In machine learning, a probabilistic classifier is a classifier, able to predict, given an observation of an input, a probability distribution over a set of classes, rather than only outputting the most class that the observation should belong to. Probabilistic classifiers provide classification that can be useful in its own right or when combining classifiers into ensembles. Formally, an "ordinary" classifier is some rule, or function, that assigns to a sample x a class label ŷ: y ^ = f The samples come from some set X, while the class labels form a finite set Y defined prior to training. Probabilistic classifiers generalize this notion of classifiers: instead of functions, they are conditional distributions Pr, meaning that for a given x ∈ X, they assign probabilities to all y ∈ Y. "Hard" classification can be done using the optimal decision rule y ^ = arg max y Pr or, in English, the predicted class is that which has the highest probability. Binary probabilistic classifiers are called binomial regression models in statistics.
In econometrics, probabilistic classification in general is called discrete choice. Some classification models, such as naive Bayes, logistic regression and multilayer perceptrons are probabilistic. Other models such as support vector machines are not, but methods exist to turn them into probabilistic classifiers; some models, such as logistic regression, are conditionally trained: they optimize the conditional probability Pr directly on a training set. Other classifiers, such as naive Bayes, are trained generatively: at training time, the class-conditional distribution Pr and the class prior Pr are found, the conditional distribution Pr is derived using Bayes' rule. Not all classification models are probabilistic, some that are, notably naive Bayes classifiers, decision trees and boosting methods, produce distorted class probability distributions. In the case of decision trees, where Pr is the proportion of training samples with label y in the leaf where x ends up, these distortions come about because learning algorithms such as C4.5 or CART explicitly aim to produce homogeneous leaves while using few samples to estimate the relevant proportion.
Calibration can be assessed using a calibration plot. A calibration plot shows the proportion of items in each class for bands of predicted probability or score. Deviations from the identity function indicate a poorly-calibrated classifier for which the predicted probabilities or scores can not be used as probabilities. In this case one can use a method to turn these scores into properly calibrated class membership probabilities. For the binary case, a common approach is to apply Platt scaling, which learns a logistic regression model on the scores. An alternative method using isotonic regression is superior to Platt's method when sufficient training data is available. In the multiclass case, one can use a reduction to binary tasks, followed by univariate calibration with an algorithm as described above and further application of the pairwise coupling algorithm by Hastie and Tibshirani. Used loss functions for probabilistic classification include log loss and the Brier score between the predicted and the true probability distributions.
The former of these is used to train logistic models. A method used to assign scores to pairs of predicted probabilities and actual discrete outcomes, so that different predictive methods can be compared, is called a scoring rule
The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical tendency of something to occur or is it a measure of how one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory. There are two broad categories of probability interpretations which can be called "physical" and "evidential" probabilities. Physical probabilities, which are called objective or frequency probabilities, are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. In such systems, a given type of event tends to occur at a persistent rate, or "relative frequency", in a long run of trials. Physical probabilities either explain, or are invoked to explain, these stable frequencies; the two main kinds of theory of physical probability are frequentist accounts and propensity accounts.
Evidential probability called Bayesian probability, can be assigned to any statement whatsoever when no random process is involved, as a way to represent its subjective plausibility, or the degree to which the statement is supported by the available evidence. On most accounts, evidential probabilities are considered to be degrees of belief, defined in terms of dispositions to gamble at certain odds; the four main evidential interpretations are the classical interpretation, the subjective interpretation, the epistemic or inductive interpretation and the logical interpretation. There are evidential interpretations of probability covering groups, which are labelled as'intersubjective'; some interpretations of probability are associated with approaches to statistical inference, including theories of estimation and hypothesis testing. The physical interpretation, for example, is taken by followers of "frequentist" statistical methods, such as Ronald Fisher, Jerzy Neyman and Egon Pearson. Statisticians of the opposing Bayesian school accept the existence and importance of physical probabilities, but consider the calculation of evidential probabilities to be both valid and necessary in statistics.
This article, focuses on the interpretations of probability rather than theories of statistical inference. The terminology of this topic is rather confusing, in part because probabilities are studied within a variety of academic fields; the word "frequentist" is tricky. To philosophers it refers to a particular theory of physical probability, one that has more or less been abandoned. To scientists, on the other hand, "frequentist probability" is just another name for physical probability; those who promote Bayesian inference view "frequentist statistics" as an approach to statistical inference that recognises only physical probabilities. The word "objective", as applied to probability, sometimes means what "physical" means here, but is used of evidential probabilities that are fixed by rational constraints, such as logical and epistemic probabilities, it is unanimously agreed. But, as to what probability is and how it is connected with statistics, there has been such complete disagreement and breakdown of communication since the Tower of Babel.
Doubtless, much of the disagreement is terminological and would disappear under sufficiently sharp analysis. The philosophy of probability presents problems chiefly in matters of epistemology and the uneasy interface between mathematical concepts and ordinary language as it is used by non-mathematicians. Probability theory is an established field of study in mathematics, it has its origins in correspondence discussing the mathematics of games of chance between Blaise Pascal and Pierre de Fermat in the seventeenth century, was formalized and rendered axiomatic as a distinct branch of mathematics by Andrey Kolmogorov in the twentieth century. In axiomatic form, mathematical statements about probability theory carry the same sort of epistemological confidence within the philosophy of mathematics as are shared by other mathematical statements; the mathematical analysis originated in observations of the behaviour of game equipment such as playing cards and dice, which are designed to introduce random and equalized elements.
This is not the only way probabilistic statements are used in ordinary human language: when people say that "it will rain", they do not mean that the outcome of rain versus not-rain is a random factor that the odds favor. When it is written that "the most probable explanation" of the name of Ludlow, Massachusetts "is that it was named after Roger Ludlow", what is meant here is not that Roger Ludlow is favored by a random factor, but rather that this is the most plausible explanation of the evidence, which admits other, less explanations. Thomas Bayes attempted to provide a logic. Though probability had somewhat mundane motivations, its modern influence and use is widespread ranging from evidence-based medicine, through Six sigma, all the w
Statistics is a branch of mathematics dealing with data collection, analysis and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics; when census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements.
In contrast, an observational study does not involve experimental manipulation. Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, inferential statistics, which draw conclusions from data that are subject to random variation. Descriptive statistics are most concerned with two sets of properties of a distribution: central tendency seeks to characterize the distribution's central or typical value, while dispersion characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets.
Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors and Type II errors. Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are subject to error. Many of these errors are classified as random or systematic, but other types of errors can be important; the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more from calculus and probability theory. In more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis.
Some definitions are: Merriam-Webster dictionary defines statistics as "a branch of mathematics dealing with the collection, analysis and presentation of masses of numerical data." Statistician Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed in relation to each other."Statistics is a mathematical body of science that pertains to the collection, interpretation or explanation, presentation of data, or as a branch of mathematics. Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty and decision making in the face of uncertainty. Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, measure-theoretic probability theory.
In applying statistics to a problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Ideally, statisticians compile data about the entire population; this may be organized by governmental statistical institutes. Descriptive statistics can be used to summarize the population data. Numerical descriptors include mean and standard deviation for continuous data types, while frequency and percentage are more useful in terms of describing categorical data; when a census is not feasible, a chosen subset of the population called. Once a sample, representative of the population is determined, data is collected for the sample members in an observational or experimental setting. Again, descriptive statistics can be used to summarize the sample data. However, the drawing of the sample has been subject to an element of randomness, hence the established numerical descriptors from the sample are due to uncertainty.
To still draw meaningful conclusions about the entire population, in
In probability theory and statistics, Bayes' theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age using Bayes' theorem, a person's age can be used to more assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person's age. One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference; when applied, the probabilities involved in Bayes' theorem may have different probability interpretations. With the Bayesian probability interpretation the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for availability of related evidence. Bayesian inference is fundamental to Bayesian statistics. Bayes' theorem is named after Reverend Thomas Bayes, who first used conditional probability to provide an algorithm that uses evidence to calculate limits on an unknown parameter, published as An Essay towards solving a Problem in the Doctrine of Chances.
In what he called a scholium, Bayes extended his algorithm to any unknown prior cause. Independently of Bayes, Pierre-Simon Laplace in 1774, in his 1812 "Théorie analytique des probabilités" used conditional probability to formulate the relation of an updated posterior probability from a prior probability, given evidence. Sir Harold Jeffreys put Laplace's formulation on an axiomatic basis. Jeffreys wrote that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry". Bayes' theorem is stated mathematically as the following equation: where A and B are events and P ≠ 0. P is a conditional probability: the likelihood of event A occurring given. P is a conditional probability: the likelihood of event B occurring given that A is true. P and P are the probabilities of observing B independently of each other. Suppose that a test for using a particular drug is 99% sensitive and 99% specific; that is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users.
Suppose that 0.5% of people are users of the drug. What is the probability that a randomly selected individual with a positive test is a drug user? P = P P P = P P P P + P P = 0.99 × 0.005 0.99 × 0.005 + 0.01 × 0.995 ≈ 33.2 % Even if an individual tests positive, it is more that they do not use the drug than that they do. This is; the number of false positives outweighs the number of true positives. For example, if 1000 individuals are tested, there are expected to be 5 users. From the 995 non-users, 0.01 × 995 ≃ 10 false positives are expected. From the 5 users, 0.99 × 5 ≈ 5 true positives are expected. Out of 15 positive results, only 5 are genuine; the importance of specificity in this example can be seen by calculating that if sensitivity is raised to 100% and specificity remains at 99% the probability of the person being a drug user only rises
Markov chain Monte Carlo
In statistics, Markov chain Monte Carlo methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by observing the chain after a number of steps; the more steps there are, the more the distribution of the sample matches the actual desired distribution. Markov chain Monte Carlo methods are used for calculating numerical approximations of multi-dimensional integrals, for example in Bayesian statistics, computational physics, computational biology, computational linguistics. In Bayesian statistics, the recent development of Markov chain Monte Carlo methods has been a key step in making it possible to compute large hierarchical models that require integrations over hundreds or thousands of unknown parameters. In rare event sampling, they are used for generating samples that populate the rare failure region. Markov chain Monte Carlo methods create samples from a multi-dimensional continuous random variable, with probability density proportional to a known function.
These samples can be used to evaluate an integral over that variable, as its expected value or variance. An ensemble of chains is developed, starting from a set of points arbitrarily chosen and sufficiently distant from each other; these chains are stochastic processes of "walkers" which move around randomly according to an algorithm which looks for places with a reasonably high contribution to the integral to move into next, assigning them higher probabilities. Random walk Monte Carlo methods are a kind of Monte Carlo method. However, whereas the random samples of the integrand used in a conventional Monte Carlo integration are statistically independent, those used in Markov chain Monte Carlo methods are autocorrelated; these algorithms create Markov chains such that they have an equilibrium distribution, proportional to the function given. While MCMC methods were created to address multi-dimensional problems better than simple Monte Carlo algorithms, when the number of dimensions rises they too tend to suffer the curse of dimensionality: the regions of higher probability tend to stretch and get lost in a increasing volume of space that gives little contribution to the desired integral.
One way to address this problem could be shortening the steps of the walker, so that it doesn't continuously try to exit the highest probability region, though this way the process would be autocorrelated and quite ineffective. More sophisticated methods use various ways of reducing the autocorrelation, while managing to keep the process in the regions that give a higher contribution to the integral; these algorithms rely on a more complicated theory, may be harder to implement, but they exhibit faster convergence. Examples of random walk Monte Carlo methods include the following: Metropolis–Hastings algorithm: This method generates a Markov chain using a proposal density for new steps and a method for rejecting some of the proposed moves, it is a general framework which includes as special cases the first and simpler MCMC and many more recent alternatives listed below. Gibbs sampling: This method requires all the conditional distributions of the target distribution to be sampled exactly; when drawing from the full-conditional distributions is not straightforward other samplers-within-Gibbs are used.
Gibbs sampling is popular because it does not require any'tuning'. Metropolis-adjusted Langevin algorithm and other methods that rely on the gradient of the log target density to propose steps that are more to be in the direction of higher probability density. Slice sampling: This method depends on the principle that one can sample from a distribution by sampling uniformly from the region under the plot of its density function, it alternates uniform sampling in the vertical direction with uniform sampling from the horizontal'slice' defined by the current vertical position. Multiple-try Metropolis: This method is a variation of the Metropolis–Hastings algorithm that allows multiple trials at each point. By making it possible to take larger steps at each iteration, it helps address the curse of dimensionality. Reversible-jump: This method is a variant of the Metropolis–Hastings algorithm that allows proposals that change the dimensionality of the space. Markov chain Monte Carlo methods that change dimensionality have long been used in statistical physics applications, where for some problems a distribution, a grand canonical ensemble is used.
But the reversible-jump variant is useful when doing Markov chain Monte Carlo or Gibbs sampling over nonparametric Bayesian models such as those involving the Dirichlet process or Chinese restaurant process, where the number of mixing components/clusters/etc. is automatically inferred from the data. Hamiltonian Monte Carlo: Tries to avoid random walk behaviour by introducing an auxiliary momentum vector and implementing Hamiltonian dynamics, so the potential energy function is the target density; the momentum samples are discarded after sampling. The end result of Hybrid Monte Carlo is. Unlike most of the current Markov chain Monte Carlo methods that ignore the previous trials, using a new algorithm the Markov chain Monte Carlo algorithm is able to use the previous steps and generate the next candidate
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, in mathematical statistics. Bayesian updating is important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, philosophy, medicine and law. In the philosophy of decision theory, Bayesian inference is related to subjective probability called "Bayesian probability". Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem: P = P ⋅ P P where H stands for any hypothesis whose probability may be affected by data. There are competing hypotheses, the task is to determine, the most probable.
P, the prior probability, is the estimate of the probability of the hypothesis H before the data E, the current evidence, is observed. The evidence E corresponds to new data. P, the posterior probability, is the probability of H given E, i.e.. This is: the probability of a hypothesis given the observed evidence. P is the probability of observing E given H, is called the likelihood; as a function of E with H fixed, it indicates the compatibility of the evidence with the given hypothesis. The likelihood function is a function of the evidence, E, while the posterior probability is a function of the hypothesis, H. P is sometimes termed the marginal likelihood or "model evidence"; this factor is the same for all possible hypotheses being considered, so this factor does not enter into determining the relative probabilities of different hypotheses. For different values of H, only the factors P and P, both in the numerator, affect the value of P – the posterior probability of a hypothesis is proportional to its prior probability and the newly acquired likelihood.
Bayes' rule can be written as follows: P = P P ⋅ P where the factor P P can be interpreted as the impact of E on the probability of H. Bayesian updating is used and computationally convenient. However, it is not the only updating rule. Ian Hacking noted that traditional "Dutch book" arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books. Hacking wrote "And neither the Dutch book argument nor any other in the personalist arsenal of proofs of the probability axioms entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian, it is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour." Indeed, there are non-Bayesian updating rules that avoid Dutch books following the publication of Richard C. Jeffrey's rule