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Information about Posterior Probability

The posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned when the relevant evidence is taken into account.

The posterior probability distribution of one random variable given the value of another can be calculated with Bayes' theorem by multiplying the prior probability distribution by the likelihood function, and then dividing by the normalizing constant, as follows:



gives the posterior probability density function for a random variable X given the data Y = y, where
  • is the prior density of X,
  • is the likelihood function as a function of x,
  • is the normalizing constant, and
  • is the posterior density of X given the data Y = y.
Probability distributions [  •  • [ edit] ]
Univariate Multivariate
Discrete: Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-MandelbrotEwens • multinomial • multivariate Polya
Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lvy • Lvy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambdaDirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: bimodal • Cantor • conditional • equilibrium • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular
In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event (i.e.
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Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B".
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Scientific evidence is evidence which serves to either support or counter a scientific theory or hypothesis. Such evidence is expected to be empirical and properly documented in accordance with scientific method such as is applicable to the particular field of inquiry.
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A random variable is an abstraction of the intuitive concept of chance into the theoretical domains of mathematics, forming the foundations of probability theory and mathematical statistics.
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Bayes' theorem (also known as Bayes' rule or Bayes' law) is a result in probability theory, which relates the conditional and marginal probability distributions of random variables.
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A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. The posterior probability is then the conditional probability of the variable taking the evidence into account.
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Likelihood as a solitary term is a shorthand for likelihood function. In non-technical usage, "likelihood" is a synonym for "probability", but throughout this article only the technical definition is used.
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The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.

Definition and examples

In probability theory, a normalizing constant
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In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.

Formally, a probability distribution has density f, if f
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probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.
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In statistics, in univariate data, each data point has only one scalar component. Or, when the statistical technique to be used, it contains only one dependent variable. The more general case is multivariate.
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A multivariate random variable or random vector is a vector X = (X1, ..., Xn) whose components are scalar-valued random variables on the same probability space (Ω, P).
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Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability .
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binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.
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Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each respectively possess energy Ei:


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A categorical distribution is the most general distribution whose sample space is the set .

It is the generalization of the Bernoulli distribution for a categorical random variable.

It should not be confused with the multinomial distribution.
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In probability theory, a compound Poisson distribution is the probability distribution of a "Poisson-distributed number" of independent identically-distributed random variables. More precisely, suppose



i.e.
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degenerate distribution is the probability distribution of a discrete random variable whose support consists of only one value. Examples include a two-headed coin and rolling a die whose sides all show the same number.
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Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. The distribution is named after Carl Friedrich Gauss, who first conjectured and studied the distribution around 1800,
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geometric distribution is either of two discrete probability distributions:
  • the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set , or
  • the probability distribution of the number Y

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hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement.
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logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution derived from the Maclaurin series expansion



From this we obtain the identity


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negative binomial distribution is a discrete probability distribution. The Pascal distribution and the Polya distribution are special cases of the negative binomial.
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In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship (see external link below).
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Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event.
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Rademacher distribution, named after Hans Rademacher is a discrete probability distribution which has a 50% chance for either 1 or -1. The probability mass function of this distribution is



The Rademacher distribution has been used in bootstrapping.
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Skellam distribution is the discrete probability distribution of the difference of two correlated or uncorrelated random variables and having Poisson distributions with different expected values and .
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discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable.
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Yule-Simon distribution is a discrete probability distribution named after Udny Yule and Herbert Simon. Simon originally called it the Yule distribution.

The probability mass function of the Yule-Simon(ρ) distribution is


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zeta distribution is a discrete probability distribution. If X is a zeta-distributed random variable with parameter s, then the probability that X takes the integer value k is given by the probability mass function


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