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Azure Stream Analytics
Microsoft Azure Stream Analytics is a serverless scalable complex event processing engine by Microsoft that enables users to develop and run real-time analytics on multiple streams of data from sources such as devices, sensors, web sites, social media, and other applications. Users can set up alerts to detect anomalies, predict trends, trigger necessary workflows when certain conditions are observed, and make data available to other downstream applications and services for presentation, archiving, or further analysis. == Query Language == Users can author real-time analytics using a simple declarative SQL-like language with embedded support for temporal logic. Callouts to custom code with JavaScript user defined functions extend the streaming logic written in SQL. Callouts to Azure Machine Learning helps with predictive scoring on streaming data. == Scalability == Azure Stream Analytics is a serverless job service on Azure that eliminates the need for infrastructure, servers, virtual machines, or managed clusters. Users only pay for the processing used for the running jobs. == IoT applications == Azure Stream Analytics integrates with Azure IoT Hub to enable real-time analytics on data from IoT devices and applications. == Real-time Dashboards == Users can build real-time dashboards with Power BI for a live command and control view. Real-time dashboards help transform live data into actionable and insightful visuals. == Data Input Sources == Stream Analytics supports three different types of input sources - Azure Event Hubs, Azure IoT Hubs, and Azure Blob Storage. Additionally, stream analytics supports Azure Blob storage as the input reference data to help augment fast moving event data streams with static data. Stream analytics supports a wide variety of output targets. Support for Power BI allows for real-time dashboarding. Event Hub, Service bus topics and queues help trigger downstream workflows. Support for Azure Table Storage, Azure SQL Databases, Azure SQL Data Warehouse, Azure SQL, Document DB, Azure Data Lake Store enable a variety of downstream analysis and archiving capabilities.
Proximal policy optimization
Proximal policy optimization (PPO) is a reinforcement learning (RL) algorithm for training an intelligent agent. Specifically, it is a policy gradient method, often used for deep RL when the policy network is very large. == History == The predecessor to PPO, Trust Region Policy Optimization (TRPO), was published in 2015. It addressed the instability issue of another algorithm, the Deep Q-Network (DQN), by using the trust region method to limit the KL divergence between the old and new policies. However, TRPO uses the Hessian matrix (a matrix of second derivatives) to enforce the trust region, but the Hessian is inefficient for large-scale problems. PPO was published in 2017. It was essentially an approximation of TRPO that does not require computing the Hessian. The KL divergence constraint was approximated by simply clipping the policy gradient. Since 2018, PPO was the default RL algorithm at OpenAI. PPO has been applied to many areas, such as controlling a robotic arm, beating professional players at Dota 2 (OpenAI Five), and playing Atari games. == TRPO == TRPO, the predecessor of PPO, is an on-policy algorithm. It can be used for environments with either discrete or continuous action spaces. The pseudocode is as follows: Input: initial policy parameters θ 0 {\textstyle \theta _{0}} , initial value function parameters ϕ 0 {\textstyle \phi _{0}} Hyperparameters: KL-divergence limit δ {\textstyle \delta } , backtracking coefficient α {\textstyle \alpha } , maximum number of backtracking steps K {\textstyle K} for k = 0 , 1 , 2 , … {\textstyle k=0,1,2,\ldots } do Collect set of trajectories D k = { τ i } {\textstyle {\mathcal {D}}_{k}=\left\{\tau _{i}\right\}} by running policy π k = π ( θ k ) {\textstyle \pi _{k}=\pi \left(\theta _{k}\right)} in the environment. Compute rewards-to-go R ^ t {\textstyle {\hat {R}}_{t}} . Compute advantage estimates, A ^ t {\textstyle {\hat {A}}_{t}} (using any method of advantage estimation) based on the current value function V ϕ k {\textstyle V_{\phi _{k}}} . Estimate policy gradient as g ^ k = 1 | D k | ∑ τ ∈ D k ∑ t = 0 T ∇ θ log π θ ( a t ∣ s t ) | θ k A ^ t {\displaystyle {\hat {g}}_{k}=\left.{\frac {1}{\left|{\mathcal {D}}_{k}\right|}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\nabla _{\theta }\log \pi _{\theta }\left(a_{t}\mid s_{t}\right)\right|_{\theta _{k}}{\hat {A}}_{t}} Use the conjugate gradient algorithm to compute x ^ k ≈ H ^ k − 1 g ^ k {\displaystyle {\hat {x}}_{k}\approx {\hat {H}}_{k}^{-1}{\hat {g}}_{k}} where H ^ k {\textstyle {\hat {H}}_{k}} is the Hessian of the sample average KL-divergence. Update the policy by backtracking line search with θ k + 1 = θ k + α j 2 δ x ^ k T H ^ k x ^ k x ^ k {\displaystyle \theta _{k+1}=\theta _{k}+\alpha ^{j}{\sqrt {\frac {2\delta }{{\hat {x}}_{k}^{T}{\hat {H}}_{k}{\hat {x}}_{k}}}}{\hat {x}}_{k}} where j ∈ { 0 , 1 , 2 , … K } {\textstyle j\in \{0,1,2,\ldots K\}} is the smallest value which improves the sample loss and satisfies the sample KL-divergence constraint. Fit value function by regression on mean-squared error: ϕ k + 1 = arg min ϕ 1 | D k | T ∑ τ ∈ D k ∑ t = 0 T ( V ϕ ( s t ) − R ^ t ) 2 {\displaystyle \phi _{k+1}=\arg \min _{\phi }{\frac {1}{\left|{\mathcal {D}}_{k}\right|T}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\left(V_{\phi }\left(s_{t}\right)-{\hat {R}}_{t}\right)^{2}} typically via some gradient descent algorithm. == PPO == The pseudocode is as follows: Input: initial policy parameters θ 0 {\textstyle \theta _{0}} , initial value function parameters ϕ 0 {\textstyle \phi _{0}} for k = 0 , 1 , 2 , … {\textstyle k=0,1,2,\ldots } do Collect set of trajectories D k = { τ i } {\textstyle {\mathcal {D}}_{k}=\left\{\tau _{i}\right\}} by running policy π k = π ( θ k ) {\textstyle \pi _{k}=\pi \left(\theta _{k}\right)} in the environment. Compute rewards-to-go R ^ t {\textstyle {\hat {R}}_{t}} . Compute advantage estimates, A ^ t {\textstyle {\hat {A}}_{t}} (using any method of advantage estimation) based on the current value function V ϕ k {\textstyle V_{\phi _{k}}} . Update the policy by maximizing the PPO-Clip objective: θ k + 1 = arg max θ 1 | D k | T ∑ τ ∈ D k ∑ t = 0 T min ( π θ ( a t ∣ s t ) π θ k ( a t ∣ s t ) A π θ k ( s t , a t ) , g ( ϵ , A π θ k ( s t , a t ) ) ) {\displaystyle \theta _{k+1}=\arg \max _{\theta }{\frac {1}{\left|{\mathcal {D}}_{k}\right|T}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\min \left({\frac {\pi _{\theta }\left(a_{t}\mid s_{t}\right)}{\pi _{\theta _{k}}\left(a_{t}\mid s_{t}\right)}}A^{\pi _{\theta _{k}}}\left(s_{t},a_{t}\right),\quad g\left(\epsilon ,A^{\pi _{\theta _{k}}}\left(s_{t},a_{t}\right)\right)\right)} typically via stochastic gradient ascent with Adam. Fit value function by regression on mean-squared error: ϕ k + 1 = arg min ϕ 1 | D k | T ∑ τ ∈ D k ∑ t = 0 T ( V ϕ ( s t ) − R ^ t ) 2 {\displaystyle \phi _{k+1}=\arg \min _{\phi }{\frac {1}{\left|{\mathcal {D}}_{k}\right|T}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\left(V_{\phi }\left(s_{t}\right)-{\hat {R}}_{t}\right)^{2}} typically via some gradient descent algorithm. Like all policy gradient methods, PPO is used for training an RL agent whose actions are determined by a differentiable policy function by gradient ascent. Intuitively, a policy gradient method takes small policy update steps, so the agent can reach higher and higher rewards in expectation. Policy gradient methods may be unstable: A step size that is too big may direct the policy in a suboptimal direction, thus having little possibility of recovery; a step size that is too small lowers the overall efficiency. To solve the instability, PPO implements a clip function that constrains the policy update of an agent from being too large, so that larger step sizes may be used without negatively affecting the gradient ascent process. === Basic concepts === To begin the PPO training process, the agent is set in an environment to perform actions based on its current input. In the early phase of training, the agent can freely explore solutions and keep track of the result. Later, with a certain amount of transition samples and policy updates, the agent will select an action to take by randomly sampling from the probability distribution P ( A | S ) {\displaystyle P(A|S)} generated by the policy network. The actions that are most likely to be beneficial will have the highest probability of being selected from the random sample. After an agent arrives at a different scenario (a new state) by acting, it is rewarded with a positive reward or a negative reward. The objective of an agent is to maximize the cumulative reward signal across sequences of states, known as episodes. === Policy gradient laws: the advantage function === The advantage function (denoted as A {\displaystyle A} ) is central to PPO, as it tries to answer the question of whether a specific action of the agent is better or worse than some other possible action in a given state. By definition, the advantage function is an estimate of the relative value for a selected action. If the output of this function is positive, it means that the action in question is better than the average return, so the possibilities of selecting that specific action will increase. The opposite is true for a negative advantage output. The advantage function can be defined as A = Q − V {\displaystyle A=Q-V} , where Q {\displaystyle Q} is the discounted sum of rewards (the total weighted reward for the completion of an episode) and V {\displaystyle V} is the baseline estimate. Since the advantage function is calculated after the completion of an episode, the program records the outcome of the episode. Therefore, calculating advantage is essentially an unsupervised learning problem. The baseline estimate comes from the value function that outputs the expected discounted sum of an episode starting from the current state. In the PPO algorithm, the baseline estimate will be noisy (with some variance), as it also uses a neural network, like the policy function itself. With Q {\displaystyle Q} and V {\displaystyle V} computed, the advantage function is calculated by subtracting the baseline estimate from the actual discounted return. If A > 0 {\displaystyle A>0} , the actual return of the action is better than the expected return from experience; if A < 0 {\displaystyle A<0} , the actual return is worse. === Ratio function === In PPO, the ratio function ( r t {\displaystyle r_{t}} ) calculates the probability of selecting action a {\displaystyle a} in state s {\displaystyle s} given the current policy network, divided by the previous probability under the old policy. In other words: If r t ( θ ) > 1 {\displaystyle r_{t}(\theta )>1} , where θ {\displaystyle \theta } are the policy network parameters, then selecting action a {\displaystyle a} in state s {\displaystyle s} is more likely based on the current policy than the previous policy. If 0 ≤ r t ( θ ) < 1 {\displaystyle 0\leq r_{t}(\theta )<1} , then selecting actio
Rectified linear unit
In the context of artificial neural networks, the rectifier or ReLU (rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: ReLU ( x ) = x + = max ( 0 , x ) = x + | x | 2 = { x if x > 0 , 0 x ≤ 0 {\displaystyle \operatorname {ReLU} (x)=x^{+}=\max(0,x)={\frac {x+|x|}{2}}={\begin{cases}x&{\text{if }}x>0,\\0&x\leq 0\end{cases}}} where x {\displaystyle x} is the input to a neuron. This is analogous to half-wave rectification in electrical engineering. ReLU is one of the most popular activation functions for artificial neural networks, and finds application in computer vision and speech recognition using deep neural nets and computational neuroscience. == History == The ReLU was first used by Alston Householder in 1941 as a mathematical abstraction of biological neural networks. Kunihiko Fukushima in 1969 used ReLU in the context of visual feature extraction in hierarchical neural networks. In 1998, Gregory Woodbury demonstrated that the rectified linear function could account for a broad range of emergent properties in the visual cortex. His work showed that a single unified model could drive the joint development of refined retinotopic maps, ocular dominance columns, and orientation selectivity. By utilizing the rectifier's "cutoff" property, Woodbury achieved a close quantitative fit to biological data, matching the spatial periodicities and topographic refinement patterns observed in macaque and cat cortical maps. Furthermore, he extended this framework to adult plasticity, accurately replicating the spatial and temporal dynamics of lesion-induced cortical reorganization. This research established that the rectified linear response was a necessary mechanism for the stable self-organisation and maintenance of complex, multi-feature neural maps. In 2000, Hahnloser et al. argued that ReLU approximates the biological relationship between neural firing rates and input current, in addition to enabling recurrent neural network dynamics to stabilise under weaker criteria. Prior to 2010, most activation functions used were the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more numerically efficient counterpart, the hyperbolic tangent. Around 2010, the use of ReLU became common again. Jarrett et al. (2009) noted that rectification by either absolute or ReLU (which they called "positive part") was critical for object recognition in convolutional neural networks (CNNs), specifically because it allows average pooling without neighboring filter outputs cancelling each other out. They hypothesized that the use of sigmoid or tanh was responsible for poor performance in previous CNNs. Nair and Hinton (2010) made a theoretical argument that the softplus activation function should be used, in that the softplus function numerically approximates the sum of an exponential number of linear models that share parameters. They then proposed ReLU as a good approximation to it. Specifically, they began by considering a single binary neuron in a Boltzmann machine that takes x {\displaystyle x} as input, and produces 1 as output with probability σ ( x ) = 1 1 + e − x {\displaystyle \sigma (x)={\frac {1}{1+e^{-x}}}} . They then considered extending its range of output by making infinitely many copies of it X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\dots } , that all take the same input, offset by an amount 0.5 , 1.5 , 2.5 , … {\displaystyle 0.5,1.5,2.5,\dots } , then their outputs are added together as ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} . They then demonstrated that ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} is approximately equal to N ( log ( 1 + e x ) , σ ( x ) ) {\displaystyle {\mathcal {N}}(\log(1+e^{x}),\sigma (x))} , which is also approximately equal to ReLU ( N ( x , σ ( x ) ) ) {\displaystyle \operatorname {ReLU} ({\mathcal {N}}(x,\sigma (x)))} , where N {\displaystyle {\mathcal {N}}} stands for the gaussian distribution. They also argued for another reason for using ReLU: that it allows "intensity equivariance" in image recognition. That is, multiplying input image by a constant k {\displaystyle k} multiplies the output also. In contrast, this is false for other activation functions like sigmoid or tanh. They found that ReLU activation allowed good empirical performance in restricted Boltzmann machines. Glorot et al (2011) argued that ReLU has the following advantages over sigmoid or tanh: ReLU is more similar to biological neurons' responses in their main operating regime. ReLU avoids vanishing gradients. ReLU is cheaper to compute. ReLU creates sparse representation naturally, because many hidden units output exactly zero for a given input. They also found empirically that deep networks trained with ReLU can achieve strong performance without unsupervised pre-training, especially on large, purely supervised tasks. In 2017, the rectified linear function became a central component of the transformer architecture introduced in the Vaswani et al paper "Attention Is All You Need". Within every transformer layer, ReLU is utilized in the position-wise feed-forward networks (FFN), defined by Equation 2 of their paper: FFN ( x ) = max ( 0 , x W 1 + b 1 ) W 2 + b 2 {\displaystyle \operatorname {FFN} (x)=\max(0,xW_{1}+b_{1})W_{2}+b_{2}} This equation is foundational to the model's capacity; while the attention mechanism determines the relationships between tokens, the ReLU-based FFN performs the majority of the numerical computation and houses the bulk of the model's parameters. The efficiency and scalability of this rectified framework triggered a global technological revolution, enabling the development of Large Language Models that have had a profound economic impact. The industrial response to this architecture—including the massive expansion of AI-specific hardware and the birth of the generative AI sector—has positioned the Transformer as a cornerstone of 21st-century infrastructure. During the post 2017 period of rapid AI advancement, the rectified linear unit function has been key to achieving increased model performance and scaling due to the fact that it zeros out responses that are immaterial for a given stimuli, preventing them from accumulating in massive scale models. It is the complete silencing of the parts of the model found to be stimuli-irrelevant during learning that allows for scaling. As the stimuli-irrelevant proportion of the model becomes more massive, these highly numerous connections within the model would inevitably accumulate during scaling no matter how small each individual response is. Therefore, the rectified linear unit function, with its absolute zeroing property, enabled the scaling to hundred billion parameter models and beyond. Early Transformer scaling giants like GPT-3 (2020) and Falcon-180B (2023) relied on the rectified linear unit function explicitly, while successors such as GPT-4 (2023) and Llama 3 (2024) utilized smoother variants like GELU or SwiGLU. These variants were used to improve training stability while fundamentally preserving the rectified principle of zeroing low responses. At the centre of modern artificial intelligence ReLU and its variants maintain absolute zero response across the bulk of the model at any one time, while maintaining approximately linear reponses for stimuli-relevant connections enabling high performance on each specific cognitive task. This feature of activation sparsity has been critical for massive scaling and performance gains of AI models right up to the present day. == Advantages == Advantages of ReLU include: Sparse activation: for example, in a randomly initialized network, only about 50% of hidden units are activated (i.e. have a non-zero output). Better gradient propagation: fewer vanishing gradient problems compared to sigmoidal activation functions that saturate in both directions. Efficiency: only requires comparison and addition. Scale-invariant (homogeneous, or "intensity equivariance"): max ( 0 , a x ) = a max ( 0 , x ) for a ≥ 0 {\displaystyle \max(0,ax)=a\max(0,x){\text{ for }}a\geq 0} . == Potential problems == Possible downsides can include: Non-differentiability at zero (however, it is differentiable anywhere else, and the value of the derivative at zero can be chosen to be 0 or 1 arbitrarily). Not zero-centered: ReLU outputs are always non-negative. This can make it harder for the network to learn during backpropagation, because gradient updates tend to push weights in one direction (positive or negative). Batch normalization can help address this. ReLU is unbounded. Redundancy of the parametrization: Because ReLU is scale-invariant, the network computes the exact same function by scaling the weights and biases in front of a ReLU activation by k {\displaystyle k} , and the weights after by 1 / k {\displaystyle 1/k} . Dying ReLU: ReLU neurons can sometimes be pushed into states
Bayesian hierarchical modeling
Bayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the posterior distribution of model parameters using the Bayesian method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the uncertainty that is present. This integration enables calculation of updated posterior over the (hyper)parameters, effectively updating prior beliefs in light of the observed data. Frequentist statistics may yield conclusions seemingly incompatible with those offered by Bayesian statistics due to the Bayesian treatment of the parameters as random variables and its use of subjective information in establishing assumptions on these parameters. As the approaches answer different questions the formal results are not technically contradictory but the two approaches disagree over which answer is relevant to particular applications. Bayesians argue that relevant information regarding decision-making and updating beliefs cannot be ignored and that hierarchical modeling has the potential to overrule classical methods in applications where respondents give multiple observational data. Moreover, the model has proven to be robust, with the posterior distribution less sensitive to the more flexible hierarchical priors. Hierarchical modeling, as its name implies, retains nested data structure, and is used when information is available at several different levels of observational units. For example, in epidemiological modeling to describe infection trajectories for multiple countries, observational units are countries, and each country has its own time-based profile of daily infected cases. In decline curve analysis to describe oil or gas production decline curve for multiple wells, observational units are oil or gas wells in a reservoir region, and each well has each own time-based profile of oil or gas production rates (usually, barrels per month). Hierarchical modeling is used to devise computation based strategies for multiparameter problems. == Philosophy == Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies a dependence of the joint probability model for these parameters. Individual degrees of belief, expressed in the form of probabilities, come with uncertainty. Amidst this is the change of the degrees of belief over time. As was stated by Professor José M. Bernardo and Professor Adrian F. Smith, "The actuality of the learning process consists in the evolution of individual and subjective beliefs about the reality." These subjective probabilities are more directly involved in the mind rather than the physical probabilities. Hence, it is with this need of updating beliefs that Bayesians have formulated an alternative statistical model which takes into account the prior occurrence of a particular event. == Bayes' theorem == The assumed occurrence of a real-world event will typically modify preferences between certain options. This is done by modifying the degrees of belief attached, by an individual, to the events defining the options. Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability θ j {\displaystyle \theta _{j}} , the survival probability will be updated with the occurrence of y, the event in which a controversial serum is created which, as believed by some, increases survival in cardiac patients. In order to make updated probability statements about θ j {\displaystyle \theta _{j}} , given the occurrence of event y, we must begin with a model providing a joint probability distribution for θ j {\displaystyle \theta _{j}} and y. This can be written as a product of the two distributions that are often referred to as the prior distribution P ( θ ) {\displaystyle P(\theta )} and the sampling distribution P ( y ∣ θ ) {\displaystyle P(y\mid \theta )} respectively: P ( θ , y ) = P ( θ ) P ( y ∣ θ ) {\displaystyle P(\theta ,y)=P(\theta )P(y\mid \theta )} Using the basic property of conditional probability, the posterior distribution will yield: P ( θ ∣ y ) = P ( θ , y ) P ( y ) = P ( y ∣ θ ) P ( θ ) P ( y ) {\displaystyle P(\theta \mid y)={\frac {P(\theta ,y)}{P(y)}}={\frac {P(y\mid \theta )P(\theta )}{P(y)}}} This equation, showing the relationship between the conditional probability and the individual events, is known as Bayes' theorem. This simple expression encapsulates the technical core of Bayesian inference which aims to deconstruct the probability, P ( θ ∣ y ) {\displaystyle P(\theta \mid y)} , relative to solvable subsets of its supportive evidence. == Exchangeability == The usual starting point of a statistical analysis is the assumption that the n values y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} are exchangeable. If no information – other than data y – is available to distinguish any of the θ j {\displaystyle \theta _{j}} 's from any others, and no ordering or grouping of the parameters can be made, one must assume symmetry of prior distribution parameters. This symmetry is represented probabilistically by exchangeability. Generally, it is useful and appropriate to model data from an exchangeable distribution as independently and identically distributed, given some unknown parameter vector θ {\displaystyle \theta } , with distribution P ( θ ) {\displaystyle P(\theta )} . === Finite exchangeability === For a fixed number n, the set y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable if the joint probability P ( y 1 , y 2 , … , y n ) {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})} is invariant under permutations of the indices. That is, for every permutation π {\displaystyle \pi } or ( π 1 , π 2 , … , π n ) {\displaystyle (\pi _{1},\pi _{2},\ldots ,\pi _{n})} of (1, 2, …, n), P ( y 1 , y 2 , … , y n ) = P ( y π 1 , y π 2 , … , y π n ) . {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})=P(y_{\pi _{1}},y_{\pi _{2}},\ldots ,y_{\pi _{n}}).} The following is an exchangeable, but not independent and identical (iid), example: Consider an urn with a red ball and a blue ball inside, with probability 1 2 {\displaystyle {\frac {1}{2}}} of drawing either. Balls are drawn without replacement, i.e. after one ball is drawn from the n {\displaystyle n} balls, there will be n − 1 {\displaystyle n-1} remaining balls left for the next draw. Let Y i = { 1 , if the i th ball is red , 0 , otherwise . {\displaystyle {\text{Let }}Y_{i}={\begin{cases}1,&{\text{if the }}i{\text{th ball is red}},\\0,&{\text{otherwise}}.\end{cases}}} The probability of selecting a red ball in the first draw and a blue ball in the second draw is equal to the probability of selecting a blue ball on the first draw and a red on the second, both of which are 1/2: P ( y 1 = 1 , y 2 = 0 ) = P ( y 1 = 0 , y 2 = 1 ) = 1 2 {\displaystyle P(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)={\frac {1}{2}}} . This makes y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} exchangeable. But the probability of selecting a red ball on the second draw given that the red ball has already been selected in the first is 0. This is not equal to the probability that the red ball is selected in the second draw, which is 1/2: P ( y 2 = 1 ∣ y 1 = 1 ) = 0 ≠ P ( y 2 = 1 ) = 1 2 {\displaystyle P(y_{2}=1\mid y_{1}=1)=0\neq P(y_{2}=1)={\frac {1}{2}}} . Thus, y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} are not independent. If x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are independent and identically distributed, then they are exchangeable, but the converse is not necessarily true. === Infinite exchangeability === Infinite exchangeability is the property that every finite subset of an infinite sequence y 1 {\displaystyle y_{1}} , y 2 , … {\displaystyle y_{2},\ldots } is exchangeable. For any n, the sequence y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable. == Hierarchical models == === Components === Bayesian hierarchical modeling makes use of two important concepts in deriving the posterior distribution, namely: Hyperparameters: parameters of the prior distribution Hyperpriors: distributions of Hyperparameters Suppose a random variable Y follows a normal distribution with parameter θ {\displaystyle \theta } as the mean and 1 as the variance, that is Y ∣ θ ∼ N ( θ , 1 ) {\displaystyle Y\mid \theta \sim N(\theta ,1)} . The tilde relation ∼ {\displaystyle \sim } can be read as "has the distribution of" or "is distributed as". Suppose also that the parameter θ {\displaystyle \theta } has a distribution given by a normal distribution with mean μ {\displaystyle \mu } and variance 1, i.e. θ ∣ μ ∼ N ( μ , 1 ) {\displaystyle \theta \mid \mu \sim N(\mu ,1)} . Furthermore, μ {\displaystyle \mu } follows another distribution given, for example, by the standard normal distribution, N ( 0 , 1 ) {\displaystyle {\text{N}}(0,1)} . The parameter μ {\dis
Sentence extraction
Sentence extraction is a technique used for automatic summarization of a text. In this shallow approach, statistical heuristics are used to identify the most salient sentences of a text. Sentence extraction is a low-cost approach compared to more knowledge-intensive deeper approaches which require additional knowledge bases such as ontologies or linguistic knowledge. In short, sentence extraction works as a filter that allows only meaningful sentences to pass. The major downside of applying sentence-extraction techniques to the task of summarization is the loss of coherence in the resulting summary. Nevertheless, sentence extraction summaries can give valuable clues to the main points of a document and are frequently sufficiently intelligible to human readers. == Procedure == Usually, a combination of heuristics is used to determine the most important sentences within the document. Each heuristic assigns a (positive or negative) score to the sentence. After all heuristics have been applied, the highest-scoring sentences are included in the summary. The individual heuristics are weighted according to their importance. === Early approaches and some sample heuristics === Seminal papers which laid the foundations for many techniques used today have been published by Hans Peter Luhn in 1958 and H. P Edmundson in 1969. Luhn proposed to assign more weight to sentences at the beginning of the document or a paragraph. Edmundson stressed the importance of title-words for summarization and was the first to employ stop-lists in order to filter uninformative words of low semantic content (e.g. most grammatical words such as of, the, a). He also distinguished between bonus words and stigma words, i.e. words that probably occur together with important (e.g. the word form significant) or unimportant information. His idea of using key-words, i.e. words which occur significantly frequently in the document, is still one of the core heuristics of today's summarizers. With large linguistic corpora available today, the tf–idf value which originated in information retrieval, can be successfully applied to identify the key words of a text: If for example the word cat occurs significantly more often in the text to be summarized (TF = "term frequency") than in the corpus (IDF means "inverse document frequency"; here the corpus is meant by document), then cat is likely to be an important word of the text; the text may in fact be a text about cats.
Variable kernel density estimation
In statistics, adaptive or "variable-bandwidth" kernel density estimation is a form of kernel density estimation in which the size of the kernels used in the estimate are varied depending upon either the location of the samples or the location of the test point. It is a particularly effective technique when the sample space is multi-dimensional. == Rationale == Given a set of samples, { x → i } {\displaystyle \lbrace {\vec {x}}_{i}\rbrace } , we wish to estimate the density, P ( x → ) {\displaystyle P({\vec {x}})} , at a test point, x → {\displaystyle {\vec {x}}} : P ( x → ) ≈ W n h D {\displaystyle P({\vec {x}})\approx {\frac {W}{nh^{D}}}} W = ∑ i = 1 n w i {\displaystyle W=\sum _{i=1}^{n}w_{i}} w i = K ( x → − x → i h ) {\displaystyle w_{i}=K\left({\frac {{\vec {x}}-{\vec {x}}_{i}}{h}}\right)} where n is the number of samples, K is the "kernel", h is its width and D is the number of dimensions in x → {\displaystyle {\vec {x}}} . The kernel can be thought of as a simple, linear filter. Using a fixed filter width may mean that in regions of low density, all samples will fall in the tails of the filter with very low weighting, while regions of high density will find an excessive number of samples in the central region with weighting close to unity. To fix this problem, we vary the width of the kernel in different regions of the sample space. There are two methods of doing this: balloon and pointwise estimation. In a balloon estimator, the kernel width is varied depending on the location of the test point. In a pointwise estimator, the kernel width is varied depending on the location of the sample. For multivariate estimators, the parameter, h, can be generalized to vary not just the size, but also the shape of the kernel. This more complicated approach will not be covered here. == Balloon estimators == A common method of varying the kernel width is to make it inversely proportional to the density at the test point: h = k [ n P ( x → ) ] 1 / D {\displaystyle h={\frac {k}{\left[nP({\vec {x}})\right]^{1/D}}}} where k is a constant. If we back-substitute the estimated PDF, and assuming a Gaussian kernel function, we can show that W is a constant: W = k D ( 2 π ) D / 2 {\displaystyle W=k^{D}(2\pi )^{D/2}} A similar derivation holds for any kernel whose normalising function is of the order hD, although with a different constant factor in place of the (2 π)D/2 term. This produces a generalization of the k-nearest neighbour algorithm. That is, a uniform kernel function will return the KNN technique. There are two components to the error: a variance term and a bias term. The variance term is given as: e 1 = P ∫ K 2 n h D {\displaystyle e_{1}={\frac {P\int K^{2}}{nh^{D}}}} . The bias term is found by evaluating the approximated function in the limit as the kernel width becomes much larger than the sample spacing. By using a Taylor expansion for the real function, the bias term drops out: e 2 = h 2 n ∇ 2 P {\displaystyle e_{2}={\frac {h^{2}}{n}}\nabla ^{2}P} An optimal kernel width that minimizes the error of each estimate can thus be derived. == Use for statistical classification == The method is particularly effective when applied to statistical classification. There are two ways we can proceed: the first is to compute the PDFs of each class separately, using different bandwidth parameters, and then compare them as in Taylor. Alternatively, we can divide up the sum based on the class of each sample: P ( j , x → ) ≈ 1 n ∑ i = 1 , c i = j n w i {\displaystyle P(j,{\vec {x}})\approx {\frac {1}{n}}\sum _{i=1,c_{i}=j}^{n}w_{i}} where ci is the class of the ith sample. The class of the test point may be estimated through maximum likelihood.