaystyle {\mathcal {S}}_{2}} be dimension reduction subspaces for y ∣ x {\displaystyle y\mid {\textbf {x}}} . If x {\displaystyle {\textbf {x}}} has density f ( a ) > 0 {\displaystyle f(a)>0} for all a ∈ Ω x {\displaystyle a\in \Omega _{x}} and f ( a ) = 0 {\displaystyle f(a)=0} everywhere else, where Ω x {\displaystyle \Omega _{x}} is convex, then the intersection S 1 ∩ S 2 {\displaystyle {\mathcal {S}}_{1}\cap {\mathcal {S}}_{2}} is also a dimension reduction subspace. It follows from this proposition that the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} exists for such x {\displaystyle {\textbf {x}}} . == Methods for dimension reduction == There are many existing methods for dimension reduction, both graphical and numeric. For example, sliced inverse regression (SIR) and sliced average variance estimation (SAVE) were introduced in the 1990s and continue to be widely used. Although SIR was originally designed to estimate an effective dimension reducing subspace, it is now understood that it estimates only the central subspace, which is generally different. More recent methods for dimension reduction include likelihood-based sufficient dimension reduction, estimating the central subspace based on the inverse third moment (or kth moment), estimating the central solution space, graphical regression, envelope model, and the principal support vector machine. For more details on these and other methods, consult the statistical literature. Principal components analysis (PCA) and similar methods for dimension reduction are not based on the sufficiency principle. === Example: linear regression === Consider the regression model y = α + β T x + ε , where ε ⊥ ⊥ x . {\displaystyle y=\alpha +\beta ^{T}{\textbf {x}}+\varepsilon ,{\text{ where }}\varepsilon \perp \!\!\!\perp {\textbf {x}}.} Note that the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} is the same as the distribution of y ∣ β T x {\displ

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  VieON

  VieON is an mobile application for television and video on demand provided by VieON Joint Stock Company (formerly Dzones), a subsidiary of DatVietVAC Media and Entertainment Group in Vietnam. The app was launched in 2020, featuring over 140 domestic and international television channels, original series, popular entertainment programs known nationwide, top-tier sports events and live streaming of major events. Additionally, VieON provides animated films, television series and television programs from various countries such as South Korea and China. == History == The application was planned for development in 2016, with the cooperation of strategic consulting partner BCG Digital Ventures from the United States. Prior to 2020, VieON was a rebranded version of VTVcab ON, a product managed by Vietnam Cable Television Corporation (VTVCab) and DatVietVAC. On June 15, 2020, after four years of research and testing, the new version of VieON was officially released by DatVietVAC Group, with Vie Channel Joint Stock Company as the business entity and service provider. This is considered the official launch date of the application. On July 21, 2023, VieON transitioned its business operations and service provision to VieON Joint Stock Company. In January 2024, VieON officially launched its global version, VieON Global, targeting Vietnamese users living abroad. == Background == According to Kantar Media Vietnam, up to 84% of Vietnamese people aged 15–54 use social media daily, and in a similar survey by Nielsen, 90% of respondents said they watch live TV weekly. Additionally, according to research organization Muvi, Southeast Asia's OTT market revenue could reach $650 million annually starting next year. Understanding this, DatVietVAC Group has planned to research and develop an OTT application, even though the Vietnamese market already has some major players such as FPT Play and the international giant Netflix. Additionally, DatVietVAC does not hide its ambition to make this application the number one entertainment channel for Vietnamese people.

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  Characteristic samples

  Characteristic samples is a concept in the field of grammatical inference, related to passive learning. In passive learning, an inference algorithm I {\displaystyle I} is given a set of pairs of strings and labels S {\displaystyle S} , and returns a representation R {\displaystyle R} that is consistent with S {\displaystyle S} . Characteristic samples consider the scenario when the goal is not only finding a representation consistent with S {\displaystyle S} , but finding a representation that recognizes a specific target language. A characteristic sample of language L {\displaystyle L} is a set of pairs of the form ( s , l ( s ) ) {\displaystyle (s,l(s))} where: l ( s ) = 1 {\displaystyle l(s)=1} if and only if s ∈ L {\displaystyle s\in L} l ( s ) = − 1 {\displaystyle l(s)=-1} if and only if s ∉ L {\displaystyle s\notin L} Given the characteristic sample S {\displaystyle S} , I {\displaystyle I} 's output on it is a representation R {\displaystyle R} , e.g. an automaton, that recognizes L {\displaystyle L} . == Formal Definition == === The Learning Paradigm associated with Characteristic Samples === There are three entities in the learning paradigm connected to characteristic samples, the adversary, the teacher and the inference algorithm. Given a class of languages C {\displaystyle \mathbb {C} } and a class of representations for the languages R {\displaystyle \mathbb {R} } , the paradigm goes as follows: The adversary A {\displaystyle A} selects a language L ∈ C {\displaystyle L\in \mathbb {C} } and reports it to the teacher The teacher T {\displaystyle T} then computes a set of strings and label them correctly according to L {\displaystyle L} , trying to make sure that the inference algorithm will compute L {\displaystyle L} The adversary can add correctly labeled words to the set in order to confuse the inference algorithm The inference algorithm I {\displaystyle I} gets the sample and computes a representation R ∈ R {\displaystyle R\in \mathbb {R} } consistent with the sample. The goal is that when the inference algorithm receives a characteristic sample for a language L {\displaystyle L} , or a sample that subsumes a characteristic sample for L {\displaystyle L} , it will return a representation that recognizes exactly the language L {\displaystyle L} . === Sample === Sample S {\displaystyle S} is a set of pairs of the form ( s , l ( s ) ) {\displaystyle (s,l(s))} such that l ( s ) ∈ { − 1 , 1 } {\displaystyle l(s)\in \{-1,1\}} ==== Sample consistent with a language ==== We say that a sample S {\displaystyle S} is consistent with language L {\displaystyle L} if for every pair ( s , l ( s ) ) {\displaystyle (s,l(s))} in S {\displaystyle S} : l ( s ) = 1 if and only if s ∈ L {\displaystyle l(s)=1{\text{ if and only if }}s\in L} l ( s ) = − 1 if and only if s ∉ L {\displaystyle l(s)=-1{\text{ if and only if }}s\notin L} === Characteristic sample === Given an inference algorithm I {\displaystyle I} and a language L {\displaystyle L} , a sample S {\displaystyle S} that is consistent with L {\displaystyle L} is called a characteristic sample of L {\displaystyle L} for I {\displaystyle I} if: I {\displaystyle I} 's output on S {\displaystyle S} is a representation R {\displaystyle R} that recognizes L {\displaystyle L} . For every sample D {\displaystyle D} that is consistent with L {\displaystyle L} and also fulfils S ⊆ D {\displaystyle S\subseteq D} , I {\displaystyle I} 's output on D {\displaystyle D} is a representation R {\displaystyle R} that recognizes L {\displaystyle L} . A Class of languages C {\displaystyle \mathbb {C} } is said to have charistaristic samples if every L ∈ C {\displaystyle L\in \mathbb {C} } has a characteristic sample. == Related Theorems == === Theorem === If equivalence is undecidable for a class C {\textstyle \mathbb {C} } over Σ {\textstyle \Sigma } of cardinality bigger than 1, then C {\textstyle \mathbb {C} } doesn't have characteristic samples. ==== Proof ==== Given a class of representations C {\textstyle \mathbb {C} } such that equivalence is undecidable, for every polynomial p ( x ) {\displaystyle p(x)} and every n ∈ N {\displaystyle n\in \mathbb {N} } , there exist two representations r 1 {\displaystyle r_{1}} and r 2 {\displaystyle r_{2}} of sizes bounded by n {\displaystyle n} , that recognize different languages but are inseparable by any string of size bounded by p ( n ) {\displaystyle p(n)} . Assuming this is not the case, we can decide if r 1 {\displaystyle r_{1}} and r 2 {\displaystyle r_{2}} are equivalent by simulating their run on all strings of size smaller than p ( n ) {\displaystyle p(n)} , contradicting the assumption that equivalence is undecidable. === Theorem === If S 1 {\displaystyle S_{1}} is a characteristic sample for L 1 {\displaystyle L_{1}} and is also consistent with L 2 {\displaystyle L_{2}} , then every characteristic sample of L 2 {\displaystyle L_{2}} , is inconsistent with L 1 {\displaystyle L_{1}} . ==== Proof ==== Given a class C {\textstyle \mathbb {C} } that has characteristic samples, let R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} be representations that recognize L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} respectively. Under the assumption that there is a characteristic sample for L 1 {\displaystyle L_{1}} , S 1 {\displaystyle S_{1}} that is also consistent with L 2 {\displaystyle L_{2}} , we'll assume falsely that there exist a characteristic sample for L 2 {\displaystyle L_{2}} , S 2 {\displaystyle S_{2}} that is consistent with L 1 {\displaystyle L_{1}} . By the definition of characteristic sample, the inference algorithm I {\displaystyle I} must return a representation which recognizes the language if given a sample that subsumes the characteristic sample itself. But for the sample S 1 ∪ S 2 {\displaystyle S_{1}\cup S_{2}} , the answer of the inferring algorithm needs to recognize both L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} , in contradiction. === Theorem === If a class is polynomially learnable by example based queries, it is learnable with characteristic samples. == Polynomialy characterizable classes == === Regular languages === The proof that DFA's are learnable using characteristic samples, relies on the fact that every regular language has a finite number of equivalence classes with respect to the right congruence relation, ∼ L {\displaystyle \sim _{L}} (where x ∼ L y {\displaystyle x\sim _{L}y} for x , y ∈ Σ ∗ {\displaystyle x,y\in \Sigma ^{}} if and only if ∀ z ∈ Σ ∗ : x z ∈ L ↔ y z ∈ L {\displaystyle \forall z\in \Sigma ^{}:xz\in L\leftrightarrow yz\in L} ). Note that if x {\displaystyle x} , y {\displaystyle y} are not congruent with respect to ∼ L {\displaystyle \sim _{L}} , there exists a string z {\displaystyle z} such that x z ∈ L {\displaystyle xz\in L} but y z ∉ L {\displaystyle yz\notin L} or vice versa, this string is called a separating suffix. ==== Constructing a characteristic sample ==== The construction of a characteristic sample for a language L {\displaystyle L} by the teacher goes as follows. Firstly, by running a depth first search on a deterministic automaton A {\displaystyle A} recognizing L {\displaystyle L} , starting from its initial state, we get a suffix closed set of words, W {\displaystyle W} , ordered in shortlex order. From the fact above, we know that for every two states in the automaton, there exists a separating suffix that separates between every two strings that the run of A {\displaystyle A} on them ends in the respective states. We refer to the set of separating suffixes as S {\displaystyle S} . The labeled set (sample) of words the teacher gives the adversary is { ( w , l ( w ) ) | w ∈ W ⋅ S ∪ W ⋅ Σ ⋅ S } {\displaystyle \{(w,l(w))|w\in W\cdot S\cup W\cdot \Sigma \cdot S\}} where l ( w ) {\displaystyle l(w)} is the correct label of w {\displaystyle w} (whether it is in L {\displaystyle L} or not). We may assume that ϵ ∈ S {\displaystyle \epsilon \in S} . ==== Constructing a deterministic automata ==== Given the sample from the adversary W {\displaystyle W} , the construction of the automaton by the inference algorithm I {\displaystyle I} starts with defining P = prefix ( W ) {\displaystyle P={\text{prefix}}(W)} and S = suffix ( W ) {\displaystyle S={\text{suffix}}(W)} , which are the set of prefixes and suffixes of W {\displaystyle W} respectively. Now the algorithm constructs a matrix M {\displaystyle M} where the elements of P {\displaystyle P} function as the rows, ordered by the shortlex order, and the elements of S {\displaystyle S} function as the columns, ordered by the shortlex order. Next, the cells in the matrix are filled in the following manner for prefix p i {\displaystyle p_{i}} and suffix s j {\displaystyle s_{j}} : If p i s j ∈ W → M i j = l ( p i s j ) {\displaystyle p_{i}s_{j}\in W\rightarrow M_{ij}=l(p_{i}s_{j})} else, M i j = 0 {\displaystyle M_{ij}=0} Now, we say row i {\displaystyle i} and t {\displaystyle t} are distinguishable if there exi

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  IDistance

  In pattern recognition, iDistance is an indexing and query processing technique for k-nearest neighbor queries on point data in multi-dimensional metric spaces. The kNN query is one of the hardest problems on multi-dimensional data, especially when the dimensionality of the data is high. iDistance is designed to process kNN queries in high-dimensional spaces efficiently and performs extremely well for skewed data distributions, which usually occur in real-life data sets. iDistance employs a two-phase search strategy involving an initial filtering of candidate regions and a subsequent refinement of results, an approach aligned with the Filter and Refine Principle (FRP). This means that the index first prunes the search space to eliminate unlikely candidates, then verifies the true nearest neighbors in a refinement step, following the general FRP paradigm used in database search algorithms. The iDistance index can also be augmented with machine learning models to learn data distributions for improved searching and storage of multi-dimensional data. == Indexing == Building the iDistance index has two steps: A number of reference points in the data space are chosen. There are various ways of choosing reference points. Using cluster centers as reference points is the most efficient way. The data points are partitioned into Voronoi cells based on well-chosen reference points. The distance between a data point and its closest reference point is calculated. This distance plus a scaling value is called the point's iDistance. By this means, points in a multi-dimensional space are mapped to one-dimensional values, and then a B+-tree can be adopted to index the points using the iDistance as the key. The figure on the right shows an example where three reference points (O1, O2, O3) are chosen. The data points are then mapped to a one-dimensional space and indexed in a B+-tree. Various extensions have been proposed to make the selection of reference points for effective query performance, including employing machine learning to learn the identification of reference points. == Query processing == To process a kNN query, the query is mapped to a number of one-dimensional range queries, which can be processed efficiently on a B+-tree. In the above figure, the query Q is mapped to a value in the B+-tree while the kNN search ``sphere" is mapped to a range in the B+-tree. The search sphere expands gradually until the k NNs are found. This corresponds to gradually expanding range searches in the B+-tree. The iDistance technique can be viewed as a way of accelerating the sequential scan. Instead of scanning records from the beginning to the end of the data file, the iDistance starts the scan from spots where the nearest neighbors can be obtained early with a very high probability. == Applications == The iDistance has been used in many applications including Image retrieval Video indexing Similarity search in P2P systems Mobile computing Recommender system == Historical background == The iDistance was first proposed by Cui Yu, Beng Chin Ooi, Kian-Lee Tan and H. V. Jagadish in 2001. Later, together with Rui Zhang, they improved the technique and performed a more comprehensive study on it in 2005.

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  Differential evolution

  Differential evolution (DE) is an evolutionary algorithm to optimize a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Such methods are commonly known as metaheuristics as they make few or no assumptions about the optimized problem and can search very large spaces of candidate solutions. However, metaheuristics such as DE do not guarantee an optimal solution is ever found. DE is used for multidimensional real-valued functions but does not use the gradient of the problem being optimized, which means DE does not require the optimization problem to be differentiable, as is required by classic optimization methods such as gradient descent and quasi-newton methods. DE can therefore also be used on optimization problems that are not even continuous, are noisy, change over time, etc. DE optimizes a problem by maintaining a population of candidate solutions and creating new candidate solutions by combining existing ones according to its simple formulae, and then keeping whichever candidate solution has the best score or fitness on the optimization problem at hand. In this way, the optimization problem is treated as a black box that merely provides a measure of quality given a candidate solution and the gradient is therefore not needed. == History == Storn and Price introduced Differential Evolution in 1995. Books have been published on theoretical and practical aspects of using DE in parallel computing, multiobjective optimization, constrained optimization, and the books also contain surveys of application areas. Surveys on the multi-faceted research aspects of DE can be found in journal articles. == Algorithm == A basic variant of the DE algorithm works by having a population of candidate solutions (called agents). These agents are moved around in the search-space by using simple mathematical formulae to combine the positions of existing agents from the population. If the new position of an agent is an improvement then it is accepted and forms part of the population, otherwise the new position is simply discarded. The process is repeated and by doing so it is hoped, but not guaranteed, that a satisfactory solution will eventually be discovered. Formally, let f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be the fitness function which must be minimized (note that maximization can be performed by considering the function h := − f {\displaystyle h:=-f} instead). The function takes a candidate solution as argument in the form of a vector of real numbers. It produces a real number as output which indicates the fitness of the given candidate solution. The gradient of f {\displaystyle f} is not known. The goal is to find a solution m {\displaystyle \mathbf {m} } for which f ( m ) ≤ f ( p ) {\displaystyle f(\mathbf {m} )\leq f(\mathbf {p} )} for all p {\displaystyle \mathbf {p} } in the search-space, which means that m {\displaystyle \mathbf {m} } is the global minimum. Let x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} designate a candidate solution (agent) in the population. The basic DE algorithm can then be described as follows: Choose the parameters NP ≥ 4 {\displaystyle {\text{NP}}\geq 4} , CR ∈ [ 0 , 1 ] {\displaystyle {\text{CR}}\in [0,1]} , and F ∈ [ 0 , 2 ] {\displaystyle F\in [0,2]} . NP : NP {\displaystyle {\text{NP}}} is the population size, i.e. the number of candidate agents or "parents". CR : The parameter CR ∈ [ 0 , 1 ] {\displaystyle {\text{CR}}\in [0,1]} is called the crossover probability. F : The parameter F ∈ [ 0 , 2 ] {\displaystyle F\in [0,2]} is called the differential weight. Typical settings are N P = 10 n {\displaystyle NP=10n} , C R = 0.9 {\displaystyle CR=0.9} and F = 0.8 {\displaystyle F=0.8} . Optimization performance may be greatly impacted by these choices; see below. Initialize all agents x {\displaystyle \mathbf {x} } with random positions in the search-space. Until a termination criterion is met (e.g. number of iterations performed, or adequate fitness reached), repeat the following: For each agent x {\displaystyle \mathbf {x} } in the population do: Pick three agents a , b {\displaystyle \mathbf {a} ,\mathbf {b} } , and c {\displaystyle \mathbf {c} } from the population at random, they must be distinct from each other as well as from agent x {\displaystyle \mathbf {x} } . ( a {\displaystyle \mathbf {a} } is called the "base" vector.) Pick a random index R ∈ { 1 , … , n } {\displaystyle R\in \{1,\ldots ,n\}} where n {\displaystyle n} is the dimensionality of the problem being optimized. Compute the agent's potentially new position y = [ y 1 , … , y n ] {\displaystyle \mathbf {y} =[y_{1},\ldots ,y_{n}]} as follows: For each i ∈ { 1 , … , n } {\displaystyle i\in \{1,\ldots ,n\}} , pick a uniformly distributed random number r i ∼ U ( 0 , 1 ) {\displaystyle r_{i}\sim U(0,1)} If r i < C R {\displaystyle r_{i} Read more →

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  Artificial intelligence content detection

  Artificial intelligence detection software aims to determine whether some content (text, image, video, or audio) was generated using artificial intelligence (AI). This software is often unreliable. == Accuracy issues == Many AI detection tools have been shown to be unreliable in detecting AI-generated text. In a 2023 study conducted by Weber-Wulff et al., researchers evaluated 14 detection tools including Turnitin and GPTZero and found that "all scored below 80% of accuracy and only 5 over 70%." They also found that these tools tend to have a bias for classifying texts more as human than as AI, and that accuracy of these tools worsens upon paraphrasing. === False positives === In AI content detection, a false positive is when human-written work is incorrectly flagged as AI-written. Many AI detection platforms claim to have a minimal level of false positives, with Turnitin claiming a less than 1% false positive rate. However, later research by The Washington Post produced much higher rates of 50%, though they used a smaller sample size. False positives in an academic setting frequently lead to accusations of academic misconduct, which can have serious consequences for a student's academic record. Additionally, studies have shown evidence that many AI detection models are prone to give false positives to work written by people whose first language is not English, and also to neurodivergent people. In June 2023, Janelle Shane wrote that portions of her book You Look Like a Thing and I Love You were flagged as AI-generated. === False negatives === A false negative is a failure to identify documents with AI-written text. False negatives often happen as a result of a detection software's sensitivity level or because evasive techniques were used when generating the work to make it sound more human. False negatives are less of a concern academically, since they aren't likely to lead to accusations and ramifications. Notably, Turnitin stated they have a 15% false negative rate. == Text detection == For text, this is usually done to prevent alleged plagiarism, often by detecting repetition of words as telltale signs that a text was AI-generated (including hallucinations). Detection systems may also rely on stylistic and structural regularities associated with LLM output, such as unusually consistent grammar, formulaic transitions, repeated discourse markers, and recurring rhetorical templates. Some tools are designed less to establish authorship provenance than to flag prose that resembles common LLM-generated style patterns. They are often used by teachers marking their students, usually on an ad hoc basis. Following the release of ChatGPT and similar AI text generative software, many educational establishments have issued policies against the use of AI by students. AI text detection software is also used by those assessing job applicants, as well as online search engines, hiring, online moderation and publishing. Current detectors may sometimes be unreliable and have incorrectly marked work by humans as originating from AI while failing to detect AI-generated work in other instances. MIT Technology Review said that the technology "struggled to pick up ChatGPT-generated text that had been slightly rearranged by humans and obfuscated by a paraphrasing tool". AI text detection software has also been shown to discriminate against non-native speakers of English. Two students from the University of California, Davis, were referred to the university's Office of Student Success and Judicial Affairs (OSSJA) after their professors scanned their essays with positive results; the first with an AI detector called GPTZero, and the second with an AI detector integration in Turnitin. However, following media coverage, and a thorough investigation, the students were cleared of any wrongdoing. In April 2023, Cambridge University and other members of the Russell Group of universities in the United Kingdom opted out of Turnitin's AI text detection tool, after expressing concerns it was unreliable. The University of Texas at Austin opted out of the system six months later. In May 2023, a professor at Texas A&M University–Commerce used ChatGPT to detect whether his students' content was written by it, which ChatGPT said was the case. As such, he threatened to fail the class despite ChatGPT not being able to detect AI-generated writing. No students were prevented from graduating because of the issue, and all but one student (who admitted to using the software) were exonerated from accusations of having used ChatGPT in their content. In July 2023, a paper titled "GPT detectors are biased against non-native English writers" was released, reporting that GPTs discriminate against non-native English authors. The paper compared seven GPT detectors against essays from both non-native English speakers and essays from United States students. The essays from non-native English speakers had an average false positive rate of 61.3%. An article by Thomas Germain, published on Gizmodo in June 2024, reported job losses among freelance writers and journalists due to AI text detection software mistakenly classifying their work as AI-generated. In September 2024, Common Sense Media reported that generative AI detectors had a 20% false positive rate for Black students, compared to 10% of Latino students and 7% of White students. To improve the reliability of AI text detection, researchers have explored digital watermarking techniques. A 2023 paper titled "A Watermark for Large Language Models" presents a method to embed imperceptible watermarks into text generated by large language models (LLMs). This watermarking approach allows content to be flagged as AI-generated with a high level of accuracy, even when text is slightly paraphrased or modified. The technique is designed to be subtle and hard to detect for casual readers, thereby preserving readability, while providing a detectable signal for those employing specialized tools. However, while promising, watermarking faces challenges in remaining robust under adversarial transformations and ensuring compatibility across different LLMs. == Anti text detection == There is software available designed to bypass AI text detection. In practice, evasion may not require specialized bypass tools. Paraphrasing, style editing, and removal of repeated discourse markers can substantially reduce the effectiveness of detectors that rely on recognizable surface patterns. A study published in August 2023 analyzed 20 abstracts from papers published in the Eye Journal, which were then paraphrased using GPT-4.0. The AI-paraphrased abstracts were examined for plagiarism using QueText and for AI-generated content using Originality.AI. The texts were then re-processed through an adversarial software called Undetectable.ai in order to reduce the AI-detection scores. The study found that the AI detection tool, Originality.AI, identified text generated by GPT-4 with a mean accuracy of 91.3%. However, after reprocessing by Undetectable.ai, the detection accuracy of Originality.ai dropped to a mean accuracy of 27.8%. Some experts also believe that techniques like digital watermarking are ineffective because they can be removed or added to trigger false positives. "A Watermark for Large Language Models" paper by Kirchenbauer et al. (2023) also addresses potential vulnerabilities of watermarking techniques. The authors outline a range of adversarial tactics, including text insertion, deletion, and substitution attacks, that could be used to bypass watermark detection. These attacks vary in complexity, from simple paraphrasing to more sophisticated approaches involving tokenization and homoglyph alterations. The study highlights the challenge of maintaining watermark robustness against attackers who may employ automated paraphrasing tools or even specific language model replacements to alter text spans iteratively while retaining semantic similarity. Experimental results show that although such attacks can degrade watermark strength, they also come at the cost of text quality and increased computational resources. == Image, video, and audio detection == Several purported AI image detection software exist, to detect AI-generated images (for example, those originating from Midjourney or DALL-E). They are not completely reliable. Industry analyses have also noted that AI-driven image recognition systems often struggle in real-world environments, where inconsistent lighting, noise and variable visual inputs reduce detection reliability, a challenge highlighted in modern agricultural quality-control research. Others claim to identify video and audio deepfakes, but this technology is also not fully reliable yet either. Despite debate around the efficacy of watermarking, Google DeepMind is actively developing a detection software called SynthID, which works by inserting a digital watermark that is invisible to the human eye into the pixels of an image.

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  Influence diagram

  An influence diagram (ID) (also called a relevance diagram, decision diagram or a decision network) is a compact graphical and mathematical representation of a decision situation. It is a generalization of a Bayesian network, in which not only probabilistic inference problems but also decision making problems (following the maximum expected utility criterion) can be modeled and solved. ID was first developed in the mid-1970s by decision analysts with an intuitive semantic that is easy to understand. It is now adopted widely and becoming an alternative to the decision tree which typically suffers from exponential growth in number of branches with each variable modeled. ID is directly applicable in team decision analysis, since it allows incomplete sharing of information among team members to be modeled and solved explicitly. Extensions of ID also find their use in game theory as an alternative representation of the game tree. == Semantics == An ID is a directed acyclic graph with three types (plus one subtype) of node and three types of arc (or arrow) between nodes. Nodes: Decision node (corresponding to each decision to be made) is drawn as a rectangle. Uncertainty node (corresponding to each uncertainty to be modeled) is drawn as an oval. Deterministic node (corresponding to special kind of uncertainty that its outcome is deterministically known whenever the outcome of some other uncertainties are also known) is drawn as a double oval. Value node (corresponding to each component of additively separable Von Neumann-Morgenstern utility function) is drawn as an octagon (or diamond). Arcs: Functional arcs (ending in value node) indicate that one of the components of additively separable utility function is a function of all the nodes at their tails. Conditional arcs (ending in uncertainty node) indicate that the uncertainty at their heads is probabilistically conditioned on all the nodes at their tails. Conditional arcs (ending in deterministic node) indicate that the uncertainty at their heads is deterministically conditioned on all the nodes at their tails. Informational arcs (ending in decision node) indicate that the decision at their heads is made with the outcome of all the nodes at their tails known beforehand. Given a properly structured ID: Decision nodes and incoming information arcs collectively state the alternatives (what can be done when the outcome of certain decisions and/or uncertainties are known beforehand) Uncertainty/deterministic nodes and incoming conditional arcs collectively model the information (what are known and their probabilistic/deterministic relationships) Value nodes and incoming functional arcs collectively quantify the preference (how things are preferred over one another). Alternative, information, and preference are termed decision basis in decision analysis, they represent three required components of any valid decision situation. Formally, the semantic of influence diagram is based on sequential construction of nodes and arcs, which implies a specification of all conditional independencies in the diagram. The specification is defined by the d {\displaystyle d} -separation criterion of Bayesian network. According to this semantic, every node is probabilistically independent on its non-successor nodes given the outcome of its immediate predecessor nodes. Likewise, a missing arc between non-value node X {\displaystyle X} and non-value node Y {\displaystyle Y} implies that there exists a set of non-value nodes Z {\displaystyle Z} , e.g., the parents of Y {\displaystyle Y} , that renders Y {\displaystyle Y} independent of X {\displaystyle X} given the outcome of the nodes in Z {\displaystyle Z} . == Example == Consider the simple influence diagram representing a situation where a decision-maker is planning their vacation. There is 1 decision node (Vacation Activity), 2 uncertainty nodes (Weather Condition, Weather Forecast), and 1 value node (Satisfaction). There are 2 functional arcs (ending in Satisfaction), 1 conditional arc (ending in Weather Forecast), and 1 informational arc (ending in Vacation Activity). Functional arcs ending in Satisfaction indicate that Satisfaction is a utility function of Weather Condition and Vacation Activity. In other words, their satisfaction can be quantified if they know what the weather is like and what their choice of activity is. (Note that they do not value Weather Forecast directly) Conditional arc ending in Weather Forecast indicates their belief that Weather Forecast and Weather Condition can be dependent. Informational arc ending in Vacation Activity indicates that they will only know Weather Forecast, not Weather Condition, when making their choice. In other words, actual weather will be known after they make their choice, and only forecast is what they can count on at this stage. It also follows semantically, for example, that Vacation Activity is independent on (irrelevant to) Weather Condition given Weather Forecast is known. == Applicability to value of information == The above example highlights the power of the influence diagram in representing an extremely important concept in decision analysis known as the value of information. Consider the following three scenarios; Scenario 1: The decision-maker could make their Vacation Activity decision while knowing what Weather Condition will be like. This corresponds to adding extra informational arc from Weather Condition to Vacation Activity in the above influence diagram. Scenario 2: The original influence diagram as shown above. Scenario 3: The decision-maker makes their decision without even knowing the Weather Forecast. This corresponds to removing informational arc from Weather Forecast to Vacation Activity in the above influence diagram. Scenario 1 is the best possible scenario for this decision situation since there is no longer any uncertainty on what they care about (Weather Condition) when making their decision. Scenario 3, however, is the worst possible scenario for this decision situation since they need to make their decision without any hint (Weather Forecast) on what they care about (Weather Condition) will turn out to be. The decision-maker is usually better off (definitely no worse off, on average) to move from scenario 3 to scenario 2 through the acquisition of new information. The most they should be willing to pay for such move is called the value of information on Weather Forecast, which is essentially the value of imperfect information on Weather Condition. The applicability of this simple ID and the value of information concept is tremendous, especially in medical decision making when most decisions have to be made with imperfect information about their patients, diseases, etc. == Related concepts == Influence diagrams are hierarchical and can be defined either in terms of their structure or in greater detail in terms of the functional and numerical relation between diagram elements. An ID that is consistently defined at all levels—structure, function, and number—is a well-defined mathematical representation and is referred to as a well-formed influence diagram (WFID). WFIDs can be evaluated using reversal and removal operations to yield answers to a large class of probabilistic, inferential, and decision questions. More recent techniques have been developed by artificial intelligence researchers concerning Bayesian network inference (belief propagation). An influence diagram having only uncertainty nodes (i.e., a Bayesian network) is also called a relevance diagram. An arc connecting node A to B implies not only that "A is relevant to B", but also that "B is relevant to A" (i.e., relevance is a symmetric relationship).

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  Bayesian network

  A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). While it is one of several forms of causal notation, causal networks are special cases of Bayesian networks. Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. Efficient algorithms can perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (e.g. speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams. == Graphical model == Formally, Bayesian networks are directed acyclic graphs (DAGs) whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Each edge represents a direct conditional dependency. Any pair of nodes that are not connected (i.e. no path connects one node to the other) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables, then the probability function could be represented by a table of 2 m {\displaystyle 2^{m}} entries, one entry for each of the 2 m {\displaystyle 2^{m}} possible parent combinations. Similar ideas may be applied to undirected, and possibly cyclic, graphs such as Markov networks. == Example == Suppose we want to model the dependencies between three variables: the sprinkler (or more appropriately, its state - whether it is on or not), the presence or absence of rain and whether the grass is wet or not. Observe that two events can cause the grass to become wet: an active sprinkler or rain. Rain has a direct effect on the use of the sprinkler (namely that when it rains, the sprinkler usually is not active). This situation can be modeled with a Bayesian network (shown to the right). Each variable has two possible values, T (for true) and F (for false). The joint probability function is, by the chain rule of probability, Pr ( G , S , R ) = Pr ( G ∣ S , R ) Pr ( S ∣ R ) Pr ( R ) {\displaystyle \Pr(G,S,R)=\Pr(G\mid S,R)\Pr(S\mid R)\Pr(R)} where G = "Grass wet (true/false)", S = "Sprinkler turned on (true/false)", and R = "Raining (true/false)". The model can answer questions about the presence of a cause given the presence of an effect (so-called inverse probability) like "What is the probability that it is raining, given the grass is wet?" by using the conditional probability formula and summing over all nuisance variables: Pr ( R = T ∣ G = T ) = Pr ( G = T , R = T ) Pr ( G = T ) = ∑ x ∈ { T , F } Pr ( G = T , S = x , R = T ) ∑ x , y ∈ { T , F } Pr ( G = T , S = x , R = y ) {\displaystyle \Pr(R=T\mid G=T)={\frac {\Pr(G=T,R=T)}{\Pr(G=T)}}={\frac {\sum _{x\in \{T,F\}}\Pr(G=T,S=x,R=T)}{\sum _{x,y\in \{T,F\}}\Pr(G=T,S=x,R=y)}}} Using the expansion for the joint probability function Pr ( G , S , R ) {\displaystyle \Pr(G,S,R)} and the conditional probabilities from the conditional probability tables (CPTs) stated in the diagram, one can evaluate each term in the sums in the numerator and denominator. For example, Pr ( G = T , S = T , R = T ) = Pr ( G = T ∣ S = T , R = T ) Pr ( S = T ∣ R = T ) Pr ( R = T ) = 0.99 × 0.01 × 0.2 = 0.00198. {\displaystyle {\begin{aligned}\Pr(G=T,S=T,R=T)&=\Pr(G=T\mid S=T,R=T)\Pr(S=T\mid R=T)\Pr(R=T)\\&=0.99\times 0.01\times 0.2\\&=0.00198.\end{aligned}}} Then the numerical results (subscripted by the associated variable values) are Pr ( R = T ∣ G = T ) = 0.00198 T T T + 0.1584 T F T 0.00198 T T T + 0.288 T T F + 0.1584 T F T + 0.0 T F F = 891 2491 ≈ 35.77 % . {\displaystyle \Pr(R=T\mid G=T)={\frac {0.00198_{TTT}+0.1584_{TFT}}{0.00198_{TTT}+0.288_{TTF}+0.1584_{TFT}+0.0_{TFF}}}={\frac {891}{2491}}\approx 35.77\%.} To answer an interventional question, such as "What is the probability that it would rain, given that we wet the grass?" the answer is governed by the post-intervention joint distribution function Pr ( S , R ∣ do ( G = T ) ) = Pr ( S ∣ R ) Pr ( R ) {\displaystyle \Pr(S,R\mid {\text{do}}(G=T))=\Pr(S\mid R)\Pr(R)} obtained by removing the factor Pr ( G ∣ S , R ) {\displaystyle \Pr(G\mid S,R)} from the pre-intervention distribution. The do operator forces the value of G to be true. The probability of rain is unaffected by the action: Pr ( R ∣ do ( G = T ) ) = Pr ( R ) . {\displaystyle \Pr(R\mid {\text{do}}(G=T))=\Pr(R).} To predict the impact of turning the sprinkler on: Pr ( R , G ∣ do ( S = T ) ) = Pr ( R ) Pr ( G ∣ R , S = T ) {\displaystyle \Pr(R,G\mid {\text{do}}(S=T))=\Pr(R)\Pr(G\mid R,S=T)} with the term Pr ( S = T ∣ R ) {\displaystyle \Pr(S=T\mid R)} removed, showing that the action affects the grass but not the rain. These predictions may not be feasible given unobserved variables, as in most policy evaluation problems. The effect of the action do ( x ) {\displaystyle {\text{do}}(x)} can still be predicted, however, whenever the back-door criterion is satisfied. It states that, if a set Z of nodes can be observed that d-separates (or blocks) all back-door paths from X to Y then Pr ( Y , Z ∣ do ( x ) ) = Pr ( Y , Z , X = x ) Pr ( X = x ∣ Z ) . {\displaystyle \Pr(Y,Z\mid {\text{do}}(x))={\frac {\Pr(Y,Z,X=x)}{\Pr(X=x\mid Z)}}.} A back-door path is one that ends with an arrow into X. Sets that satisfy the back-door criterion are called "sufficient" or "admissible." For example, the set Z = R is admissible for predicting the effect of S = T on G, because R d-separates the (only) back-door path S ← R → G. However, if S is not observed, no other set d-separates this path and the effect of turning the sprinkler on (S = T) on the grass (G) cannot be predicted from passive observations. In that case P(G | do(S = T)) is not "identified". This reflects the fact that, lacking interventional data, the observed dependence between S and G is due to a causal connection or is spurious (apparent dependence arising from a common cause, R). (see Simpson's paradox) To determine whether a causal relation is identified from an arbitrary Bayesian network with unobserved variables, one can use the three rules of "do-calculus" and test whether all do terms can be removed from the expression of that relation, thus confirming that the desired quantity is estimable from frequency data. Using a Bayesian network can save considerable amounts of memory over exhaustive probability tables, if the dependencies in the joint distribution are sparse. For example, a naive way of storing the conditional probabilities of 10 two-valued variables as a table requires storage space for 2 10 = 1024 {\displaystyle 2^{10}=1024} values. If no variable's local distribution depends on more than three parent variables, the Bayesian network representation stores at most 10 ⋅ 2 3 = 80 {\displaystyle 10\cdot 2^{3}=80} values. One advantage of Bayesian networks is that it is intuitively easier for a human to understand (a sparse set of) direct dependencies and local distributions than complete joint distributions. == Inference and learning == Bayesian networks perform three main inference tasks: Inferring unobserved variables Parameter learning for the probability distributions of each node in the network Structure learning of the graphical network === Inferring unobserved variables === Because a Bayesian network is a complete model for its variables and their relationships, it can be used to answer probabilistic queries about them. For example, the network can be used to update knowledge of the state of a subset of variables when other variables (the evidence variables) are observed. This process of computing the posterior distribution of variables given evidence is called probabilistic inference. The posterior gives a universal sufficient statistic for detection applications, when choosing values for the variable subset that minimize some expected loss function, for instance the probability of decision error. A Bayesian network can thus be considered a mechanism for automatically applying Bayes' theorem to complex problems. The most common exact inference methods are: variable elimination, which eliminates (by integration or summation) the non-observed non-query variables one by one by distributing the sum over the prod

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