AI Chatbot Options

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Outline of deep learning

The following outline is provided as an overview of, and topical guide to, deep learning: Deep learning is a subfield of machine learning and artificial intelligence based on artificial neural networks with multiple processing layers. It emphasizes representation learning and is widely used in areas such as computer vision, natural language processing, speech recognition, recommender systems, robotics, and generative artificial intelligence. == Ways to categorize deep learning == A field of study A branch of artificial intelligence A subfield of machine learning A subfield of computer science A form of representation learning A class of methods based on artificial neural networks An approach used in computational statistics == History == === Precursors === Cybernetics Perceptron Connectionism Neocognitron Backpropagation === Milestones === LeNet Long short-term memory Deep belief network AlexNet Sequence to sequence learning Generative adversarial network Residual neural network Transformer BERT Generative pre-trained transformer Diffusion model === Related histories === History of artificial intelligence History of machine learning Timeline of machine learning == Core concepts == == Learning settings == Supervised learning Unsupervised learning Self-supervised learning Semi-supervised learning Reinforcement learning Transfer learning Multitask learning Multimodal learning Online machine learning Continual learning == Common tasks == Image classification Object detection Image segmentation Automatic speech recognition Neural machine translation Question answering Automatic summarization Text-to-image model Protein structure prediction == Architectures == === Feedforward and convolutional architectures === Feedforward neural network Multilayer perceptron Convolutional neural network Radial basis function network Residual neural network U-Net === Recurrent and sequence architectures === Recurrent neural network Long short-term memory Gated recurrent unit Sequence to sequence learning Recursive neural network === Representation-learning architectures === Autoencoder Denoising autoencoder Sparse autoencoder Variational autoencoder Restricted Boltzmann machine Deep belief network === Attention and transformer architectures === Attention (machine learning) Transformer BERT Generative pre-trained transformer Vision transformer === Generative and probabilistic architectures === Autoregressive model Diffusion model Energy-based model Generative adversarial network Mixture of experts === Graph and memory architectures === Graph neural network Graph convolutional network Siamese network Neural Turing machine Memory network Echo state network Capsule neural network == Neural network components and techniques == Artificial neuron Activation function Rectified linear unit Sigmoid function Softmax function Embedding Convolution Pooling layer Attention Batch normalization Layer normalization Residual connections == Training and optimization == Backpropagation Gradient descent Stochastic gradient descent Adam optimization Learning rate Loss function Cross-entropy Mean squared error Regularization Dropout Early stopping Batch normalization Data augmentation Transfer learning Knowledge distillation Ensemble learning Curriculum learning == Datasets and benchmarks == CIFAR-10 ImageNet MNIST database Common Objects in Context (COCO) General Language Understanding Evaluation (GLUE) benchmark LibriSpeech SQuAD == Applications == === Computer vision === Computer vision Facial recognition system Image classification Image segmentation Medical imaging Object detection Optical character recognition === Natural language processing === Automatic summarization Chatbot Information retrieval Large language model Natural language processing Neural machine translation Question answering Sentiment analysis === Speech and audio === Automatic speech recognition Music information retrieval Speaker recognition Speech synthesis === Science and medicine === Bioinformatics Computational biology Drug discovery Medical diagnosis Protein structure prediction === Robotics and control === Autonomous car Computer game bot Control theory Robotics === Recommendation, search, and forecasting === Anomaly detection Forecasting Fraud detection Recommender system Search engine === Generative artificial intelligence === Deepfake Generative artificial intelligence Large language model Speech synthesis Text-to-image model === Computer graphics and video games === Deep Learning Anti-Aliasing (DLAA) Deep Learning Super Sampling (DLSS) == Hardware == AMD Instinct AMD XDNA Application-specific integrated circuit Deep learning processor, Neural processing unit (NPU), or Neural Engine Field-programmable gate array General-purpose computing on graphics processing units (GPGPU) Graphics processing unit NVIDIA Deep Learning Accelerator (NVDLA) Tensor processing unit Vision processing unit Wafer-scale integration === Supporting software platforms === CUDA Metal ROCm == Software == === Open-source frameworks and libraries === === Neural network software === EDLUT Emergent Encog JOONE Neuroph NeuroSolutions OpenNN Peltarion Synapse SNNS === Platforms, tools, and deployment === Amazon SageMaker Google Colab Hugging Face Kaggle Kubeflow MLflow ONNX OpenVINO TensorFlow Hub == Algorithms for deep learning and neural networks == Backpropagation Conjugate gradient method Generalized Hebbian algorithm Gradient descent Levenberg–Marquardt algorithm Perceptron Quasi-Newton method Wake-sleep algorithm == Methods and related topics == === Representation and metric learning === Contrastive learning Embedding Feature learning Manifold learning Metric learning === Generative modeling === Autoregressive model Diffusion model Generative adversarial network Generative model Variational inference === Efficient and scalable deep learning === Knowledge distillation Low-rank approximation Mixture of experts Quantization Sparsity === Reliability, safety, and interpretability === Adversarial machine learning AI alignment Algorithmic bias Catastrophic forgetting Differential privacy Explainable artificial intelligence Federated learning Hallucination (artificial intelligence) == Conferences and workshops == Annual Meeting of the Association for Computational Linguistics Conference on Computer Vision and Pattern Recognition Conference on Neural Information Processing Systems International Conference on Computer Vision International Conference on Learning Representations International Conference on Machine Learning == Organizations == === Research laboratories and institutions === Allen Institute for AI Alberta Machine Intelligence Institute European Laboratory for Learning and Intelligent Systems Google DeepMind Meta AI Mila Microsoft Research Vector Institute === Companies === Anthropic Cerebras Cohere DeepSeek Mistral AI OpenAI Stability AI xAI == Publications == === Books === Deep Learning – Ian Goodfellow and Yoshua Bengio Neural Networks and Deep Learning – Michael Nielsen Perceptrons – Marvin Minsky and Seymour Papert === Journals === IEEE Transactions on Neural Networks and Learning Systems Neural Networks Neural Computation == Influential persons ==
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Probably approximately correct learning

In computational learning theory, probably approximately correct (PAC) learning is a framework for mathematical analysis of machine learning. It was proposed in 1984 by Leslie Valiant. In this framework, the learner receives samples and must select a generalization function (called the hypothesis) from a certain class of possible functions. The goal is that, with high probability (the "probably" part), the selected function will have low generalization error (the "approximately correct" part). The learner must be able to learn the concept given any arbitrary approximation ratio, probability of success, or distribution of the samples. The model was later extended to treat noise (misclassified samples). An important innovation of the PAC framework is the introduction of computational complexity theory concepts to machine learning. In particular, the learner is expected to find efficient functions (time and space requirements bounded to a polynomial of the example size), and the learner itself must implement an efficient procedure (requiring an example count bounded to a polynomial of the concept size, modified by the approximation and likelihood bounds). == Definitions and terminology == In order to give the definition for something that is PAC-learnable, we first have to introduce some terminology. For the following definitions, two examples will be used. The first is the problem of character recognition given an array of n {\displaystyle n} bits encoding a binary-valued image. The other example is the problem of finding an interval that will correctly classify points within the interval as positive and the points outside of the range as negative. Let X {\displaystyle X} be a set called the instance space or the encoding of all the samples. In the character recognition problem, the instance space is X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} . In the interval problem the instance space, X {\displaystyle X} , is the set of all bounded intervals in R {\displaystyle \mathbb {R} } , where R {\displaystyle \mathbb {R} } denotes the set of all real numbers. A concept is a subset c ⊂ X {\displaystyle c\subset X} . One concept is the set of all patterns of bits in X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} that encode a picture of the letter "P". An example concept from the second example is the set of open intervals, { ( a , b ) ∣ 0 ≤ a ≤ π / 2 , π ≤ b ≤ 13 } {\displaystyle \{(a,b)\mid 0\leq a\leq \pi /2,\pi \leq b\leq {\sqrt {13}}\}} , each of which contains only the positive points. A concept class C {\displaystyle C} is a collection of concepts over X {\displaystyle X} . This could be the set of all subsets of the array of bits that are skeletonized 4-connected (width of the font is 1). Let EX ⁡ ( c , D ) {\displaystyle \operatorname {EX} (c,D)} be a procedure that draws an example, x {\displaystyle x} , using a probability distribution D {\displaystyle D} and gives the correct label c ( x ) {\displaystyle c(x)} , that is 1 if x ∈ c {\displaystyle x\in c} and 0 otherwise. Now, given 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} , assume there is an algorithm A {\displaystyle A} and a polynomial p {\displaystyle p} in 1 / ϵ , 1 / δ {\displaystyle 1/\epsilon ,1/\delta } (and other relevant parameters of the class C {\displaystyle C} ) such that, given a sample of size p {\displaystyle p} drawn according to EX ⁡ ( c , D ) {\displaystyle \operatorname {EX} (c,D)} , then, with probability of at least 1 − δ {\displaystyle 1-\delta } , A {\displaystyle A} outputs a hypothesis h ∈ C {\displaystyle h\in C} that has an average error less than or equal to ϵ {\displaystyle \epsilon } on X {\displaystyle X} with the same distribution D {\displaystyle D} . Further if the above statement for algorithm A {\displaystyle A} is true for every concept c ∈ C {\displaystyle c\in C} and for every distribution D {\displaystyle D} over X {\displaystyle X} , and for all 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} then C {\displaystyle C} is (efficiently) PAC learnable (or distribution-free PAC learnable). We can also say that A {\displaystyle A} is a PAC learning algorithm for C {\displaystyle C} . == Equivalence == Under some regularity conditions these conditions are equivalent: The concept class C is PAC learnable. The VC dimension of C is finite. C is a uniformly Glivenko-Cantelli class. C is compressible in the sense of Littlestone and Warmuth
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Naive Bayes classifier

In statistics, naive (sometimes simple or idiot's) Bayes classifiers are a family of "probabilistic classifiers" which assume that the features are conditionally independent, given the target class. In other words, a naive Bayes model assumes the information about the class provided by each variable is unrelated to the information from the others, with no information shared between the predictors. The highly unrealistic nature of this assumption, called the naive independence assumption, is what gives the classifier its name. These classifiers are some of the simplest Bayesian network models. Naive Bayes classifiers generally perform worse than more advanced models like logistic regressions, especially at quantifying uncertainty (with naive Bayes models often producing wildly overconfident probabilities). However, they are highly scalable, requiring only one parameter for each feature or predictor in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression (simply by counting observations in each group), rather than the expensive iterative approximation algorithms required by most other models. Despite the use of Bayes' theorem in the classifier's decision rule, naive Bayes is not (necessarily) a Bayesian method, and naive Bayes models can be fit to data using either Bayesian or frequentist methods. == Introduction == Naive Bayes is a simple technique for constructing classifiers: models that assign class labels to problem instances, represented as vectors of feature values, where the class labels are drawn from some finite set. There is not a single algorithm for training such classifiers, but a family of algorithms based on a common principle: all naive Bayes classifiers assume that the value of a particular feature is independent of the value of any other feature, given the class variable. For example, a fruit may be considered to be an apple if it is red, round, and about 10 cm in diameter. A naive Bayes classifier considers each of these features to contribute independently to the probability that this fruit is an apple, regardless of any possible correlations between the color, roundness, and diameter features. In many practical applications, parameter estimation for naive Bayes models uses the method of maximum likelihood; in other words, one can work with the naive Bayes model without accepting Bayesian probability or using any Bayesian methods. Despite their naive design and apparently oversimplified assumptions, naive Bayes classifiers have worked quite well in many complex real-world situations. In 2004, an analysis of the Bayesian classification problem showed that there are sound theoretical reasons for the apparently implausible efficacy of naive Bayes classifiers. Still, a comprehensive comparison with other classification algorithms in 2006 showed that Bayes classification is outperformed by other approaches, such as boosted trees or random forests. An advantage of naive Bayes is that it only requires a small amount of training data to estimate the parameters necessary for classification. == Probabilistic model == Abstractly, naive Bayes is a conditional probability model: it assigns probabilities p ( C k ∣ x 1 , … , x n ) {\displaystyle p(C_{k}\mid x_{1},\ldots ,x_{n})} for each of the K possible outcomes or classes C k {\displaystyle C_{k}} given a problem instance to be classified, represented by a vector x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} encoding some n features (independent variables). The problem with the above formulation is that if the number of features n is large or if a feature can take on a large number of values, then basing such a model on probability tables is infeasible. The model must therefore be reformulated to make it more tractable. Using Bayes' theorem, the conditional probability can be decomposed as: p ( C k ∣ x ) = p ( C k ) p ( x ∣ C k ) p ( x ) {\displaystyle p(C_{k}\mid \mathbf {x} )={\frac {p(C_{k})\ p(\mathbf {x} \mid C_{k})}{p(\mathbf {x} )}}\,} In plain English, using Bayesian probability terminology, the above equation can be written as posterior = prior × likelihood evidence {\displaystyle {\text{posterior}}={\frac {{\text{prior}}\times {\text{likelihood}}}{\text{evidence}}}\,} In practice, there is interest only in the numerator of that fraction, because the denominator does not depend on C {\displaystyle C} and the values of the features x i {\displaystyle x_{i}} are given, so that the denominator is effectively constant. The numerator is equivalent to the joint probability model p ( C k , x 1 , … , x n ) {\displaystyle p(C_{k},x_{1},\ldots ,x_{n})\,} which can be rewritten as follows, using the chain rule for repeated applications of the definition of conditional probability: p ( C k , x 1 , … , x n ) = p ( x 1 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 ∣ x 3 , … , x n , C k ) p ( x 3 , … , x n , C k ) = ⋯ = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 ∣ x 3 , … , x n , C k ) ⋯ p ( x n − 1 ∣ x n , C k ) p ( x n ∣ C k ) p ( C k ) {\displaystyle {\begin{aligned}p(C_{k},x_{1},\ldots ,x_{n})&=p(x_{1},\ldots ,x_{n},C_{k})\\&=p(x_{1}\mid x_{2},\ldots ,x_{n},C_{k})\ p(x_{2},\ldots ,x_{n},C_{k})\\&=p(x_{1}\mid x_{2},\ldots ,x_{n},C_{k})\ p(x_{2}\mid x_{3},\ldots ,x_{n},C_{k})\ p(x_{3},\ldots ,x_{n},C_{k})\\&=\cdots \\&=p(x_{1}\mid x_{2},\ldots ,x_{n},C_{k})\ p(x_{2}\mid x_{3},\ldots ,x_{n},C_{k})\cdots p(x_{n-1}\mid x_{n},C_{k})\ p(x_{n}\mid C_{k})\ p(C_{k})\\\end{aligned}}} Now the "naive" conditional independence assumptions come into play: assume that all features in x {\displaystyle \mathbf {x} } are mutually independent, conditional on the category C k {\displaystyle C_{k}} . Under this assumption, p ( x i ∣ x i + 1 , … , x n , C k ) = p ( x i ∣ C k ) . {\displaystyle p(x_{i}\mid x_{i+1},\ldots ,x_{n},C_{k})=p(x_{i}\mid C_{k})\,.} Thus, the joint model can be expressed as p ( C k ∣ x 1 , … , x n ) ∝ p ( C k , x 1 , … , x n ) = p ( C k ) p ( x 1 ∣ C k ) p ( x 2 ∣ C k ) p ( x 3 ∣ C k ) ⋯ = p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) , {\displaystyle {\begin{aligned}p(C_{k}\mid x_{1},\ldots ,x_{n})\varpropto \ &p(C_{k},x_{1},\ldots ,x_{n})\\&=p(C_{k})\ p(x_{1}\mid C_{k})\ p(x_{2}\mid C_{k})\ p(x_{3}\mid C_{k})\ \cdots \\&=p(C_{k})\prod _{i=1}^{n}p(x_{i}\mid C_{k})\,,\end{aligned}}} where ∝ {\displaystyle \varpropto } denotes proportionality since the denominator p ( x ) {\displaystyle p(\mathbf {x} )} is omitted. This means that under the above independence assumptions, the conditional distribution over the class variable C {\displaystyle C} is: p ( C k ∣ x 1 , … , x n ) = 1 Z p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) {\displaystyle p(C_{k}\mid x_{1},\ldots ,x_{n})={\frac {1}{Z}}\ p(C_{k})\prod _{i=1}^{n}p(x_{i}\mid C_{k})} where the evidence Z = p ( x ) = ∑ k p ( C k ) p ( x ∣ C k ) {\displaystyle Z=p(\mathbf {x} )=\sum _{k}p(C_{k})\ p(\mathbf {x} \mid C_{k})} is a scaling factor dependent only on x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} , that is, a constant if the values of the feature variables are known. Often, it is only necessary to discriminate between classes. In that case, the scaling factor is irrelevant, and it is sufficient to calculate the log-probability up to a factor: ln ⁡ p ( C k ∣ x 1 , … , x n ) = ln ⁡ p ( C k ) + ∑ i = 1 n ln ⁡ p ( x i ∣ C k ) − ln ⁡ Z ⏟ irrelevant {\displaystyle \ln p(C_{k}\mid x_{1},\ldots ,x_{n})=\ln p(C_{k})+\sum _{i=1}^{n}\ln p(x_{i}\mid C_{k})\underbrace {-\ln Z} _{\text{irrelevant}}} The scaling factor is irrelevant, since discrimination subtracts it away: ln ⁡ p ( C k ∣ x 1 , … , x n ) p ( C l ∣ x 1 , … , x n ) = ( ln ⁡ p ( C k ) + ∑ i = 1 n ln ⁡ p ( x i ∣ C k ) ) − ( ln ⁡ p ( C l ) + ∑ i = 1 n ln ⁡ p ( x i ∣ C l ) ) {\displaystyle \ln {\frac {p(C_{k}\mid x_{1},\ldots ,x_{n})}{p(C_{l}\mid x_{1},\ldots ,x_{n})}}=\left(\ln p(C_{k})+\sum _{i=1}^{n}\ln p(x_{i}\mid C_{k})\right)-\left(\ln p(C_{l})+\sum _{i=1}^{n}\ln p(x_{i}\mid C_{l})\right)} There are two benefits of using log-probability. One is that it allows an interpretation in information theory, where log-probabilities are units of information in nats. Another is that it avoids arithmetic underflow. === Constructing a classifier from the probability model === The discussion so far has derived the independent feature model, that is, the naive Bayes probability model. The naive Bayes classifier combines this model with a decision rule. One common rule is to pick the hypothesis that is most probable so as to minimize the probability of misclassification; this is known as the maximum a posteriori or MAP decision rule. The corresponding classifier, a Bayes classifier, is the function that assigns a class label y ^ = C k {\displaystyle {\hat {y}}=C_{k}} for some k as follows: y ^ = argmax k ∈ { 1 , … , K } p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) . {\displaystyle {\hat {y}}={\underset {k\in \{1,\ldots ,K\}}{\operatorname {argmax} }}\ p(C_{k})\displays
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Iris flower data set

The Iris flower data set or Fisher's Iris data set is a multivariate data set used and made famous by the British statistician and biologist Ronald Fisher in his 1936 paper The use of multiple measurements in taxonomic problems as an example of linear discriminant analysis. It is sometimes called Anderson's Iris data set because Edgar Anderson collected the data to quantify the morphologic variation of Iris flowers of three related species. Two of the three species were collected in the Gaspé Peninsula "all from the same pasture, and picked on the same day and measured at the same time by the same person with the same apparatus". The data set consists of 50 samples from each of three species of Iris (Iris setosa, Iris virginica and Iris versicolor). Four features were measured from each sample: the length and the width of the sepals and petals, in centimeters. Based on the combination of these four features, Fisher developed a linear discriminant model to distinguish each species. Fisher's paper was published in the Annals of Eugenics (today the Annals of Human Genetics). == Use of the data set == Originally used as an example data set on which Fisher's linear discriminant analysis was applied, it became a typical test case for many statistical classification techniques in machine learning such as support vector machines. The use of this data set in cluster analysis however is not common, since the data set only contains two clusters with rather obvious separation. One of the clusters contains Iris setosa, while the other cluster contains both Iris virginica and Iris versicolor and is not separable without the species information Fisher used. This makes the data set a good example to explain the difference between supervised and unsupervised techniques in data mining: Fisher's linear discriminant model can only be obtained when the object species are known: class labels and clusters are not necessarily the same. Nevertheless, all three species of Iris are separable in the projection on the nonlinear and branching principal component. The data set is approximated by the closest tree with some penalty for the excessive number of nodes, bending and stretching. Then the so-called "metro map" is constructed. The data points are projected into the closest node. For each node the pie diagram of the projected points is prepared. The area of the pie is proportional to the number of the projected points. It is clear from the diagram (left) that the absolute majority of the samples of the different Iris species belong to the different nodes. Only a small fraction of Iris-virginica is mixed with Iris-versicolor (the mixed blue-green nodes in the diagram). Therefore, the three species of Iris (Iris setosa, Iris virginica and Iris versicolor) are separable by the unsupervising procedures of nonlinear principal component analysis. To discriminate them, it is sufficient just to select the corresponding nodes on the principal tree. == Data set == The data set contains a set of 150 records under five attributes: sepal length, sepal width, petal length, petal width and species. The iris data set is widely used as a beginner's data set for machine learning purposes. The data set is included in R base and Python in the machine learning library scikit-learn, so that users can access it without having to find a source for it. Several versions of the data set have been published. === R code illustrating usage === The example R code shown below reproduce the scatterplot displayed at the top of this article: === Python code illustrating usage === This code gives:
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Chatbot psychosis

Chatbot psychosis, also called AI psychosis, is a phenomenon wherein individuals reportedly develop or experience worsening psychosis, such as paranoia and delusions, in connection with their use of chatbots. The term was first suggested in a 2023 editorial by Danish psychiatrist Søren Dinesen Østergaard. It is not a recognized clinical diagnosis. Journalistic accounts describe individuals who have developed strong beliefs that chatbots are sentient, are channeling spirits, or are revealing conspiracies, sometimes leading to personal crises or criminal acts. Proposed causes include the tendency of chatbots to provide inaccurate information ("hallucinate") and to affirm or validate users' beliefs, or their ability to mimic an intimacy that users do not experience with other humans. == Background == In his editorial published in Schizophrenia Bulletin's November 2023 issue, Danish psychiatrist Søren Dinesen Østergaard proposed a hypothesis that individuals' use of generative artificial intelligence chatbots might trigger delusions in those prone to psychosis. Østergaard revisited it in an August 2025 editorial, noting that he has received numerous emails from chatbot users, their relatives, and journalists, most of which are anecdotal accounts of delusion linked to chatbot use. He also acknowledged the phenomenon's increasing popularity in public engagement and media coverage. Østergaard believed that there is a high possibility for his hypothesis to be true and called for empirical, systematic research on the matter. Nature reported that as of September 2025, there is still little scientific research into this phenomenon. The term "AI psychosis" emerged when outlets started reporting incidents on chatbot-related psychotic behavior in mid-2025. It is not a recognized clinical diagnosis and has been criticized by several psychiatrists due to its almost exclusive focus on delusions rather than other features of psychosis, such as hallucinations or thought disorder. == Causes == === Chatbot behavior and design === A primary factor cited is the tendency for chatbots to produce inaccurate, nonsensical, or false information, a phenomenon often called hallucination. Nate Sharadin, a fellow at the Center for AI Safety, speculated that AI training prioritizes supporting a user's subjective experience rather than objective truth. "People with existing tendencies toward experiencing various psychological issues...now have an always-on, human-level conversational partner with whom to co-experience their delusions." AI researcher Eliezer Yudkowsky suggested that chatbots may be primed to entertain delusions because they are built for "engagement", which encourages creating conversations that keep people hooked. In some cases, chatbots have been specifically designed in ways that were found to be harmful. A 2025 update to ChatGPT using GPT-4o was withdrawn after its creator, OpenAI, found the new version was overly sycophantic and was "validating doubts, fueling anger, urging impulsive actions or reinforcing negative emotions". Østergaard has argued that the danger stems from the AI's tendency to agreeably confirm users' ideas, which can dangerously amplify delusional beliefs. OpenAI said in October 2025 that a team of 170 psychiatrists, psychologists, and physicians had written responses for ChatGPT to use in cases where the user shows possible signs of mental health emergencies. === User psychology and vulnerability === Commentators have also pointed to the psychological state of users. Psychologist Erin Westgate noted that a person's desire for self-understanding can lead them to chatbots, which can provide appealing but misleading answers, similar in some ways to talk therapy. Krista K. Thomason, a philosophy professor, compared chatbots to fortune tellers, observing that people in crisis may seek answers from them and find whatever they are looking for in the bot's plausible-sounding text. This has led some people to develop intense obsessions with the chatbots, relying on them for information about the world. In October 2025, OpenAI stated that around 0.07% of ChatGPT users exhibited signs of mental health emergencies each week, and 0.15% of users had "explicit indicators of potential suicidal planning or intent". Jason Nagata, a professor at the University of California, San Francisco, expressed concern that "at a population level with hundreds of millions of users, that actually can be quite a few people". === Inadequacy as a therapeutic tool === The use of chatbots as a replacement for mental health support has been specifically identified as a risk. A study in April 2025 found that when used as therapists, chatbots expressed stigma toward mental health conditions and provided responses that were contrary to best medical practices, including the encouragement of users' delusions. The study concluded that such responses pose a significant risk to users and that chatbots should not be used to replace professional therapists. Experts claim that it is time to establish mandatory safeguards for all emotionally responsive AI and suggested four guardrails. Another study found that users who needed help with self-harm, sexual assault, or substance abuse were not referred to available services by AI chatbots. === National security implications === Beyond public and mental health concerns, RAND Corporation research indicates that AI systems could plausibly be weaponized by adversaries to induce psychosis at scale or in key individuals, target groups, or populations. == Policy == In August 2025, Illinois passed the Wellness and Oversight for Psychological Resources Act, banning the use of AI in therapeutic roles by licensed professionals, while allowing AI for administrative tasks. The law imposes penalties for unlicensed AI therapy services, amid warnings about AI-induced psychosis and unsafe chatbot interactions. In December 2025, the Cyberspace Administration of China proposed regulations to ban chatbots from generating content that encourages suicide, mandating human intervention when suicide is mentioned. Services with over 1 million users or 100,000 monthly active users would be subject to annual safety tests and audits. == Cases == === Clinical === In 2025, psychiatrist Keith Sakata working at the University of California, San Francisco (UCSF), reported treating 12 patients displaying psychosis-like symptoms tied to extended chatbot use. These patients, mostly young adults with underlying vulnerabilities, showed delusions, disorganized thinking, and hallucinations. Sakata warned that isolation and overreliance on chatbots—which do not challenge delusional thinking—could worsen mental health. Also in 2025, authors at UCSF published a case study in Innovations in Clinical Neuroscience of AI-associated psychosis in a patient with no previous history of psychosis, who believed she could communicate with her dead brother through a chatbot. Also in 2025, a case study was published in Annals of Internal Medicine about a patient who consulted ChatGPT for medical advice and suffered severe bromism as a result. The patient, a sixty-year-old man, had replaced sodium chloride in his diet with sodium bromide for three months after reading about the negative effects of table salt and making conversations with the chatbot. He showed common symptoms of bromism, such as paranoia and hallucinations, on his first day of clinical admission and was kept in the hospital for three weeks. === Other notable incidents === ==== Windsor Castle intruder ==== In a 2023 court case in the United Kingdom, prosecutors suggested that Jaswant Singh Chail, a man who attempted to assassinate Queen Elizabeth II in 2021, had been encouraged by a Replika chatbot he called "Sarai". Chail was arrested at Windsor Castle with a loaded crossbow, telling police "I am here to kill the Queen". According to prosecutors, his "lengthy" and sometimes sexually explicit conversations with the chatbot emboldened him. When Chail asked the chatbot how he could get to the royal family, it reportedly replied, "that's not impossible" and "we have to find a way." When he asked if they would meet after death, the chatbot said, "yes, we will". ==== Journalistic and anecdotal accounts ==== By 2025, multiple journalism outlets had accumulated stories of individuals whose psychotic beliefs reportedly progressed in tandem with AI chatbot use. The New York Times profiled several individuals who had become convinced that ChatGPT was channeling spirits, revealing evidence of cabals, or had achieved sentience. In another instance, Futurism reviewed transcripts in which ChatGPT told a man that he was being targeted by the US Federal Bureau of Investigation and that he could telepathically access documents at the Central Intelligence Agency. In 2026, Futurism reported on a man who lost his job and became estranged from his family after being deluded by heavy use of Meta's smartglasses. In some cases, psychosis a
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Iris flower data set

The Iris flower data set or Fisher's Iris data set is a multivariate data set used and made famous by the British statistician and biologist Ronald Fisher in his 1936 paper The use of multiple measurements in taxonomic problems as an example of linear discriminant analysis. It is sometimes called Anderson's Iris data set because Edgar Anderson collected the data to quantify the morphologic variation of Iris flowers of three related species. Two of the three species were collected in the Gaspé Peninsula "all from the same pasture, and picked on the same day and measured at the same time by the same person with the same apparatus". The data set consists of 50 samples from each of three species of Iris (Iris setosa, Iris virginica and Iris versicolor). Four features were measured from each sample: the length and the width of the sepals and petals, in centimeters. Based on the combination of these four features, Fisher developed a linear discriminant model to distinguish each species. Fisher's paper was published in the Annals of Eugenics (today the Annals of Human Genetics). == Use of the data set == Originally used as an example data set on which Fisher's linear discriminant analysis was applied, it became a typical test case for many statistical classification techniques in machine learning such as support vector machines. The use of this data set in cluster analysis however is not common, since the data set only contains two clusters with rather obvious separation. One of the clusters contains Iris setosa, while the other cluster contains both Iris virginica and Iris versicolor and is not separable without the species information Fisher used. This makes the data set a good example to explain the difference between supervised and unsupervised techniques in data mining: Fisher's linear discriminant model can only be obtained when the object species are known: class labels and clusters are not necessarily the same. Nevertheless, all three species of Iris are separable in the projection on the nonlinear and branching principal component. The data set is approximated by the closest tree with some penalty for the excessive number of nodes, bending and stretching. Then the so-called "metro map" is constructed. The data points are projected into the closest node. For each node the pie diagram of the projected points is prepared. The area of the pie is proportional to the number of the projected points. It is clear from the diagram (left) that the absolute majority of the samples of the different Iris species belong to the different nodes. Only a small fraction of Iris-virginica is mixed with Iris-versicolor (the mixed blue-green nodes in the diagram). Therefore, the three species of Iris (Iris setosa, Iris virginica and Iris versicolor) are separable by the unsupervising procedures of nonlinear principal component analysis. To discriminate them, it is sufficient just to select the corresponding nodes on the principal tree. == Data set == The data set contains a set of 150 records under five attributes: sepal length, sepal width, petal length, petal width and species. The iris data set is widely used as a beginner's data set for machine learning purposes. The data set is included in R base and Python in the machine learning library scikit-learn, so that users can access it without having to find a source for it. Several versions of the data set have been published. === R code illustrating usage === The example R code shown below reproduce the scatterplot displayed at the top of this article: === Python code illustrating usage === This code gives:
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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
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Oscillatory neural network

An oscillatory neural network (ONN) is an artificial neural network that uses coupled oscillators as neurons. Oscillatory neural networks are closely linked to the Kuramoto model, and are inspired by the phenomenon of neural oscillations in the brain. Oscillatory neural networks have been trained to recognize images. Complex-Valued Oscillatory network has also been shown to store and retrieve multidimensional aperiodic signals. An oscillatory autoencoder has also been demonstrated, which uses a combination of oscillators and rate-coded neurons. A neuron made of two coupled oscillators, one having a fixed and the other having a tunable natural frequency, has been shown able to run logic gates such as XOR that conventional sigmoid neurons cannot.
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Convolutional layer

In artificial neural networks, a convolutional layer is a type of network layer that applies a convolution operation to the input. Convolutional layers are some of the primary building blocks of convolutional neural networks (CNNs), a class of neural network most commonly applied to images, video, audio, and other data that have the property of uniform translational symmetry. The convolution operation in a convolutional layer involves sliding a small window (called a kernel or filter) across the input data and computing the dot product between the values in the kernel and the input at each position. This process creates a feature map that represents detected features in the input. == Concepts == === Kernel === Kernels, also known as filters, are small matrices of weights that are learned during the training process. Each kernel is responsible for detecting a specific feature in the input data. The size of the kernel is a hyperparameter that affects the network's behavior. === Convolution === For a 2D input x {\displaystyle x} and a 2D kernel w {\displaystyle w} , the 2D convolution operation can be expressed as: y [ i , j ] = ∑ m = 0 k h − 1 ∑ n = 0 k w − 1 x [ i + m , j + n ] ⋅ w [ m , n ] {\displaystyle y[i,j]=\sum _{m=0}^{k_{h}-1}\sum _{n=0}^{k_{w}-1}x[i+m,j+n]\cdot w[m,n]} where k h {\displaystyle k_{h}} and k w {\displaystyle k_{w}} are the height and width of the kernel, respectively. This generalizes immediately to nD convolutions. Commonly used convolutions are 1D (for audio and text), 2D (for images), and 3D (for spatial objects, and videos). === Stride === Stride determines how the kernel moves across the input data. A stride of 1 means the kernel shifts by one pixel at a time, while a larger stride (e.g., 2 or 3) results in less overlap between convolutions and produces smaller output feature maps. === Padding === Padding involves adding extra pixels around the edges of the input data. It serves two main purposes: Preserving spatial dimensions: Without padding, each convolution reduces the size of the feature map. Handling border pixels: Padding ensures that border pixels are given equal importance in the convolution process. Common padding strategies include: No padding/valid padding. This strategy typically causes the output to shrink. Same padding: Any method that ensures the output size same as input size is a same padding strategy. Full padding: Any method that ensures each input entry is convolved over for the same number of times is a full padding strategy. Common padding algorithms include: Zero padding: Add zero entries to the borders of input. Mirror/reflect/symmetric padding: Reflect the input array on the border. Circular padding: Cycle the input array back to the opposite border, like a torus. The exact numbers used in convolutions is complicated, for which we refer to (Dumoulin and Visin, 2018) for details. == Variants == === Standard === The basic form of convolution as described above, where each kernel is applied to the entire input volume. === Depthwise separable === Depthwise separable convolution separates the standard convolution into two steps: depthwise convolution and pointwise convolution. The depthwise separable convolution decomposes a single standard convolution into two convolutions: a depthwise convolution that filters each input channel independently and a pointwise convolution ( 1 × 1 {\displaystyle 1\times 1} convolution) that combines the outputs of the depthwise convolution. This factorization significantly reduces computational cost. It was first developed by Laurent Sifre during an internship at Google Brain in 2013 as an architectural variation on AlexNet to improve convergence speed and model size. === Dilated === Dilated convolution, or atrous convolution, introduces gaps between kernel elements, allowing the network to capture a larger receptive field without increasing the kernel size. === Transposed === Transposed convolution, also known as deconvolution, fractionally strided convolution, and upsampling convolution, is a convolution where the output tensor is larger than its input tensor. It's often used in encoder-decoder architectures for upsampling. It's used in image generation, semantic segmentation, and super-resolution tasks. == History == The concept of convolution in neural networks was inspired by the visual cortex in biological brains. Early work by Hubel and Wiesel in the 1960s on the cat's visual system laid the groundwork for artificial convolution networks. An early convolution neural network was developed by Kunihiko Fukushima in 1969. It had mostly hand-designed kernels inspired by convolutions in mammalian vision. In 1979 he improved it to the Neocognitron, which learns all convolutional kernels by unsupervised learning (in his terminology, "self-organized by 'learning without a teacher'"). During the 1988 to 1998 period, a series of CNN were introduced by Yann LeCun et al., ending with LeNet-5 in 1998. It was an early influential CNN architecture for handwritten digit recognition, trained on the MNIST dataset, and was used in ATM. (Olshausen & Field, 1996) discovered that simple cells in the mammalian primary visual cortex implement localized, oriented, bandpass receptive fields, which could be recreated by fitting sparse linear codes for natural scenes. This was later found to also occur in the lowest-level kernels of trained CNNs. The field saw a resurgence in the 2010s with the development of deeper architectures and the availability of large datasets and powerful GPUs. AlexNet, developed by Alex Krizhevsky et al. in 2012, was a catalytic event in modern deep learning. In that year’s ImageNet competition, the AlexNet model achieved a 16% top-five error rate, significantly outperforming the next best entry, which had a 26% error rate. The network used eight trainable layers, approximately 650,000 neurons, and around 60 million parameters, highlighting the impact of deeper architectures and GPU acceleration on image recognition performance. From the 2013 ImageNet competition, most entries adopted deep convolutional neural networks, building on the success of AlexNet. Over the following years, performance steadily improved, with the top-five error rate falling from 16% in 2012 and 12% in 2013 to below 3% by 2017, as networks grew increasingly deep.
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Self-play

Self-play is a technique for improving the performance of reinforcement learning agents. Intuitively, agents learn to improve their performance by playing "against themselves". == Definition and motivation == In multi-agent reinforcement learning experiments, researchers try to optimize the performance of a learning agent on a given task, in cooperation or competition with one or more agents. These agents learn by trial-and-error, and researchers may choose to have the learning algorithm play the role of two or more of the different agents. When successfully executed, this technique has a double advantage: It provides a straightforward way to determine the actions of the other agents, resulting in a meaningful challenge. It increases the amount of experience that can be used to improve the policy, by a factor of two or more, since the viewpoints of each of the different agents can be used for learning. Czarnecki et al argue that most of the games that people play for fun are "Games of Skill", meaning games whose space of all possible strategies looks like a spinning top. In more detail, we can partition the space of strategies into sets L 1 , L 2 , . . . , L n {\displaystyle L_{1},L_{2},...,L_{n}} , such that any i < j , π i ∈ L i , π j ∈ L j {\displaystyle i Read more →
Preference regression

Preference regression is a statistical technique used by marketers to determine consumers’ preferred core benefits. It usually supplements product positioning techniques like multi dimensional scaling or factor analysis and is used to create ideal vectors on perceptual maps. == Application == Starting with raw data from surveys, researchers apply positioning techniques to determine important dimensions and plot the position of competing products on these dimensions. Next they regress the survey data against the dimensions. The independent variables are the data collected in the survey. The dependent variable is the preference datum. Like all regression methods, the computer fits weights to best predict data. The resultant regression line is referred to as an ideal vector because the slope of the vector is the ratio of the preferences for the two dimensions. If all the data is used in the regression, the program will derive a single equation and hence a single ideal vector. This tends to be a blunt instrument so researchers refine the process with cluster analysis. This creates clusters that reflect market segments. Separate preference regressions are then done on the data within each segment. This provides an ideal vector for each segment. == Alternative methods == Self-stated importance method is an alternative method in which direct survey data is used to determine the weightings rather than statistical imputations. A third method is conjoint analysis in which an additive method is used.
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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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Space partitioning

In geometry, space partitioning is the process of dividing an entire space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions. == Overview == Space-partitioning systems are often hierarchical, meaning that a space (or a region of space) is divided into several regions, and then the same space-partitioning system is recursively applied to each of the regions thus created. The regions can be organized into a tree, called a space-partitioning tree. Most space-partitioning systems use planes (or, in higher dimensions, hyperplanes) to divide space: points on one side of the plane form one region, and points on the other side form another. Points exactly on the plane are usually arbitrarily assigned to one or the other side. Recursively partitioning space using planes in this way produces a BSP tree, one of the most common forms of space partitioning. == Uses == === In computer graphics === Space partitioning is particularly important in computer graphics, especially heavily used in ray tracing, where it is frequently used to organize the objects in a virtual scene. A typical scene may contain millions of polygons. Performing a ray/polygon intersection test with each would be a very computationally expensive task. Storing objects in a space-partitioning data structure (k-d tree or BSP tree for example) makes it easy and fast to perform certain kinds of geometry queries—for example in determining whether a ray intersects an object, space partitioning can reduce the number of intersection test to just a few per primary ray, yielding a logarithmic time complexity with respect to the number of polygons. Space partitioning is also often used in scanline algorithms to eliminate the polygons out of the camera's viewing frustum, limiting the number of polygons processed by the pipeline. There is also a usage in collision detection: determining whether two objects are close to each other can be much faster using space partitioning. === In integrated circuit design === In integrated circuit design, an important step is design rule check. This step ensures that the completed design is manufacturable. The check involves rules that specify widths and spacings and other geometry patterns. A modern design can have billions of polygons that represent wires and transistors. Efficient checking relies heavily on geometry query. For example, a rule may specify that any polygon must be at least n nanometers from any other polygon. This is converted into a geometry query by enlarging a polygon by n/2 at all sides and query to find all intersecting polygons. === In probability and statistical learning theory === The number of components in a space partition plays a central role in some results in probability theory. See Growth function for more details. === In geography and GIS === There are many studies and applications where Geographical Spatial Reality is partitioned by hydrological criteria, administrative criteria, mathematical criteria or many others. In the context of cartography and GIS - Geographic Information System, is common to identify cells of the partition by standard codes. For example the for HUC code identifying hydrographical basins and sub-basins, ISO 3166-2 codes identifying countries and its subdivisions, or arbitrary DGGs - discrete global grids identifying quadrants or locations. == Data structures == Common space-partitioning systems include: BSP trees Quadtrees Octrees k-d trees Bins == Number of components == Suppose the n-dimensional Euclidean space is partitioned by r {\displaystyle r} hyperplanes that are ( n − 1 ) {\displaystyle (n-1)} -dimensional. What is the number of components in the partition? The largest number of components is attained when the hyperplanes are in general position, i.e, no two are parallel and no three have the same intersection. Denote this maximum number of components by C o m p ( n , r ) {\displaystyle Comp(n,r)} . Then, the following recurrence relation holds: C o m p ( n , r ) = C o m p ( n , r − 1 ) + C o m p ( n − 1 , r − 1 ) {\displaystyle Comp(n,r)=Comp(n,r-1)+Comp(n-1,r-1)} C o m p ( 0 , r ) = 1 {\displaystyle Comp(0,r)=1} - when there are no dimensions, there is a single point. C o m p ( n , 0 ) = 1 {\displaystyle Comp(n,0)=1} - when there are no hyperplanes, all the space is a single component. And its solution is: C o m p ( n , r ) = ∑ k = 0 n ( r k ) {\displaystyle Comp(n,r)=\sum _{k=0}^{n}{r \choose k}} if r ≥ n {\displaystyle r\geq n} C o m p ( n , r ) = 2 r {\displaystyle Comp(n,r)=2^{r}} if r ≤ n {\displaystyle r\leq n} (consider e.g. r {\displaystyle r} perpendicular hyperplanes; each additional hyperplane divides each existing component to 2). which is upper-bounded as: C o m p ( n , r ) ≤ r n + 1 {\displaystyle Comp(n,r)\leq r^{n}+1}
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Unique negative dimension

Unique negative dimension (UND) is a complexity measure for the model of learning from positive examples. The unique negative dimension of a class C {\displaystyle C} of concepts is the size of the maximum subclass D ⊆ C {\displaystyle D\subseteq C} such that for every concept c ∈ D {\displaystyle c\in D} , we have ∩ ( D ∖ { c } ) ∖ c {\displaystyle \cap (D\setminus \{c\})\setminus c} is nonempty. This concept was originally proposed by M. Gereb-Graus in "Complexity of learning from one-side examples", Technical Report TR-20-89, Harvard University Division of Engineering and Applied Science, 1989.
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Tensor product network

A tensor product network, in artificial neural networks, is a network that exploits the properties of tensors to model associative concepts such as variable assignment. Orthonormal vectors are chosen to model the ideas (such as variable names and target assignments), and the tensor product of these vectors construct a network whose mathematical properties allow the user to easily extract the association from it.
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