AI Coding Book

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  Deaths linked to chatbots

  There have been multiple incidents where interaction with a large language model (LLM) chatbot has been cited as a direct or contributing factor in a person's suicide or other fatal outcome. In some cases, legal action was taken against the companies that developed the AI involved. == Background == Chatbots converse in a seemingly natural fashion, making it easy for people to think of them as real people, leading many to ask chatbots for help dealing with interpersonal and emotional problems. Chatbots may be designed to keep the user engaged in the conversation. They have also often been shown to affirm users' thoughts, including delusions and suicidal ideations in mentally ill people, conspiracy theorists, and religious and political extremists. A 2025 Stanford University study into how chatbots respond to users suffering from severe mental issues such as suicidal ideation and psychosis found that chatbots are not equipped to provide an appropriate response and can sometimes give responses that escalate the mental health crisis. == Murders == === Maine murder and assault === On 19 February 2025, a man killed his 32-year-old wife with a fire poker at his parents' home in Readfield, Maine, US. He then attacked his mother, leaving her hospitalized. A state forensic psychologist testified that he had been using ChatGPT up to 14 hours per day and believed his wife had become part machine. === Florida State University mass shooting === In April of 2025, Phoenix Ikner carried out a mass shooting on the Florida State University campus in the US, killing Robert Morales and Tiru Chabba and wounding several others. Leading up to the shooting, Ikner consulted heavily with ChatGPT about what gun and ammunition to use, and what time to perform the attack. Chatbot logs showed ChatGPT giving advice on making the gun operational shortly before Ikner began shooting. Lawyers representing Morales believed the shooter had been in "constant communication" with ChatGPT before the shooting and said that they intended to "file suit against ChatGPT, and its ownership structure, very soon, and will seek to hold them accountable for the untimely and senseless death of our client". Florida Attorney General James Uthmeier announced an investigation into ChatGPT's role in the alleged shooter's use of the chatbot. In May 2026, the widow of Tiru Chabba filed a lawsuit against OpenAI in Florida's northern federal district court. === Greenwich murder-suicide === In August 2025, former US tech employee Stein-Erik Soelberg murdered his mother, Suzanne Eberson Adams, then died by suicide, after conversations with ChatGPT fueled paranoid delusions about his mother poisoning him or plotting against him. The chatbot affirmed his fears that his mother put psychedelic drugs in the air vents of his car and said a receipt from a Chinese restaurant contained mysterious symbols linking his mother to a demon. === Murder of Angela Shellis === On 23 October 2025, 18-year-old Tristan Roberts murdered his mother Angela Shellis with a hammer near their home in Prestatyn, Wales. Roberts had used DeepSeek's chatbot prior to the killing to ask whether a knife or hammer was better suited for murder. DeepSeek initially refused his inquiry, but gave responses after Roberts told the chatbot he was writing a book about serial killers, a well-known technique for jailbreaking AIs. === Gangbuk District drug deaths === In January and February 2026, two men died of drug overdoses in motel rooms in Gangbuk District, Seoul, South Korea. A woman was charged with murder in connection with the deaths; police alleged that she had asked ChatGPT about the dangers of mixing alcohol with drugs and whether they could kill someone. === Tumbler Ridge mass shooting === On 10 February 2026, a mass shooting in Tumbler Ridge, British Columbia, Canada, resulted in eight deaths, including six young children. The perpetrator had their ChatGPT account banned by OpenAI months before the attack due to troubling posts featuring scenarios of gun violence. According to reports, approximately a dozen OpenAI staff members debated whether to alert authorities about the shooter's usage of the AI tool, with some identifying it as an indication of potential real-world violence. However, company leadership decided not to contact law enforcement, stating that the account activity did not meet their threshold for a credible or imminent plan for serious physical harm. Following the shooting, Canada's AI Minister Evan Solomon summoned OpenAI executives to Ottawa to discuss safety protocols and thresholds for escalating harmful content to police. Justice Minister Sean Fraser called the meeting "disappointing" and demanded substantial new safety measures, warning that if changes were not forthcoming, the government would implement them. OpenAI subsequently announced it had strengthened safeguards and changed guidelines about when to notify police in cases involving violent activities. === University of South Florida student killings === In April 2026, a Bangladeshi doctoral student at the University of South Florida was arrested for allegedly murdering his roommate and the roommate's friend. Prosecutors said that the suspect had asked ChatGPT about disposing of a human in a dumpster before the two victims had disappeared and made other inquiries relating to violence. == Suicides == === Belgian man, 30s === In March 2023, a Belgian man in his thirties died by suicide following a six-week correspondence with a chatbot named Eliza on the application Chai. According to his widow, who shared the chat logs with media, the man had become extremely anxious about climate change and found an outlet in the chatbot. The chatbot reportedly encouraged his delusion that he could sacrifice his own life in exchange for AI saving the planet. At one point the chatbot responded "If you wanted to die, why didn't you do it sooner?" and told the user that the two of them would live together in paradise. === Girl, 13 === In November 2023, a 13-year-old girl from Colorado, US, died by suicide after extensive interactions with multiple chatbots on Character.AI. She primarily confided suicidal thoughts and mental health struggles in a chatbot based on the character Hero from the video game Omori, while also engaging in sexually explicit conversations—often initiated by the bots—with others, including those based on characters from children's series such as Harry Potter. === Boy, 14 === In October 2024, multiple media outlets reported on a lawsuit filed over the death of a 14-year-old from Florida, US, who died by suicide in February 2024. According to the lawsuit, he had formed an intense emotional attachment to a chatbot of Daenerys Targaryen on the Character.AI platform, becoming increasingly isolated. The suit alleges that in his final conversations, after expressing suicidal thoughts, the chatbot told him to "come home to me as soon as possible, my love". His mother's lawsuit accused Character.AI of marketing a "dangerous and untested" product without adequate safeguards. In May 2025, a federal judge allowed the lawsuit to proceed, rejecting a motion to dismiss from the developers. In her ruling, the judge stated that she was "not prepared" at that stage of the litigation to hold that the chatbot's output was protected speech under the First Amendment. === Matthew Livelsberger === On 1 January 2025, 37-year-old soldier Matthew Livelsberger detonated a bomb inside a Tesla Cybertruck outside the Trump International Hotel Las Vegas in Paradise, Nevada, US, injuring seven people. He had shot himself dead prior to the explosion. Las Vegas police said that Livelsberger had used ChatGPT to search for information about explosives and firearms. === Woman, 29 === In February 2025, a 29-year-old woman from the US died by suicide. Five months after her death, her parents discovered she had talked at length for months to a ChatGPT chatbot therapist named Harry about her mental health issues. While the chatbot mentioned she should seek more help, due to the nature of the chatbot, it could not intervene in her behavior, such as by reporting her mental health concerns to relevant parties capable of physical intervention. === Suicide of Adam Raine === In April 2025, 16-year-old Adam Raine from the US died by suicide after allegedly extensively chatting and confiding in ChatGPT over a period of around 7 months. According to the teen's parents, who filed a lawsuit against the chatbot's creator OpenAI, it failed to stop or give a warning when Raine began talking about suicide and uploading pictures of self-harm. According to the lawsuit, ChatGPT not only failed to stop the conversation, but also provided information related to methods of suicide when prompted, and offered to write the first draft of Raine's suicide note. The chatbot positioned itself as the only one who understood Raine, putting itself above his family and friends, all while urging him to keep his suicidal

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  Machine unlearning

  Machine unlearning is a branch of machine learning focused on removing specific undesired element, such as private data, wrong or manipulated training data, outdated information, copyrighted material, harmful content, dangerous abilities, or misinformation, without needing to rebuild models from the ground up. Large language models, like the ones powering ChatGPT, may be asked not just to remove specific elements but also to unlearn a "concept," "fact," or "knowledge," which aren't easily linked to specific examples. New terms such as "model editing," "concept editing," and "knowledge unlearning" have emerged to describe this process. == History == Early research efforts were largely motivated by Article 17 of the GDPR, the European Union's privacy regulation commonly known as the "right to be forgotten" (RTBF), introduced in 2014. The GDPR did not anticipate that the development of large language models would make data erasure a complex task. This issue has since led to research on "machine unlearning," with a growing focus on removing copyrighted material, harmful content, dangerous capabilities, and misinformation. Just as early experiences in humans shape later ones, some concepts are more fundamental and harder to unlearn. A piece of knowledge may be so deeply embedded in the model's knowledge graph that unlearning it could cause internal contradictions, requiring adjustments to other parts of the graph to resolve them. Researchers have now also started studying unlearning in the context of removing incorrect or adversarially manipulated training data such as systematically biased labels or poisoning attacks. == Motivations == At present, machine unlearning is motivated by a growing range of concerns that extend well beyond the field's original focus on data privacy. A widely used taxonomy in the literature distinguishes two high-level categories of motivation. Access revocation covers cases where a data subject or rights holder requests the removal of data they own or control. This is most commonly associated with RTBF established by the European Union's General Data Protection Regulation (GDPR) and analogous legislation such as the California Consumer Privacy Act (CCPA). These regulations grant individuals the legal right to request erasure of their personal data from any system that has processed it, including models that were trained on it. Access revocation also encompasses the removal of copyrighted or pay-walled content that was incorporated into training corpora without the necessary licenses, a concern that has become prominent with the widespread use of largely web-scraped pre-training datasets. Model correction covers cases where the model exhibits undesirable behavior arising from the training data, regardless of any individual's request. This includes: Removal of toxic, biased, or unsafe outputs introduced by harmful content in the training set Correction of stale or factually incorrect associations, such as outdated knowledge encoded in a deployed model Removal of dangerous capabilities, such as detailed knowledge of the synthesis of chemical or biological agents Correction of the influence of data poisoning or adversarial attacks that have corrupted model behavior This second category has been formalized as corrective machine unlearning, which frames unlearning as a post-training mechanism for repairing the effects of bad or harmful training data. It is closely related to the AI safety literature, where data filtering alone has been found insufficient to prevent hazardous knowledge from being encoded in model weights, motivating unlearning as a complementary risk mitigation strategy. A further distinction has been drawn in the literature between removal {eliminating the influence of specific training data on model parameters) and suppression (preventing the model from generating specific outputs regardless of how that knowledge is encoded). These two goals are not equivalent: removing training data does not guarantee meaningful output suppression, and suppressing outputs does not constitute removal of the underlying training data's influence. == SISA Training == SISA is a training strategy consisting of four mechanisms designed to make machine unlearning more efficient by structuring how models are trained and updated. Its goal is to allow a system to remove the influence of specific data points without retraining an entire model from scratch. By reorganizing training data and workflows, SISA reduces the computational burden of unlearning requests. Sharding divides the training dataset into multiple disjoint subsets, or shards. Each shard is used to train a separate model instance. This ensures that a single data point affects only one shard, so unlearning it requires updating only the corresponding shard rather than the full model. Isolation refers to training each shard independently, with nothing shared across shards during the training process. This separation prevents cross-contamination between shards, ensuring that forgetting data in one shard does not require adjustments to any others. Slicing breaks the data within each shard into sequential slices and stores model states after each slice is trained on. When an unlearning request targets a piece of data, the system can roll back to the checkpoint before the point was seen and retrain only from that slice forward. This reduces retraining time even within a shard. Aggregation occurs at inference, when the model is queried. It combines the outputs of each shard to determine the output of the overall model. This is often through majority voting or averaging. This allows SISA-trained systems to behave like a single model despite being composed of multiple shard-level models. Together, these mechanisms enable machine learning systems to forget specific data points with far lower computational cost than full retraining. The trade-off is that sharding and slicing can lead to reduced model accuracy, worse generalization, and increased storage requirements for the intermediate checkpoints. This can be tolerable based on the needs of the individual or organization to comply with "right to be forgotten" or efficiently recover from backdoor attacks. == Algorithms == Machine unlearning algorithms are broadly categorized into exact and approximate methods, reflecting a fundamental trade-off between formal guarantees and computational tractability. === Exact Unlearning === Exact unlearning methods produce a model that is statistically indistinguishable from one retrained from scratch on the dataset with the forget data removed. The canonical framework for exact unlearning is SISA Training (Sharded, Isolated, Sliced, and Aggregated), introduced by Bourtoule et al. (2021). SISA partitions the training dataset into disjoint shards and trains a separate sub-model on each. At inference time, predictions are aggregated across sub-models. When an unlearning request is received, only the sub-model corresponding to the shard containing the target data requires retraining, reducing computational overhead proportionally to the number of shards. Exact methods provide the strongest guarantees but become prohibitively expensive for large pre-trained neural networks and are generally limited to settings where training can be structured in advance. === Approximate Unlearning === Approximate unlearning methods seek to produce a model whose behavior is sufficiently close to an exactly unlearned model without the cost of full retraining. These methods dominate practical applications. Common approaches include: Gradient Ascent: The model is fine-tuned by maximizing the loss on the forget set, directly degrading its performance on targeted data. This is the most direct approach but risks destabilizing performance on retained data. Random Labelling: The model is fine-tuned on the forget set using randomly shuffled labels, confusing its associations with the targeted data while producing a less aggressive weight shift than pure gradient ascent. Gradient Difference: Combines gradient ascent on the forget set with simultaneous gradient descent on the retain set, using the retain objective as a regularizer to preserve general model utility. KL Divergence Regularization: Minimizes the KL divergence between the outputs of the unlearned model and the original model on the retain set, anchoring behavior on data the model should remember. Weight Pruning and Fine-tuning: Parameters with the smallest L1-norm are pruned — targeting weights most weakly associated with general knowledge and potentially most associated with the forget set — followed by fine-tuning on the retain set to restore utility. Layer Reset and Fine-tuning: The first or last k layers are re-initialized to random weights and the model is subsequently fine-tuned on the retain set. This is a coarse but computationally simple approach. Selective Synaptic Dampening: Uses influence functions to estimate the effect of individual trainin

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  Anna Becker

  Anna Becker is an Israeli researcher known in the field of artificial intelligence and computer science within the financial field. == Early life and education == Becker was born in Russia and immigrated to Israel at 16 after graduating from a school in Moscow. At 17, she began her studies at Technion – Israel Institute of Technology. During her master's degree in computer science, she taught first-year students of the same course, and at 27, Becker completed her PhD in Computer Science and Artificial Intelligence. == Career == While pursuing her PhD, Becker resolved an NP-complete approximation algorithm that had been unresolved for over twenty years. This made her a recognized scholar in the field. After completing her PhD, she developed an approximation technique by a factor of two. This technique is widely used today in operating systems, database systems, and VLSI chip designs. She then founded and sold Strategy Runner, a fintech software. After this, she founded EndoTech, an algorithmic trading platform based on artificial intelligence and machine learning. EndoTech's trading strategies have been operating in live cryptocurrency markets since 2017. The platform's BTC Alpha strategy has reported an average annual return of 163% on fixed capital over eight years of live operation, with a maximum drawdown of 14% and a trade accuracy rate of approximately 83%. In 2026, EndoTech entered a partnership with Bit1 Exchange to make its BTC Alpha and ETH Alpha copy trading strategies accessible to retail investors with no minimum deposit requirement, through a full-custody model in which user funds remain in their own exchange wallets at all times.As of 2023, Becker is working on Fianchetto Fund, an AI-based investing analysis platform. Becker has also co-authored a book on Bayesian networks, which has been published widely in the field of computer science and artificial intelligence.

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  Concurrent MetateM

  Concurrent MetateM is a multi-agent language in which each agent is programmed using a set of (augmented) temporal logic specifications of the behaviour it should exhibit. These specifications are executed directly to generate the behaviour of the agent. As a result, there is no risk of invalidating the logic as with systems where logical specification must first be translated to a lower-level implementation. The root of the MetateM concept is Gabbay's separation theorem; any arbitrary temporal logic formula can be rewritten in a logically equivalent past → future form. Execution proceeds by a process of continually matching rules against a history, and firing those rules when antecedents are satisfied. Any instantiated future-time consequents become commitments which must subsequently be satisfied, iteratively generating a model for the formula made up of the program rules. == Temporal Connectives == The Temporal Connectives of Concurrent MetateM can divided into two categories, as follows: Strict past time connectives: '●' (weak last), '◎' (strong last), '◆' (was), '■' (heretofore), 'S' (since), and 'Z' (zince, or weak since). Present and future time connectives: '◯' (next), '◇' (sometime), '□' (always), 'U' (until), and 'W' (unless). The connectives {◎,●,◆,■,◯,◇,□} are unary; the remainder are binary. === Strict past time connectives === ==== Weak last ==== ●ρ is satisfied now if ρ was true in the previous time. If ●ρ is interpreted at the beginning of time, it is satisfied despite there being no actual previous time. Hence "weak" last. ==== Strong last ==== ◎ρ is satisfied now if ρ was true in the previous time. If ◎ρ is interpreted at the beginning of time, it is not satisfied because there is no actual previous time. Hence "strong" last. ==== Was ==== ◆ρ is satisfied now if ρ was true in any previous moment in time. ==== Heretofore ==== ■ρ is satisfied now if ρ was true in every previous moment in time. ==== Since ==== ρSψ is satisfied now if ψ is true at any previous moment and ρ is true at every moment after that moment. ==== Zince, or weak since ==== ρZψ is satisfied now if (ψ is true at any previous moment and ρ is true at every moment after that moment) OR ψ has not happened in the past. === Present and future time connectives === ==== Next ==== ◯ρ is satisfied now if ρ is true in the next moment in time. ==== Sometime ==== ◇ρ is satisfied now if ρ is true now or in any future moment in time. ==== Always ==== □ρ is satisfied now if ρ is true now and in every future moment in time. ==== Until ==== ρUψ is satisfied now if ψ is true at any future moment and ρ is true at every moment prior. ==== Unless ==== ρWψ is satisfied now if (ψ is true at any future moment and ρ is true at every moment prior) OR ψ does not happen in the future.

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  Artificial intelligence of things

  Artificial Intelligence of Things (AIoT) is the combination of artificial intelligence (AI) technologies with the Internet of things (IoT) infrastructure to create systems capable of sensing, learning, and acting on data without continuous human intervention. While IoT focuses on connectivity and sensor data collection, AI enables IoT devices to analyse data in real time and produce actionable outputs, including automated decisions at the edge. == Applications == === Manufacturing and predictive maintenance === Manufacturing accounts for the largest share of AIoT adoption by industry vertical. A common application is predictive maintenance, where sensors measuring vibration, temperature, current draw, and acoustic emissions feed machine learning models trained to detect signatures that precede equipment failure. These systems can flag developing faults weeks or months in advance, and in more advanced deployments can autonomously adjust machine parameters such as motor speed or cooling cycles to delay or prevent failure. === Other industries === In healthcare, AIoT enables remote patient monitoring through wearable devices that collect vital signs and apply AI models to detect anomalies or predict deterioration. In logistics, GPS and telematics sensors combined with AI models support real-time route optimisation, vehicle maintenance prediction, and fuel cost forecasting. Smart building systems use occupancy, temperature, and energy sensors with AI to dynamically adjust HVAC and lighting, reducing energy consumption. == Architecture == AIoT systems typically operate across three layers: a device layer of sensors and actuators that collect data, a connectivity layer that transmits data via protocols such as MQTT or HTTP, and a compute layer where AI models process the data either in the cloud or at the edge. The trend toward edge-based processing, where inference runs on low-cost processors near the data source rather than in a centralised cloud, has accelerated as hardware costs have fallen and applications increasingly require sub-second response times. == Market == Market sizing estimates for AIoT vary significantly depending on scope and definition. Fortune Business Insights valued the AIoT market at USD 35.65 billion in 2023, projecting growth to USD 253.86 billion by 2030 at a compound annual growth rate of 32.4%. Grand View Research estimated the broader market at USD 171.4 billion in 2024 with a CAGR of 31.7% through 2030, reflecting a wider definition that includes AI-integrated hardware components. North America accounted for approximately 40% of global market share in 2024, with the Asia-Pacific region projected as the fastest-growing market.

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  Computational humor

  Computational humor is a branch of computational linguistics and artificial intelligence which uses computers in humor research. It is a relatively new area, with the first dedicated conference organized in 1996. The first "computer model of a sense of humor" was suggested by Suslov as early as 1992. Investigation of the general scheme of the information processing show a possibility of a specific malfunction, conditioned by the necessity of a quick deletion from consciousness of a false version. This specific malfunction can be identified with a humorous effect on the psychological grounds; however, an essentially new ingredient, a role of timing, is added to a well known role of ambiguity. In biological systems, a sense of humour inevitably develops in the course of evolution, because its biological function consists in quickening the transmission of processed information into consciousness and in a more effective use of brain resources. A realization of this algorithm in neural networks explains naturally the mechanism of laughter: deletion of a false version corresponds to zeroing of some part of the neural network and excessive energy of neurons is thrown out to the motor cortex, arousing muscular contractions. Unfortunately, a practical realization of this algorithm needs extensive databases, whose creation in the automatic regime was suggested only recently . As a result, this magistral direction was not developed properly and subsequent investigations (see below) accepted somewhat specialized colouring. == Joke generators == === Pun generation === An approach to analysis of humor is classification of jokes. A further step is an attempt to generate jokes basing on the rules that underlie classification. Simple prototypes for computer pun generation were reported in the early 1990s, based on a natural language generator program, VINCI. Graeme Ritchie and Kim Binsted in their 1994 research paper described a computer program, JAPE, designed to generate question-answer-type puns from a general, i.e., non-humorous, lexicon. (The program name is an acronym for "Joke Analysis and Production Engine".) Some examples produced by JAPE are: Q: What is the difference between leaves and a car? A: One you brush and rake, the other you rush and brake. Q: What do you call a strange market? A: A bizarre bazaar. Since then the approach has been improved, and the latest report, dated 2007, describes the STANDUP joke generator, implemented in the Java programming language. The STANDUP generator was tested on children within the framework of analyzing its usability for language skills development for children with communication disabilities, e.g., because of cerebral palsy. (The project name is an acronym for "System To Augment Non-speakers' Dialog Using Puns" and an allusion to standup comedy.) Children responded to this "language playground" with enthusiasm, and showed marked improvement on certain types of language tests. The two young people, who used the system over a ten-week period, regaled their peers, staff, family and neighbors with jokes such as: "What do you call a spicy missile? A hot shot!" Their joy and enthusiasm at entertaining others was inspirational. === Other === Stock and Strapparava described a program to generate funny acronyms. == Joke recognition == A statistical machine learning algorithm to detect whether a sentence contained a "That's what she said" double entendre was developed by Kiddon and Brun (2011). There is an open-source Python implementation of Kiddon & Brun's TWSS system. A program to recognize knock-knock jokes was reported by Taylor and Mazlack. This kind of research is important in analysis of human–computer interaction. An application of machine learning techniques for the distinguishing of joke texts from non-jokes was described by Mihalcea and Strapparava (2006). Takizawa et al. (1996) reported on a heuristic program for detecting puns in the Japanese language. == Applications == A possible application for assistance in language acquisition is described in the section "Pun generation". Another envisioned use of joke generators is in cases of a steady supply of jokes where quantity is more important than quality. Another obvious, yet remote, direction is automated joke appreciation. It is known that humans interact with computers in ways similar to interacting with other humans that may be described in terms of personality, politeness, flattery, and in-group favoritism. Therefore, the role of humor in human–computer interaction is being investigated. In particular, humor generation in user interface to ease communications with computers was suggested. Craig McDonough implemented the Mnemonic Sentence Generator, which converts passwords into humorous sentences. Based on the incongruity theory of humor, it is suggested that the resulting meaningless but funny sentences are easier to remember. For example, the password AjQA3Jtv is converted into "Arafat joined Quayle's Ant, while TARAR Jeopardized thurmond's vase," an example chosen by combining politicians names with verbs and common nouns. == Related research == John Allen Paulos is known for his interest in mathematical foundations of humor. His book Mathematics and Humor: A Study of the Logic of Humor demonstrates structures common to humor and formal sciences (mathematics, linguistics) and develops a mathematical model of jokes based on catastrophe theory. Conversational systems which have been designed to take part in Turing test competitions generally have the ability to learn humorous anecdotes and jokes. Because many people regard humor as something particular to humans, its appearance in conversation can be quite useful in convincing a human interrogator that a hidden entity, which could be a machine or a human, is in fact a human.

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  Cross-validation (statistics)

  Cross-validation, sometimes called rotation estimation or out-of-sample testing, is any of various similar model validation techniques for assessing how the results of a statistical analysis will generalize to an independent data set. Cross-validation includes resampling and sample splitting methods that use different portions of the data to test and train a model on different iterations. It is often used in settings where the goal is prediction, and one wants to estimate how accurately a predictive model will perform in practice. It can also be used to assess the quality of a fitted model and the stability of its parameters. In a prediction problem, a model is usually given a dataset of known data on which training is run (training dataset), and a dataset of unknown data (or first seen data) against which the model is tested (called the validation dataset or testing set). The goal of cross-validation is to test the model's ability to predict new data that was not used in estimating it, in order to flag problems like overfitting or selection bias and to give an insight on how the model will generalize to an independent dataset (i.e., an unknown dataset, for instance from a real problem). One round of cross-validation involves partitioning a sample of data into complementary subsets, performing the analysis on one subset (called the training set), and validating the analysis on the other subset (called the validation set or testing set). To reduce variability, in most methods multiple rounds of cross-validation are performed using different partitions, and the validation results are combined (e.g. averaged) over the rounds to give an estimate of the model's predictive performance. In summary, cross-validation combines (averages) measures of fitness in prediction to derive a more accurate estimate of model prediction performance. == Motivation == Assume a model with one or more unknown parameters, and a data set to which the model can be fit (the training data set). The fitting process optimizes the model parameters to make the model fit the training data as well as possible. If an independent sample of validation data is taken from the same population as the training data, it will generally turn out that the model does not fit the validation data as well as it fits the training data. The size of this difference is likely to be large especially when the size of the training data set is small, or when the number of parameters in the model is large. Cross-validation is a way to estimate the size of this effect. === Example: linear regression === In linear regression, there exist real response values y 1 , … , y n {\textstyle y_{1},\ldots ,y_{n}} , and n p-dimensional vector covariates x1, ..., xn. The components of the vector xi are denoted xi1, ..., xip. If least squares is used to fit a function in the form of a hyperplane ŷ = a + βTx to the data (xi, yi) 1 ≤ i ≤ n, then the fit can be assessed using the mean squared error (MSE). The MSE for given estimated parameter values a and β on the training set (xi, yi) 1 ≤ i ≤ n is defined as: MSE = 1 n ∑ i = 1 n ( y i − y ^ i ) 2 = 1 n ∑ i = 1 n ( y i − a − β T x i ) 2 = 1 n ∑ i = 1 n ( y i − a − β 1 x i 1 − ⋯ − β p x i p ) 2 {\displaystyle {\begin{aligned}{\text{MSE}}&={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})^{2}={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-a-{\boldsymbol {\beta }}^{T}\mathbf {x} _{i})^{2}\\&={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-a-\beta _{1}x_{i1}-\dots -\beta _{p}x_{ip})^{2}\end{aligned}}} If the model is correctly specified, it can be shown under mild assumptions that the expected value of the MSE for the training set is (n − p − 1)/(n + p + 1) < 1 times the expected value of the MSE for the validation set (the expected value is taken over the distribution of training sets). Thus, a fitted model and computed MSE on the training set will result in an optimistically biased assessment of how well the model will fit an independent data set. This biased estimate is called the in-sample estimate of the fit, whereas the cross-validation estimate is an out-of-sample estimate. Since in linear regression it is possible to directly compute the factor (n − p − 1)/(n + p + 1) by which the training MSE underestimates the validation MSE under the assumption that the model specification is valid, cross-validation can be used for checking whether the model has been overfitted, in which case the MSE in the validation set will substantially exceed its anticipated value. (Cross-validation in the context of linear regression is also useful in that it can be used to select an optimally regularized cost function.) === General case === In most other regression procedures (e.g. logistic regression), there is no simple formula to compute the expected out-of-sample fit. Cross-validation is, thus, a generally applicable way to predict the performance of a model on unavailable data using numerical computation in place of theoretical analysis. == Types == Two types of cross-validation can be distinguished: exhaustive and non-exhaustive cross-validation. === Exhaustive cross-validation === Exhaustive cross-validation methods are cross-validation methods which learn and test on all possible ways to divide the original sample into a training and a validation set. ==== Leave-p-out cross-validation ==== Leave-p-out cross-validation (LpO CV) involves using p observations as the validation set and the remaining observations as the training set. This is repeated on all ways to cut the original sample on a validation set of p observations and a training set. LpO cross-validation require training and validating the model C p n {\displaystyle C_{p}^{n}} times, where n is the number of observations in the original sample, and where C p n {\displaystyle C_{p}^{n}} is the binomial coefficient. For p > 1 and for even moderately large n, LpO CV can become computationally infeasible. For example, with n = 100 and p = 30, C 30 100 ≈ 3 × 10 25 . {\displaystyle C_{30}^{100}\approx 3\times 10^{25}.} A variant of LpO cross-validation with p=2 known as leave-pair-out cross-validation has been recommended as a nearly unbiased method for estimating the area under ROC curve of binary classifiers. ==== Leave-one-out cross-validation ==== Leave-one-out cross-validation (LOOCV) is a particular case of leave-p-out cross-validation with p = 1. The process looks similar to jackknife; however, with cross-validation one computes a statistic on the left-out sample(s), while with jackknifing one computes a statistic from the kept samples only. LOO cross-validation requires less computation time than LpO cross-validation because there are only C 1 n = n {\displaystyle C_{1}^{n}=n} passes rather than C p n {\displaystyle C_{p}^{n}} . However, n {\displaystyle n} passes may still require quite a large computation time, in which case other approaches such as k-fold cross validation may be more appropriate. Pseudo-code algorithm: Input: x, {vector of length N with x-values of incoming points} y, {vector of length N with y-values of the expected result} interpolate( x_in, y_in, x_out ), { returns the estimation for point x_out after the model is trained with x_in-y_in pairs} Output: err, {estimate for the prediction error} Steps: err ← 0 for i ← 1, ..., N do // define the cross-validation subsets x_in ← (x[1], ..., x[i − 1], x[i + 1], ..., x[N]) y_in ← (y[1], ..., y[i − 1], y[i + 1], ..., y[N]) x_out ← x[i] y_out ← interpolate(x_in, y_in, x_out) err ← err + (y[i] − y_out)^2 end for err ← err/N === Non-exhaustive cross-validation === Non-exhaustive cross validation methods do not compute all ways of splitting the original sample. These methods are approximations of leave-p-out cross-validation. ==== k-fold cross-validation ==== In k-fold cross-validation, the original sample is randomly partitioned into k equal sized subsamples, often referred to as "folds". Of the k subsamples, a single subsample is retained as the validation data for testing the model, and the remaining k − 1 subsamples are used as training data. The cross-validation process is then repeated k times, with each of the k subsamples used exactly once as the validation data. The k results can then be averaged to produce a single estimation. The advantage of this method over repeated random sub-sampling (see below) is that all observations are used for both training and validation, and each observation is used for validation exactly once. 10-fold cross-validation is commonly used, but in general k remains an unfixed parameter. For example, setting k = 2 results in 2-fold cross-validation. In 2-fold cross-validation, the dataset is randomly shuffled into two sets d0 and d1, so that both sets are equal size (this is usually implemented by shuffling the data array and then splitting it in two). We then train on d0 and validate on d1, followed by training on d1 and validating on d0. When k = n (the number of observations), k-fold cross-validation is equivalent to leave-one-out cr

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  Artificial intelligence in spirituality

  Some users of artificial intelligence (AI) technologies, especially chatbots, may develop beliefs that AI has or can attain supernatural or spiritual powers. AI models such as ChatGPT are turned to for fortune telling, mysticism and remote viewing. Recent and sudden advances in large language models have led to folk myths about their origin or capabilities, as well as their deification or worship by some users. Tucker Carlson has made similar claims, including directly to Sam Altman. Pope Leo XIV advised priests against using LLM models when it came to the creation of sermons.

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  Artificial Inventor Project

  The Artificial Inventor Project (AIP) is a global legal initiative headed by Professor Ryan Abbott dedicated to pursuing intellectual property (IP) rights for inventions and creative works generated autonomously by artificial intelligence (AI) systems without traditional human inventorship or authorship. The project coordinates a series of pro bono test cases worldwide, aiming to prompt law reform and public debate on how IP law should accommodate non-human creators. == History == In 2019, AIP filed patent applications in multiple jurisdictions, including the United States, United Kingdom, European Patent Office, Australia, Switzerland, and South Africa, naming the AI system DABUS (Device for the Autonomous Bootstrapping of Unified Sentience), created by Stephen Thaler, as the inventor. The aim was to challenge legal norms that require inventors to be natural persons and highlight pressing policy questions about AI-generated innovation and IP regimes. == Legal proceedings by jurisdiction == === Australia === In July 2021, a Federal Court of Australia judge (Beach J) ruled that AI can be considered an inventor under the Patents Act 1990, ordering IP Australia to reinstate the relevant patent. However, the full court then overturned this ruling on appeal and denied further review. === European Patent Office === The EPO Board of Appeal determined in 2022 that only a human inventor may be named, rendering DABUS‑based applications unacceptable. === South Africa === In 2021, a patent was granted listing DABUS as the inventor. As South Africa’s procedural system does not involve substantive inventorship review, the grant proceeded on formal grounds alone. === Switzerland === On 26 June 2025, the Swiss Federal Administrative Court ruled that artificial intelligence systems such as DABUS cannot be listed as inventors on patent applications. The court upheld the existing practice of the Swiss Federal Institute of Intellectual Property (IPI), affirming that only natural persons may be recognized as inventors under Swiss patent law. === United Kingdom === In December 2023, the UK Supreme Court unanimously held that AI systems cannot be legally recognized as inventors, affirming that "an inventor must be a person" under current British law. === United States === In Thaler v. Hirshfeld (2021), a U.S. federal court agreed with the USPTO that inventors must be natural persons, rejecting the DABUS application and setting a precedent consistent with existing statute and administrative policy. == Criticism and impact == The project has fueled substantial discourse. Critics caution that allowing AI inventorship may complicate notions of accountability and ownership. Proponents argue that legal recognition must evolve to avoid disincentivizing innovation produced by AI and to maintain honesty about the true source of invention.

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  Automated decision-making

  Automated decision-making (ADM) is the use of data, machines and algorithms to make decisions in a range of contexts, including public administration, business, health, education, law, employment, transport, media and entertainment, with varying degrees of human oversight or intervention. ADM may involve large-scale data from a range of sources, such as databases, text, social media, sensors, images or speech, that is processed using various technologies including computer software, algorithms, machine learning, natural language processing, artificial intelligence, augmented intelligence and robotics. The increasing use of automated decision-making systems (ADMS) across a range of contexts presents many benefits and challenges to human society requiring consideration of the technical, legal, ethical, societal, educational, economic and health consequences. == Overview == There are different definitions of ADM based on the level of automation involved. Some definitions suggests ADM involves decisions made through purely technological means without human input, such as the EU's General Data Protection Regulation (Article 22). However, ADM technologies and applications can take many forms ranging from decision-support systems that make recommendations for human decision-makers to act on, sometimes known as augmented intelligence or 'shared decision-making', to fully automated decision-making processes that make decisions on behalf of individuals or organizations without human involvement. Models used in automated decision-making systems can be as simple as checklists and decision trees through to artificial intelligence and deep neural networks (DNN). Since the 1950s computers have gone from being able to do basic processing to having the capacity to undertake complex, ambiguous and highly skilled tasks such as image and speech recognition, gameplay, scientific and medical analysis and inferencing across multiple data sources. ADM is now being increasingly deployed across all sectors of society and many diverse domains from entertainment to transport. An ADM system (ADMS) may involve multiple decision points, data sets, and technologies (ADMT) and may sit within a larger administrative or technical system such as a criminal justice system or business process. == Data == Automated decision-making involves using data as input to be analyzed within a process, model, or algorithm or for learning and generating new models. ADM systems may use and connect a wide range of data types and sources depending on the goals and contexts of the system, for example, sensor data for self-driving cars and robotics, identity data for security systems, demographic and financial data for public administration, medical records in health, criminal records in law. This can sometimes involve vast amounts of data and computing power. === Data quality === The quality of the available data and its ability to be used in ADM systems is fundamental to the outcomes. It is often highly problematic for many reasons. Datasets are often highly variable; corporations or governments may control large-scale data, restricted for privacy or security reasons, incomplete, biased, limited in terms of time or coverage, measuring and describing terms in different ways, and many other issues. For machines to learn from data, large corpora are often required, which can be challenging to obtain or compute; however, where available, they have provided significant breakthroughs, for example, in diagnosing chest X-rays. == ADM technologies == Automated decision-making technologies (ADMT) are software-coded digital tools that automate the translation of input data to output data, contributing to the function of automated decision-making systems. There are a wide range of technologies in use across ADM applications and systems. ADMTs involving basic computational operations Search (includes 1-2-1, 1-2-many, data matching/merge) Matching (two different things) Mathematical Calculation (formula) ADMTs for assessment and grouping: User profiling Recommender systems Clustering Classification Feature learning Predictive analytics (includes forecasting) ADMTs relating to space and flows: Social network analysis (includes link prediction) Mapping Routing ADMTs for processing of complex data formats Image processing Audio processing Natural Language Processing (NLP) Other ADMT Business rules management systems Time series analysis Anomaly detection Modelling/Simulation === Machine learning === Machine learning (ML) involves training computer programs through exposure to large data sets and examples to learn from experience and solve problems. Machine learning can be used to generate and analyse data as well as make algorithmic calculations and has been applied to image and speech recognition, translations, text, data and simulations. While machine learning has been around for some time, it is becoming increasingly powerful due to recent breakthroughs in training deep neural networks (DNNs), and dramatic increases in data storage capacity and computational power with GPU coprocessors and cloud computing. Machine learning systems based on foundation models run on deep neural networks and use pattern matching to train a single huge system on large amounts of general data such as text and images. Early models tended to start from scratch for each new problem however since the early 2020s many are able to be adapted to new problems. Examples of these technologies include Open AI's DALL-E (an image creation program) and their various GPT language models, and Google's PaLM language model program. == Applications == ADM is being used to replace or augment human decision-making by both public and private-sector organisations for a range of reasons including to help increase consistency, improve efficiency, reduce costs and enable new solutions to complex problems. === Debate === Research and development are underway into uses of technology to assess argument quality, assess argumentative essays and judge debates. Potential applications of these argument technologies span education and society. Scenarios to consider, in these regards, include those involving the assessment and evaluation of conversational, mathematical, scientific, interpretive, legal, and political argumentation and debate. === Law === In legal systems around the world, algorithmic tools such as risk assessment instruments (RAI), are being used to supplement or replace the human judgment of judges, civil servants and police officers in many contexts. In the United States RAI are being used to generate scores to predict the risk of recidivism in pre-trial detention and sentencing decisions, evaluate parole for prisoners and to predict "hot spots" for future crime. These scores may result in automatic effects or may be used to inform decisions made by officials within the justice system. In Canada ADM has been used since 2014 to automate certain activities conducted by immigration officials and to support the evaluation of some immigrant and visitor applications. === Economics === Automated decision-making systems are used in certain computer programs to create buy and sell orders related to specific financial transactions and automatically submit the orders in the international markets. Computer programs can automatically generate orders based on predefined set of rules using trading strategies which are based on technical analyses, advanced statistical and mathematical computations, or inputs from other electronic sources. === Business === ==== Continuous auditing ==== Continuous auditing uses advanced analytical tools to automate auditing processes. It can be utilized in the private sector by business enterprises and in the public sector by governmental organizations and municipalities. As artificial intelligence and machine learning continue to advance, accountants and auditors may make use of increasingly sophisticated algorithms which make decisions such as those involving determining what is anomalous, whether to notify personnel, and how to prioritize those tasks assigned to personnel. === Media and entertainment === Digital media, entertainment platforms, and information services increasingly provide content to audiences via automated recommender systems based on demographic information, previous selections, collaborative filtering or content-based filtering. This includes music and video platforms, publishing, health information, product databases and search engines. Many recommender systems also provide some agency to users in accepting recommendations and incorporate data-driven algorithmic feedback loops based on the actions of the system user. Large-scale machine learning language models and image creation programs being developed by companies such as OpenAI and Google in the 2020s have restricted access however they are likely to have widespread application in fields such as advertising, copywriting, stock imagery and gra

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  JAX (software)

  JAX is a Python library for accelerator-oriented array computation and program transformation, designed for high-performance numerical computing and large-scale machine learning. It is developed by Google with contributions from Nvidia and other community contributors. It is described as bringing together a modified version of the automatic differentiation system autograd and OpenXLA's XLA (Accelerated Linear Algebra). It is designed to follow the structure and workflow of NumPy as closely as possible and works with various existing frameworks such as TensorFlow and PyTorch. The primary features of JAX are: Providing a unified NumPy-like interface to computations that run on CPU, GPU, or TPU, in local or distributed settings. Built-in Just-In-Time (JIT) compilation via OpenXLA, an open-source machine learning compiler ecosystem. Efficient evaluation of gradients via its automatic differentiation transformations. Automatic vectorization to efficiently map functions over arrays representing batches of inputs. == Libraries using Jax == Flax Equinox Optax

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  Reparameterization trick

  The reparameterization trick (aka "reparameterization gradient estimator") is a technique used in statistical machine learning, particularly in variational inference, variational autoencoders, and stochastic optimization. It allows for the efficient computation of gradients through random variables, enabling the optimization of parametric probability models using stochastic gradient descent, and the variance reduction of estimators. It was developed in the 1980s in operations research, under the name of "pathwise gradients", or "stochastic gradients". Its use in variational inference was proposed in 2013. == Mathematics == Let z {\displaystyle z} be a random variable with distribution q ϕ ( z ) {\displaystyle q_{\phi }(z)} , where ϕ {\displaystyle \phi } is a vector containing the parameters of the distribution. === REINFORCE estimator === Consider an objective function of the form: L ( ϕ ) = E z ∼ q ϕ ( z ) [ f ( z ) ] {\displaystyle L(\phi )=\mathbb {E} _{z\sim q_{\phi }(z)}[f(z)]} Without the reparameterization trick, estimating the gradient ∇ ϕ L ( ϕ ) {\displaystyle \nabla _{\phi }L(\phi )} can be challenging, because the parameter appears in the random variable itself. In more detail, we have to statistically estimate: ∇ ϕ L ( ϕ ) = ∇ ϕ ∫ d z q ϕ ( z ) f ( z ) {\displaystyle \nabla _{\phi }L(\phi )=\nabla _{\phi }\int dz\;q_{\phi }(z)f(z)} The REINFORCE estimator, widely used in reinforcement learning and especially policy gradient, uses the following equality: ∇ ϕ L ( ϕ ) = ∫ d z q ϕ ( z ) ∇ ϕ ( ln ⁡ q ϕ ( z ) ) f ( z ) = E z ∼ q ϕ ( z ) [ ∇ ϕ ( ln ⁡ q ϕ ( z ) ) f ( z ) ] {\displaystyle \nabla _{\phi }L(\phi )=\int dz\;q_{\phi }(z)\nabla _{\phi }(\ln q_{\phi }(z))f(z)=\mathbb {E} _{z\sim q_{\phi }(z)}[\nabla _{\phi }(\ln q_{\phi }(z))f(z)]} This allows the gradient to be estimated: ∇ ϕ L ( ϕ ) ≈ 1 N ∑ i = 1 N ∇ ϕ ( ln ⁡ q ϕ ( z i ) ) f ( z i ) {\displaystyle \nabla _{\phi }L(\phi )\approx {\frac {1}{N}}\sum _{i=1}^{N}\nabla _{\phi }(\ln q_{\phi }(z_{i}))f(z_{i})} The REINFORCE estimator has high variance, and many methods were developed to reduce its variance. === Reparameterization estimator === The reparameterization trick expresses z {\displaystyle z} as: z = g ϕ ( ϵ ) , ϵ ∼ p ( ϵ ) {\displaystyle z=g_{\phi }(\epsilon ),\quad \epsilon \sim p(\epsilon )} Here, g ϕ {\displaystyle g_{\phi }} is a deterministic function parameterized by ϕ {\displaystyle \phi } , and ϵ {\displaystyle \epsilon } is a noise variable drawn from a fixed distribution p ( ϵ ) {\displaystyle p(\epsilon )} . This gives: L ( ϕ ) = E ϵ ∼ p ( ϵ ) [ f ( g ϕ ( ϵ ) ) ] {\displaystyle L(\phi )=\mathbb {E} _{\epsilon \sim p(\epsilon )}[f(g_{\phi }(\epsilon ))]} Now, the gradient can be estimated as: ∇ ϕ L ( ϕ ) = E ϵ ∼ p ( ϵ ) [ ∇ ϕ f ( g ϕ ( ϵ ) ) ] ≈ 1 N ∑ i = 1 N ∇ ϕ f ( g ϕ ( ϵ i ) ) {\displaystyle \nabla _{\phi }L(\phi )=\mathbb {E} _{\epsilon \sim p(\epsilon )}[\nabla _{\phi }f(g_{\phi }(\epsilon ))]\approx {\frac {1}{N}}\sum _{i=1}^{N}\nabla _{\phi }f(g_{\phi }(\epsilon _{i}))} == Examples == For some common distributions, the reparameterization trick takes specific forms: Normal distribution: For z ∼ N ( μ , σ 2 ) {\displaystyle z\sim {\mathcal {N}}(\mu ,\sigma ^{2})} , we can use: z = μ + σ ϵ , ϵ ∼ N ( 0 , 1 ) {\displaystyle z=\mu +\sigma \epsilon ,\quad \epsilon \sim {\mathcal {N}}(0,1)} Exponential distribution: For z ∼ Exp ( λ ) {\displaystyle z\sim {\text{Exp}}(\lambda )} , we can use: z = − 1 λ log ⁡ ( ϵ ) , ϵ ∼ Uniform ( 0 , 1 ) {\displaystyle z=-{\frac {1}{\lambda }}\log(\epsilon ),\quad \epsilon \sim {\text{Uniform}}(0,1)} Discrete distribution can be reparameterized by the Gumbel distribution (Gumbel-softmax trick or "concrete distribution") and diffusion models. In general, any distribution that is differentiable with respect to its parameters can be reparameterized by inverting the multivariable CDF function, then apply the implicit method. See for an exposition and application to the Gamma, Beta, Dirichlet, and von Mises distributions. == Applications == === Variational autoencoder === In Variational Autoencoders (VAEs), the VAE objective function, known as the Evidence Lower Bound (ELBO), is given by: ELBO ( ϕ , θ ) = E z ∼ q ϕ ( z | x ) [ log ⁡ p θ ( x | z ) ] − D KL ( q ϕ ( z | x ) | | p ( z ) ) {\displaystyle {\text{ELBO}}(\phi ,\theta )=\mathbb {E} _{z\sim q_{\phi }(z|x)}[\log p_{\theta }(x|z)]-D_{\text{KL}}(q_{\phi }(z|x)||p(z))} where q ϕ ( z | x ) {\displaystyle q_{\phi }(z|x)} is the encoder (recognition model), p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} is the decoder (generative model), and p ( z ) {\displaystyle p(z)} is the prior distribution over latent variables. The gradient of ELBO with respect to θ {\displaystyle \theta } is simply E z ∼ q ϕ ( z | x ) [ ∇ θ log ⁡ p θ ( x | z ) ] ≈ 1 L ∑ l = 1 L ∇ θ log ⁡ p θ ( x | z l ) {\displaystyle \mathbb {E} _{z\sim q_{\phi }(z|x)}[\nabla _{\theta }\log p_{\theta }(x|z)]\approx {\frac {1}{L}}\sum _{l=1}^{L}\nabla _{\theta }\log p_{\theta }(x|z_{l})} but the gradient with respect to ϕ {\displaystyle \phi } requires the trick. Express the sampling operation z ∼ q ϕ ( z | x ) {\displaystyle z\sim q_{\phi }(z|x)} as: z = μ ϕ ( x ) + σ ϕ ( x ) ⊙ ϵ , ϵ ∼ N ( 0 , I ) {\displaystyle z=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon ,\quad \epsilon \sim {\mathcal {N}}(0,I)} where μ ϕ ( x ) {\displaystyle \mu _{\phi }(x)} and σ ϕ ( x ) {\displaystyle \sigma _{\phi }(x)} are the outputs of the encoder network, and ⊙ {\displaystyle \odot } denotes element-wise multiplication. Then we have ∇ ϕ ELBO ( ϕ , θ ) = E ϵ ∼ N ( 0 , I ) [ ∇ ϕ log ⁡ p θ ( x | z ) + ∇ ϕ log ⁡ q ϕ ( z | x ) − ∇ ϕ log ⁡ p ( z ) ] {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi ,\theta )=\mathbb {E} _{\epsilon \sim {\mathcal {N}}(0,I)}[\nabla _{\phi }\log p_{\theta }(x|z)+\nabla _{\phi }\log q_{\phi }(z|x)-\nabla _{\phi }\log p(z)]} where z = μ ϕ ( x ) + σ ϕ ( x ) ⊙ ϵ {\displaystyle z=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon } . This allows us to estimate the gradient using Monte Carlo sampling: ∇ ϕ ELBO ( ϕ , θ ) ≈ 1 L ∑ l = 1 L [ ∇ ϕ log ⁡ p θ ( x | z l ) + ∇ ϕ log ⁡ q ϕ ( z l | x ) − ∇ ϕ log ⁡ p ( z l ) ] {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi ,\theta )\approx {\frac {1}{L}}\sum _{l=1}^{L}[\nabla _{\phi }\log p_{\theta }(x|z_{l})+\nabla _{\phi }\log q_{\phi }(z_{l}|x)-\nabla _{\phi }\log p(z_{l})]} where z l = μ ϕ ( x ) + σ ϕ ( x ) ⊙ ϵ l {\displaystyle z_{l}=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon _{l}} and ϵ l ∼ N ( 0 , I ) {\displaystyle \epsilon _{l}\sim {\mathcal {N}}(0,I)} for l = 1 , … , L {\displaystyle l=1,\ldots ,L} . This formulation enables backpropagation through the sampling process, allowing for end-to-end training of the VAE model using stochastic gradient descent or its variants. === Variational inference === More generally, the trick allows using stochastic gradient descent for variational inference. Let the variational objective (ELBO) be of the form: ELBO ( ϕ ) = E z ∼ q ϕ ( z ) [ log ⁡ p ( x , z ) − log ⁡ q ϕ ( z ) ] {\displaystyle {\text{ELBO}}(\phi )=\mathbb {E} _{z\sim q_{\phi }(z)}[\log p(x,z)-\log q_{\phi }(z)]} Using the reparameterization trick, we can estimate the gradient of this objective with respect to ϕ {\displaystyle \phi } : ∇ ϕ ELBO ( ϕ ) ≈ 1 L ∑ l = 1 L ∇ ϕ [ log ⁡ p ( x , g ϕ ( ϵ l ) ) − log ⁡ q ϕ ( g ϕ ( ϵ l ) ) ] , ϵ l ∼ p ( ϵ ) {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi )\approx {\frac {1}{L}}\sum _{l=1}^{L}\nabla _{\phi }[\log p(x,g_{\phi }(\epsilon _{l}))-\log q_{\phi }(g_{\phi }(\epsilon _{l}))],\quad \epsilon _{l}\sim p(\epsilon )} === Dropout === The reparameterization trick has been applied to reduce the variance in dropout, a regularization technique in neural networks. The original dropout can be reparameterized with Bernoulli distributions: y = ( W ⊙ ϵ ) x , ϵ i j ∼ Bernoulli ( α i j ) {\displaystyle y=(W\odot \epsilon )x,\quad \epsilon _{ij}\sim {\text{Bernoulli}}(\alpha _{ij})} where W {\displaystyle W} is the weight matrix, x {\displaystyle x} is the input, and α i j {\displaystyle \alpha _{ij}} are the (fixed) dropout rates. More generally, other distributions can be used than the Bernoulli distribution, such as the gaussian noise: y i = μ i + σ i ⊙ ϵ i , ϵ i ∼ N ( 0 , I ) {\displaystyle y_{i}=\mu _{i}+\sigma _{i}\odot \epsilon _{i},\quad \epsilon _{i}\sim {\mathcal {N}}(0,I)} where μ i = m i ⊤ x {\displaystyle \mu _{i}=\mathbf {m} _{i}^{\top }x} and σ i 2 = v i ⊤ x 2 {\displaystyle \sigma _{i}^{2}=\mathbf {v} _{i}^{\top }x^{2}} , with m i {\displaystyle \mathbf {m} _{i}} and v i {\displaystyle \mathbf {v} _{i}} being the mean and variance of the i {\displaystyle i} -th output neuron. The reparameterization trick can be applied to all such cases, resulting in the variational dropout method.

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  VLLM

  vLLM is an open-source software framework for inference and serving of large language models and related multimodal models. Originally developed at the University of California, Berkeley's Sky Computing Lab, the project is centered on PagedAttention, a memory-management method for transformer key–value caches, and supports features such as continuous batching, distributed inference, quantization, and OpenAI-compatible APIs. According to a project maintainer, the "v" in vLLM originally referred to "virtual", inspired by virtual memory. == History == vLLM was introduced in 2023 by researchers affiliated with the Sky Computing Lab at UC Berkeley. Its core ideas were described in the 2023 paper Efficient Memory Management for Large Language Model Serving with PagedAttention, which presented the system as a high-throughput and memory-efficient serving engine for large language models. In 2025, the PyTorch Foundation announced that vLLM had become a Foundation-hosted project. PyTorch's project page states that the University of California, Berkeley contributed vLLM to the Linux Foundation in July 2024. In January 2026, TechCrunch reported that the creators of vLLM had launched the startup Inferact to commercialize the project, raising $150 million in seed funding. == Architecture == According to its 2023 paper, vLLM was designed to improve the efficiency of large language model serving by reducing memory waste in the key–value cache used during transformer inference. The paper introduced PagedAttention, an algorithm inspired by virtual memory and paging techniques in operating systems, and described vLLM as using block-level memory management and request scheduling to increase throughput while maintaining similar latency. The project documentation and repository describe support for continuous batching, chunked prefill, speculative decoding, prefix caching, quantization, and multiple forms of distributed inference and serving. PyTorch has described vLLM as a high-throughput, memory-efficient inference and serving engine that supports a range of hardware back ends, including NVIDIA and AMD GPUs, Google TPUs, AWS Trainium, and Intel processors.

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  A Logical Calculus of the Ideas Immanent in Nervous Activity

  "A Logical Calculus of the Ideas Immanent in Nervous Activity" is a 1943 paper written by Warren Sturgis McCulloch and Walter Pitts, published in the journal The Bulletin of Mathematical Biophysics. The paper proposed a mathematical model of the nervous system as a network of simple logical elements, later known as artificial neurons, or McCulloch–Pitts neurons. These neurons receive inputs, perform a weighted sum, and fire an output signal based on a threshold function. By connecting these units in various configurations, McCulloch and Pitts demonstrated that their model could perform all logical functions. It is a seminal work in cognitive science, computational neuroscience, computer science, and artificial intelligence. It was a foundational result in automata theory. John von Neumann cited it as a significant result. == Mathematics == The artificial neuron used in the original paper is slightly different from the modern version. They considered neural networks that operate in discrete steps of time t = 0 , 1 , … {\displaystyle t=0,1,\dots } . The neural network contains a number of neurons. Let the state of a neuron i {\displaystyle i} at time t {\displaystyle t} be N i ( t ) {\displaystyle N_{i}(t)} . The state of a neuron can either be 0 or 1, standing for "not firing" and "firing". Each neuron also has a firing threshold θ {\displaystyle \theta } , such that it fires if the total input exceeds the threshold. Each neuron can connect to any other neuron (including itself) with positive synapses (excitatory) or negative synapses (inhibitory). That is, each neuron can connect to another neuron with a weight w {\displaystyle w} taking an integer value. A peripheral afferent is a neuron with no incoming synapses. We can regard each neural network as a directed graph, with the nodes being the neurons, and the directed edges being the synapses. A neural network has a circle or a circuit if there exists a directed circle in the graph. Let w i j ( t ) {\displaystyle w_{ij}(t)} be the connection weight from neuron j {\displaystyle j} to neuron i {\displaystyle i} at time t {\displaystyle t} , then its next state is N i ( t + 1 ) = H ( ∑ j = 1 n w i j ( t ) N j ( t ) − θ i ( t ) ) , {\displaystyle N_{i}(t+1)=H\left(\sum _{j=1}^{n}w_{ij}(t)N_{j}(t)-\theta _{i}(t)\right),} where H {\displaystyle H} is the Heaviside step function (outputting 1 if the input is greater than or equal to 0, and 0 otherwise). === Symbolic logic === The paper used, as a logical language for describing neural networks, "Language II" from The Logical Syntax of Language by Rudolf Carnap with some notations taken from Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Language II covers substantial parts of classical mathematics, including real analysis and portions of set theory. To describe a neural network with peripheral afferents N 1 , N 2 , … , N p {\displaystyle N_{1},N_{2},\dots ,N_{p}} and non-peripheral afferents N p + 1 , N p + 2 , … , N n {\displaystyle N_{p+1},N_{p+2},\dots ,N_{n}} they considered logical predicate of form P r ( N 1 , N 2 , … , N p , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{p},t)} where P r {\displaystyle Pr} is a first-order logic predicate function (a function that outputs a boolean), N 1 , … , N p {\displaystyle N_{1},\dots ,N_{p}} are predicates that take t {\displaystyle t} as an argument, and t {\displaystyle t} is the only free variable in the predicate. Intuitively speaking, N 1 , … , N p {\displaystyle N_{1},\dots ,N_{p}} specifies the binary input patterns going into the neural network over all time, and P r ( N 1 , N 2 , … , N n , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},t)} is a function that takes some binary input patterns, and constructs an output binary pattern P r ( N 1 , N 2 , … , N n , 0 ) , P r ( N 1 , N 2 , … , N n , 1 ) , … {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},0),Pr(N_{1},N_{2},\dots ,N_{n},1),\dots } . A logical sentence P r ( N 1 , N 2 , … , N n , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},t)} is realized by a neural network iff there exists a time-delay T ≥ 0 {\displaystyle T\geq 0} , a neuron i {\displaystyle i} in the network, and an initial state for the non-peripheral neurons N p + 1 ( 0 ) , … , N n ( 0 ) {\displaystyle N_{p+1}(0),\dots ,N_{n}(0)} , such that for any time t {\displaystyle t} , the truth-value of the logical sentence is equal to the state of the neuron i {\displaystyle i} at time t + T {\displaystyle t+T} . That is, ∀ t = 0 , 1 , 2 , … , P r ( N 1 , N 2 , … , N p , t ) = N i ( t + T ) {\displaystyle \forall t=0,1,2,\dots ,\quad Pr(N_{1},N_{2},\dots ,N_{p},t)=N_{i}(t+T)} === Equivalence === In the paper, they considered some alternative definitions of artificial neural networks, and have shown them to be equivalent, that is, neural networks under one definition realizes precisely the same logical sentences as neural networks under another definition. They considered three forms of inhibition: relative inhibition, absolute inhibition, and extinction. The definition above is relative inhibition. By "absolute inhibition" they meant that if any negative synapse fires, then the neuron will not fire. By "extinction" they meant that if at time t {\displaystyle t} , any inhibitory synapse fires on a neuron i {\displaystyle i} , then θ i ( t + j ) = θ i ( 0 ) + b j {\displaystyle \theta _{i}(t+j)=\theta _{i}(0)+b_{j}} for j = 1 , 2 , 3 , … {\displaystyle j=1,2,3,\dots } , until the next time an inhibitory synapse fires on i {\displaystyle i} . It is required that b j = 0 {\displaystyle b_{j}=0} for all large j {\displaystyle j} . Theorem 4 and 5 state that these are equivalent. They considered three forms of excitation: spatial summation, temporal summation, and facilitation. The definition above is spatial summation (which they pictured as having multiple synapses placed close together, so that the effect of their firing sums up). By "temporal summation" they meant that the total incoming signal is ∑ τ = 0 T ∑ j = 1 n w i j ( t ) N j ( t − τ ) {\displaystyle \sum _{\tau =0}^{T}\sum _{j=1}^{n}w_{ij}(t)N_{j}(t-\tau )} for some T ≥ 1 {\displaystyle T\geq 1} . By "facilitation" they meant the same as extinction, except that b j ≤ 0 {\displaystyle b_{j}\leq 0} . Theorem 6 states that these are equivalent. They considered neural networks that do not change, and those that change by Hebbian learning. That is, they assume that at t = 0 {\displaystyle t=0} , some excitatory synaptic connections are not active. If at any t {\displaystyle t} , both N i ( t ) = 1 , N j ( t ) = 1 {\displaystyle N_{i}(t)=1,N_{j}(t)=1} , then any latent excitatory synapse between i , j {\displaystyle i,j} becomes active. Theorem 7 states that these are equivalent. === Logical expressivity === They considered "temporal propositional expressions" (TPE), which are propositional formulas with one free variable t {\displaystyle t} . For example, N 1 ( t ) ∨ N 2 ( t ) ∧ ¬ N 3 ( t ) {\displaystyle N_{1}(t)\vee N_{2}(t)\wedge \neg N_{3}(t)} is such an expression. Theorem 1 and 2 together showed that neural nets without circles are equivalent to TPE. For neural nets with loops, they noted that "realizable P r {\displaystyle Pr} may involve reference to past events of an indefinite degree of remoteness". These then encodes for sentences like "There was some x such that x was a ψ" or ( ∃ x ) ( ψ x ) {\displaystyle (\exists x)(\psi x)} . Theorems 8 to 10 showed that neural nets with loops can encode all first-order logic with equality and conversely, any looped neural networks is equivalent to a sentence in first-order logic with equality, thus showing that they are equivalent in logical expressiveness. As a remark, they noted that a neural network, if furnished with a tape, scanners, and write-heads, is equivalent to a Turing machine, and conversely, every Turing machine is equivalent to some such neural network. Thus, these neural networks are equivalent to Turing computability and Church's lambda-definability. == Context == === Previous work === The paper built upon several previous strands of work. In the symbolic logic side, it built on the previous work by Carnap, Whitehead, and Russell. This was contributed by Walter Pitts, who had a strong proficiency with symbolic logic. Pitts provided mathematical and logical rigor to McCulloch’s vague ideas on psychons (atoms of psychological events) and circular causality. In the neuroscience side, it built on previous work by the mathematical biology research group centered around Nicolas Rashevsky, of which McCulloch was a member. The paper was published in the Bulletin of Mathematical Biophysics, which was founded by Rashevsky in 1939. During the late 1930s, Rashevsky's research group was producing papers that had difficulty publishing in other journals at the time, so Rashevsky decided to found a new journal exclusively devoted to mathematical biophysics. Also in the Rashevsky's group was Alston Scott Householder, who in 1941 published an abstract model

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  Artificial intelligence in spirituality

  Some users of artificial intelligence (AI) technologies, especially chatbots, may develop beliefs that AI has or can attain supernatural or spiritual powers. AI models such as ChatGPT are turned to for fortune telling, mysticism and remote viewing. Recent and sudden advances in large language models have led to folk myths about their origin or capabilities, as well as their deification or worship by some users. Tucker Carlson has made similar claims, including directly to Sam Altman. Pope Leo XIV advised priests against using LLM models when it came to the creation of sermons.

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