In probability theory, the chain rule [1] (also called the general product rule [2] [3] ) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities. The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.
For two events and , the chain rule states that
where denotes the conditional probability of given .
An Urn A has 1 black ball and 2 white balls and another Urn B has 1 black ball and 3 white balls. Suppose we pick an urn at random and then select a ball from that urn. Let event be choosing the first urn, i.e. , where is the complementary event of . Let event be the chance we choose a white ball. The chance of choosing a white ball, given that we have chosen the first urn, is The intersection then describes choosing the first urn and a white ball from it. The probability can be calculated by the chain rule as follows:
For events whose intersection has not probability zero, the chain rule states
For , i.e. four events, the chain rule reads
We randomly draw 4 cards without replacement from deck with 52 cards. What is the probability that we have picked 4 aces?
First, we set . Obviously, we get the following probabilities
Applying the chain rule,
Let be a probability space. Recall that the conditional probability of an given is defined as
Then we have the following theorem.
Chain rule — Let be a probability space. Let . Then
The formula follows immediately by recursion
where we used the definition of the conditional probability in the first step.
For two discrete random variables , we use the eventsand in the definition above, and find the joint distribution as
or
where is the probability distribution of and conditional probability distribution of given .
Let be random variables and . By the definition of the conditional probability,
and using the chain rule, where we set , we can find the joint distribution as
For , i.e. considering three random variables. Then, the chain rule reads
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: CS1 maint: location missing publisher (link)Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample.
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In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.
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Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s).
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted , is a family of continuous multivariate probability distributions parameterized by a vector of positive reals. It is a multivariate generalization of the beta distribution, hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.
In probability theory, if a large number of events are all independent of one another and each has probability less than 1, then there is a positive probability that none of the events will occur. The Lovász local lemma allows one to relax the independence condition slightly: As long as the events are "mostly" independent from one another and aren't individually too likely, then there will still be a positive probability that none of them occurs. It is most commonly used in the probabilistic method, in particular to give existence proofs.
In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution. It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector , and an observation drawn from a multinomial distribution with probability vector p and number of trials n. The Dirichlet parameter vector captures the prior belief about the situation and can be seen as a pseudocount: observations of each outcome that occur before the actual data is collected. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, machine learning, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution.
In probability theory and theoretical computer science, McDiarmid's inequality is a concentration inequality which bounds the deviation between the sampled value and the expected value of certain functions when they are evaluated on independent random variables. McDiarmid's inequality applies to functions that satisfy a bounded differences property, meaning that replacing a single argument to the function while leaving all other arguments unchanged cannot cause too large of a change in the value of the function.
A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe or checkers (draughts). In other words, it extends the notion of a discrete game, where the players choose from a finite set of pure strategies. The continuous game concepts allows games to include more general sets of pure strategies, which may be uncountably infinite.
In probability theory, particularly information theory, the conditional mutual information is, in its most basic form, the expected value of the mutual information of two random variables given the value of a third.
Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations, and conditional probability distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory. Conditioning leads to a non-random result if the condition is completely specified; otherwise, if the condition is left random, the result of conditioning is also random.
In probability theory, a Markov kernel is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space.
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(A|B) or occasionally PB(A). This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the "given" one happening (how many times A occurs rather than not assuming B has occurred): .
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In computational statistics, the pseudo-marginal Metropolis–Hastings algorithm is a Monte Carlo method to sample from a probability distribution. It is an instance of the popular Metropolis–Hastings algorithm that extends its use to cases where the target density is not available analytically. It relies on the fact that the Metropolis–Hastings algorithm can still sample from the correct target distribution if the target density in the acceptance ratio is replaced by an estimate. It is especially popular in Bayesian statistics, where it is applied if the likelihood function is not tractable.
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