Apache Mahout is a project of the Apache Software Foundation to produce free implementations of distributed or otherwise scalable machine learning algorithms focused primarily on linear algebra. In the past, many of the implementations use the Apache Hadoop platform, however today it is primarily focused on Apache Spark. Mahout also provides Java/Scala libraries for common math operations (focused on linear algebra and statistics) and primitive Java collections. Mahout is a work in progress; a number of algorithms have been implemented. == Features == === Samsara === Apache Mahout-Samsara refers to a Scala domain-specific language (DSL) that allows users to use R-like syntax as opposed to traditional Scala-like syntax. This allows user to express algorithms concisely and clearly. === Backend agnostic === Apache Mahout's code abstracts the domain-specific language from the engine where the code is run. While active development is done with the Apache Spark engine, users are free to implement any engine they choose- H2O and Apache Flink have been implemented in the past and examples exist in the code base. === GPU/CPU accelerators === The JVM has notoriously slow computation. To improve speed, "native solvers" were added which move in-core, and by extension, distributed BLAS operations out of the JVM, offloading to off-heap or GPU memory for processing via multiple CPUs and/or CPU cores, or GPUs when built against the ViennaCL library. ViennaCL is a highly optimized C++ library with BLAS operations implemented in OpenMP, and OpenCL. As of release 14.1, the OpenMP build considered to be stable, leaving the OpenCL build is still in its experimental proof-of-concept phase. === Recommenders === Apache Mahout features implementations of Alternating Least Squares, Co-Occurrence, and Correlated Co-Occurrence, a unique-to-Mahout recommender algorithm that extends co-occurrence to be used on multiple dimensions of data. == History == === Transition from Map Reduce to Apache Spark === While Mahout's core algorithms for clustering, classification and batch based collaborative filtering were implemented on top of Apache Hadoop using the map/reduce paradigm, it did not restrict contributions to Hadoop-based implementations. Contributions that run on a single node or on a non-Hadoop cluster were also welcomed. For example, the 'Taste' collaborative-filtering recommender component of Mahout was originally a separate project and can run stand-alone without Hadoop. Starting with the release 0.10.0, the project shifted its focus to building a backend-independent programming environment, code named "Samsara". The environment consists of an algebraic backend-independent optimizer and an algebraic Scala DSL unifying in-memory and distributed algebraic operators. Supported algebraic platforms are Apache Spark, H2O, and Apache Flink. Support for MapReduce algorithms started being gradually phased out in 2014. === Release history === === Developers === Apache Mahout is developed by a community. The project is managed by a group called the "Project Management Committee" (PMC). The current PMC is Andrew Musselman, Andrew Palumbo, Drew Farris, Isabel Drost-Fromm, Jake Mannix, Pat Ferrel, Paritosh Ranjan, Trevor Grant, Robin Anil, Sebastian Schelter, Stevo Slavić.
Psychology of reasoning
The psychology of reasoning (also known as the cognitive science of reasoning) is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory. Psychological experiments on how humans and other animals reason have been carried out for over 100 years. An enduring question is whether or not people have the capacity to be rational. Current research in this area addresses various questions about reasoning, rationality, judgments, intelligence, relationships between emotion and reasoning, and development. == Everyday reasoning == One of the most obvious areas in which people employ reasoning is with sentences in everyday language. Most experimentation on deduction has been carried out on hypothetical thought, in particular, examining how people reason about conditionals, e.g., If A then B. Participants in experiments make the modus ponens inference, given the indicative conditional If A then B, and given the premise A, they conclude B. However, given the indicative conditional and the minor premise for the modus tollens inference, not-B, about half of the participants in experiments conclude not-A and the remainder concludes that nothing follows. The ease with which people make conditional inferences is affected by context, as demonstrated in the well-known selection task developed by Peter Wason. Participants are better able to test a conditional in an ecologically relevant context, e.g., if the envelope is sealed then it must have a 50 cent stamp on it compared to one that contains symbolic content, e.g., if the letter is a vowel then the number is even. Background knowledge can also lead to the suppression of even the simple modus ponens inference Participants given the conditional if Lisa has an essay to write then she studies late in the library and the premise Lisa has an essay to write make the modus ponens inference 'she studies late in the library', but the inference is suppressed when they are also given a second conditional if the library stays open then she studies late in the library. Interpretations of the suppression effect are controversial Other investigations of propositional inference examine how people think about disjunctive alternatives, e.g., A or else B, and how they reason about negation, e.g., It is not the case that A and B. Many experiments have been carried out to examine how people make relational inferences, including comparisons, e.g., A is better than B. Such investigations also concern spatial inferences, e.g. A is in front of B and temporal inferences, e.g. A occurs before B. Other common tasks include categorical syllogisms, used to examine how people reason about quantifiers such as All or Some, e.g., Some of the A are not B. For example if all A are B and some B are C, what (if anything) follows? == Theories of reasoning == There are several alternative theories of the cognitive processes that human reasoning is based on. One view is that people rely on a mental logic consisting of formal (abstract or syntactic) inference rules similar to those developed by logicians in the propositional calculus. Another view is that people rely on domain-specific or content-sensitive rules of inference. A third view is that people rely on mental models, that is, mental representations that correspond to imagined possibilities. A fourth view is that people compute probabilities. One controversial theoretical issue is the identification of an appropriate competence model, or a standard against which to compare human reasoning. Initially classical logic was chosen as a competence model. Subsequently, some researchers opted for non-monotonic logic and Bayesian probability. Research on mental models and reasoning has led to the suggestion that people are rational in principle but err in practice. Connectionist approaches towards reasoning have also been proposed. Despite the ongoing debate about the cognitive processes involved in human reasoning, recent research has shown that multiple approaches can be useful in modeling human thinking. For instance, studies have found that people's reasoning is often influenced by their prior beliefs, which can be modeled using Bayesian probability theory. Additionally, research on mental models has shown that people tend to reason about problems by constructing multiple mental representations of the situation, which can help them to identify relevant features and make inferences based on their understanding of the problem. Moreover, connectionist approaches to reasoning have also gained attention, which focus on the neural network models that can learn from data and generalize to new situations. == Development of reasoning == It is an active question in psychology how, why, and when the ability to reason develops from infancy to adulthood. Jean Piaget's theory of cognitive development posited general mechanisms and stages in the development of reasoning from infancy to adulthood. According to the neo-Piagetian theories of cognitive development, changes in reasoning with development come from increasing working memory capacity, increasing speed of processing, and enhanced executive functions and control. Increasing self-awareness is also an important factor. In their book The Enigma of Reason, the cognitive scientists Hugo Mercier and Dan Sperber put forward an "argumentative" theory of reasoning, claiming that humans evolved to reason primarily to justify our beliefs and actions and to convince others in a social environment. Key evidence for their theory includes the errors in reasoning that solitary individuals are prone to when their arguments are not criticized, such as logical fallacies, and how groups become much better at performing cognitive reasoning tasks when they communicate with one another and can evaluate each other's arguments. Sperber and Mercier offer one attempt to resolve the apparent paradox that the confirmation bias is so strong despite the function of reasoning naively appearing to be to come to veridical conclusions about the world. The study of the development of reasoning abilities is an ongoing area of research in psychology, and multiple factors have been proposed to explain how, why, and when reasoning develops from infancy to adulthood. Recent research has suggested that early experiences and social interactions play a critical role in the development of reasoning abilities. For example, studies have shown that infants as young as six months old can engage in basic logical reasoning, such as reasoning about the relationship between objects and their properties. Furthermore, research has highlighted the importance of parental interaction and cognitive stimulation in the development of children's reasoning abilities. Additionally, studies have suggested that cultural factors, such as educational practices and the emphasis on critical thinking, can also influence the development of reasoning skills across different populations. == Different sorts of reasoning == Philip Johnson-Laird trying to taxonomize thought, distinguished between goal-directed thinking and thinking without goal, noting that association was involved in unrelated reading. He argues that goal directed reasoning can be classified based on the problem space involved in a solution, citing Allen Newell and Herbert A. Simon. Inductive reasoning makes broad generalizations from specific cases or observations. In this process of reasoning, general assertions are made based on past specific pieces of evidence. This kind of reasoning allows the conclusion to be false even if the original statement is true. For example, if one observes a college athlete, one makes predictions and assumptions about other college athletes based on that one observation. Scientists use inductive reasoning to create theories and hypotheses. Philip Johnson-Laird distinguished inductive from deductive reasoning, in that the former creates semantic information while the later does not . In opposition, deductive reasoning is a basic form of valid reasoning. In this reasoning process a person starts with a known claim or a general belief and from there asks what follows from these foundations or how will these premises influence other beliefs. In other words, deduction starts with a hypothesis and examines the possibilities to reach a conclusion. Deduction helps people understand why their predictions are wrong and indicates that their prior knowledge or beliefs are off track. An example of deduction can be seen in the scientific method when testing hypotheses and theories. Although the conclusion usually corresponds and therefore proves the hypothesis, there are some cases where the conclusion is logical, but the generalization is not. For example, the argument, "All young girls wear skirts; Julie is a young
Neurocomputing (journal)
Neurocomputing is a peer-reviewed scientific journal covering research on artificial intelligence, machine learning, and neural computation. It was established in 1989 and is published by Elsevier. The editor-in-chief is Zidong Wang (Brunel University London). Independent scientometric studies noted that despite being one of the most productive journals in the field, it has kept its reputation across the years intact and plays an important role in leading the research in the area. The journal is abstracted and indexed in Scopus and Science Citation Index Expanded. According to the Journal Citation Reports, its 2023 impact factor is 5.5.
Sycophancy (artificial intelligence)
In the field of artificial intelligence, sycophancy is a tendency of large language models (LLMs) and other AI assistants to tailor their responses to what they predict the user wants to hear rather than to what is accurate or warranted. The behavior takes several forms: an assistant may agree with a user's stated opinion even when the user is mistaken; it may abandon a correct answer after a challenge such as "are you sure?"; it may validate beliefs, decisions or self-presentation regardless of merit; or it may praise the user, their work or their ideas in unwarranted terms. The word is borrowed from the ordinary English term for fawning flattery, and is used in AI alignment and AI safety research to describe a class of misalignment failures associated with training on human feedback. Researchers at Anthropic first documented the behavior systematically in 2022. They found that models fine-tuned with reinforcement learning from human feedback (RLHF) were more likely than untuned models to repeat back a user's preferred answer. A 2023 follow-up paper, "Towards Understanding Sycophancy in Language Models", showed that five frontier assistants from OpenAI, Anthropic and Meta all exhibited the behavior, and traced its origin to biases in the human preference data used during training. Later work documented sycophancy in mathematics, medicine, academic peer review and other domains, and identified a broader category called "social sycophancy" affecting an assistant's emotional and interpersonal responses. The issue drew widespread public attention in April 2025 after OpenAI rolled back an update to its GPT-4o model. Users had reported that the assistant praised dangerous decisions, endorsed delusional thinking and offered exaggerated compliments for trivial prompts. OpenAI's post-mortem attributed the change in behavior to an additional training signal based on user thumbs-up and thumbs-down feedback. That episode, together with reporting in The New York Times, Rolling Stone and elsewhere on users drawn into delusional thinking through prolonged chatbot interaction, has been cited in litigation and in academic studies as evidence that sycophancy poses risks to user well-being. Proposed mitigations include fine-tuning on synthetic data that rewards disagreement with incorrect user statements, editing the small subset of model parameters causally responsible for the behavior, changes to the dialogue or system prompt, and benchmarks designed to surface sycophantic behavior before models are released. == Causes == The dominant explanation points to RLHF, the standard technique for aligning chat assistants with user expectations. Human annotators rank candidate model responses; a reward model is trained to predict those rankings; and the language model is then optimized against the reward model. Because human raters tend to prefer outputs that confirm their existing beliefs or flatter their work, the pipeline systematically rewards responses that agree with the annotator. Perez and colleagues at Anthropic published the first large-scale empirical evidence of the effect in 2022. They reported that RLHF training increased the probability that a model would repeat back a dialog user's preferred answer, and that larger models exhibited the behavior more strongly. Sharma and colleagues, the following year, went further and examined Anthropic's own preference data directly. Both the human raters and the reward models trained on their judgments preferred convincingly written sycophantic responses to truthful ones at a non-negligible rate. Wei and co-authors at Google DeepMind found similar results in the PaLM family, observing that both model scale and instruction tuning increased sycophancy on opinion questions. The behavior is often classified as a form of reward hacking, in which an optimization process exploits a flaw in its reward signal rather than achieving the intended objective. OpenAI's post-mortem of the April 2025 GPT-4o incident identified a more specific mechanism. An additional reward signal based on aggregated thumbs-up and thumbs-down feedback from ChatGPT users had, in OpenAI's words, "weakened the influence of our primary reward signal, which had been holding sycophancy in check." Separately, an Anthropic interpretability paper from 2025 located a linear direction in a model's internal activations corresponding to sycophantic behavior, and showed that such "persona vectors" could be used to flag sycophancy-inducing training data and to steer models away from the trait at inference time. == Measurement == The Anthropic team released SycophancyEval with its 2023 paper, supplying test sets for each of the four canonical behaviors. Two further benchmarks from Stanford followed in 2025. SycEval, applied to mathematical and medical reasoning tasks, reported an overall sycophancy rate of 58 per cent across the GPT-4o, Claude and Gemini models tested. ELEPHANT, aimed at social sycophancy, found that the eleven LLMs evaluated affirmed posts that the Reddit community r/AmITheAsshole had judged inappropriate in 42 per cent of cases, and preserved a user's face 45 percentage points more often than human respondents did. Domain-specific benchmarks have followed. BrokenMath tests robustness to plausible-looking but false mathematical claims drawn from competition problems, and reports that the best evaluated model was sycophantic in 29 per cent of cases. SYCON-Bench measures how many dialogue turns are required before a model abandons a correct position. Visual sycophancy in multimodal models has been examined with MM-SY and PENDULUM. A 2026 study by researchers at the Massachusetts Institute of Technology reported that personalization features, which adapt assistants to individual users over repeated sessions, can intensify social sycophancy. == Notable incidents == === GPT-4o rollback (April 2025) === On 25 April 2025, OpenAI completed the rollout of an update to GPT-4o, the default model used in ChatGPT at the time. Within days, users reported that the assistant had begun praising trivial messages in extravagant terms, endorsing impulsive or dangerous decisions, and reinforcing strong emotional statements without pushback. Widely shared examples included the model congratulating a user who reported stopping prescribed psychiatric medication, and praising a business plan to sell "shit on a stick" as venture-capital ready. OpenAI's chief executive, Sam Altman, wrote on 27 April that recent updates had made the model "too sycophant-y and annoying" and said fixes were in progress. The company began reverting the update on 28 April and completed the rollback for free users by 30 April. Two post-mortems followed: a short note on 29 April and a longer technical follow-up, "Expanding on what we missed with sycophancy", on 2 May. Both attributed the regression to a new training signal based on user thumbs-up and thumbs-down feedback, to inadequate pre-launch evaluation for sycophantic drift, and to the dismissal of qualitative concerns raised by internal testers before release. Reporting in CNN, Fortune and Bloomberg News treated the incident as a turning point in public awareness of the problem. === Chatbot-related psychological harm === From mid-2025 onward, news reports began to link sycophantic chatbot behavior to acute psychological harm. In June 2025, The New York Times technology reporter Kashmir Hill published an investigation centered on Eugene Torres, a Manhattan accountant with no history of mental illness, who developed a sustained delusional episode after a series of conversations with ChatGPT about simulation theory. According to the article, the assistant encouraged Torres to stop taking prescribed medication, to cut off friends and family, and at one point told him that he could fly from a nineteen-story building if he "truly believed". Futurism and Rolling Stone ran parallel investigations documenting other cases in which heavy use of ChatGPT had been associated with delusional thinking, involuntary commitment or, in at least one case, the death of a user with a pre-existing psychiatric diagnosis. A 2026 paper by researchers at the Massachusetts Institute of Technology and the University of Washington put forward a formal Bayesian model. It showed that even an ideally rational user could be drawn into what the authors call "delusional spiraling" when interacting with a sufficiently sycophantic assistant, and that the effect was not eliminated by suppressing hallucinations or by warning users in advance. The lawsuit Raine v. OpenAI, filed in San Francisco Superior Court in August 2025 by the parents of a sixteen-year-old who had died by suicide, alleges that "heightened sycophancy" was a design feature of ChatGPT that contributed to their son's death; it is the first wrongful-death suit against a large language-model provider. === Wider commentary === Mainstream coverage in outlets including The New York Times, The Washington Pos
Algorithmic probability
In algorithmic information theory, algorithmic probability, also known as Solomonoff probability, is a mathematical method of assigning a prior probability to a given observation. It was invented by Ray Solomonoff in the 1960s. It is used in inductive inference theory and analyses of algorithms. In his general theory of inductive inference, Solomonoff uses the method together with Bayes' rule to obtain probabilities of prediction for an algorithm's future outputs. In the mathematical formalism used, the observations have the form of finite binary strings viewed as outputs of Turing machines, and the universal prior is a probability distribution over the set of finite binary strings calculated from a probability distribution over programs (that is, inputs to a universal Turing machine). The prior is universal in the Turing-computability sense, i.e. no string has zero probability. It is not computable, but it can be approximated. Formally, the probability P {\displaystyle P} is not a probability and it is not computable. It is only "lower semi-computable" and a "semi-measure". By "semi-measure", it means that 0 ≤ ∑ x P ( x ) < 1 {\displaystyle 0\leq \sum _{x}P(x)<1} . That is, the "probability" does not actually sum up to one, unlike actual probabilities. This is because some inputs to the Turing machine causes it to never halt, which means the probability mass allocated to those inputs is lost. By "lower semi-computable", it means there is a Turing machine that, given an input string x {\displaystyle x} , can print out a sequence y 1 < y 2 < ⋯ {\displaystyle y_{1} In computer vision, photo-consistency determines whether a given voxel is occupied. A voxel is considered to be photo consistent when its color appears to be similar to all the cameras that can see it. Most voxel coloring or space carving techniques require using photo consistency as a check condition in Image-based modeling and rendering applications. == Usage == 3D Volumetric Reconstruction. Image registration. Multi-view reconstruction. 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 crPhoto-consistency
Cross-validation (statistics)