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In
Bayesian probability Bayesian probability is an Probability interpretations, interpretation of the concept of probability, in which, instead of frequentist probability, frequency or propensity probability, propensity of some phenomenon, probability is interpreted as re ...
theory, if the
posterior distribution The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
p(\theta \mid x) is in the same probability distribution family as the
prior probability distribution In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken int ...
p(\theta), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the
likelihood function The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood funct ...
p(x \mid \theta). A conjugate prior is an algebraic convenience, giving a
closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
for the posterior; otherwise,
numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
may be necessary. Further, conjugate priors may give intuition by more transparently showing how a likelihood function updates a prior distribution. The concept, as well as the term "conjugate prior", were introduced by
Howard Raiffa Howard Raiffa (; January 24, 1924 – July 8, 2016) was an American academic who was the Frank P. Ramsey Professor (Emeritus) of Managerial Economics, a joint chair held by the Business School and Harvard Kennedy School at Harvard University. He w ...
and
Robert Schlaifer Robert Osher Schlaifer (13 September 1914 – 24 July 1994) was a pioneer of Bayesian decision theory. At the time of his death he was William Ziegler Professor of Business Administration Emeritus of the Harvard Business School. In 1961 he was ...
in their work on
Bayesian decision theory In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the po ...
.
Howard Raiffa Howard Raiffa (; January 24, 1924 – July 8, 2016) was an American academic who was the Frank P. Ramsey Professor (Emeritus) of Managerial Economics, a joint chair held by the Business School and Harvard Kennedy School at Harvard University. He w ...
and
Robert Schlaifer Robert Osher Schlaifer (13 September 1914 – 24 July 1994) was a pioneer of Bayesian decision theory. At the time of his death he was William Ziegler Professor of Business Administration Emeritus of the Harvard Business School. In 1961 he was ...
. ''Applied Statistical Decision Theory''. Division of Research, Graduate School of Business Administration, Harvard University, 1961.
A similar concept had been discovered independently by
George Alfred Barnard George Alfred Barnard (23 September 1915 – 9 August 2002) was a British statistician known particularly for his work on the foundations of statistics and on quality control. Biography George Barnard was born in Walthamstow, Lon ...
.Jeff Miller et al
Earliest Known Uses of Some of the Words of Mathematics
Electronic document, revision of November 13, 2005, retrieved December 2, 2005.


Example

The form of the conjugate prior can generally be determined by inspection of the
probability density In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
or
probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
of a distribution. For example, consider a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
which consists of the number of successes s in n
Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...
s with unknown probability of success q in ,1 This random variable will follow the
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
, with a probability mass function of the form :p(s) = q^s (1-q)^ The usual conjugate prior is the
beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
with parameters (\alpha, \beta): :p(q) = where \alpha and \beta are chosen to reflect any existing belief or information (\alpha=1 and \beta=1 would give a uniform distribution) and \Beta(\alpha,\beta) is the
Beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^( ...
acting as a normalising constant. In this context, \alpha and \beta are called ''
hyperparameter In Bayesian statistics, a hyperparameter is a parameter of a prior distribution; the term is used to distinguish them from parameters of the model for the underlying system under analysis. For example, if one is using a beta distribution to mo ...
s'' (parameters of the prior), to distinguish them from parameters of the underlying model (here q). A typical characteristic of conjugate priors is that the dimensionality of the hyperparameters is one greater than that of the parameters of the original distribution. If all parameters are scalar values, then there will be one more hyperparameter than parameter; but this also applies to vector-valued and matrix-valued parameters. (See the general article on the
exponential family In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...
, and also consider the
Wishart distribution In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928. It is a family of probability distributions define ...
, conjugate prior of the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
of a
multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
, for an example where a large dimensionality is involved.) If we sample this random variable and get s successes and f = n - s failures, then we have :\begin P(s, f \mid q=x) &= x^s(1-x)^f,\\ P(q=x) &= ,\\ P(q=x \mid s,f) &= \frac\\ & = \\ & = , \end which is another Beta distribution with parameters (\alpha + s, \beta + f). This posterior distribution could then be used as the prior for more samples, with the hyperparameters simply adding each extra piece of information as it comes.


Interpretations


Pseudo-observations

It is often useful to think of the hyperparameters of a conjugate prior distribution corresponding to having observed a certain number of ''pseudo-observations'' with properties specified by the parameters. For example, the values \alpha and \beta of a
beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
can be thought of as corresponding to \alpha-1 successes and \beta-1 failures if the posterior mode is used to choose an optimal parameter setting, or \alpha successes and \beta failures if the posterior mean is used to choose an optimal parameter setting. In general, for nearly all conjugate prior distributions, the hyperparameters can be interpreted in terms of pseudo-observations. This can help provide intuition behind the often messy update equations and help choose reasonable hyperparameters for a prior.


Analogy with eigenfunctions

Conjugate priors are analogous to
eigenfunctions In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
in
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
in that they are distributions on which the "conditioning operator" acts in a well-understood way, thinking of the process of changing from the prior to the posterior as an operator. In both eigenfunctions and conjugate priors, the operator preserves a ''finite-dimensional'' space: the output is of the same form (in the same space) as the input. This greatly simplifies the analysis, as it otherwise considers an infinite-dimensional space (space of all functions, space of all distributions). However, the processes are only analogous, not identical: conditioning is not linear, as the space of distributions is not closed under linear combination, only
convex combination In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other word ...
, and the posterior is only of the same ''form'' as the prior, not a scalar multiple. Just as one can easily analyze how a linear combination of eigenfunctions evolves under the application of an operator (because, for these functions, the operator is
diagonalized In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
), one can easily analyze how a convex combination of conjugate priors evolves under conditioning; this is called using a ''
hyperprior In Bayesian statistics, a hyperprior is a prior distribution on a hyperparameter, that is, on a parameter of a prior distribution. As with the term ''hyperparameter,'' the use of ''hyper'' is to distinguish it from a prior distribution of a param ...
,'' and corresponds to using a
mixture density In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collectio ...
of conjugate priors, rather than a single conjugate prior.


Dynamical system

One can think of conditioning on conjugate priors as defining a kind of (discrete time)
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
: from a given set of hyperparameters, incoming data updates these hyperparameters, so one can see the change in hyperparameters as a kind of "time evolution" of the system, corresponding to "learning". Starting at different points yields different flows over time. This is again analogous with the dynamical system defined by a linear operator, but note that since different samples lead to different inferences, this is not simply dependent on time but rather on data over time. For related approaches, see
Recursive Bayesian estimation In probability theory, statistics, and machine learning, recursive Bayesian estimation, also known as a Bayes filter, is a general probabilistic approach for estimating an unknown probability density function (PDF) recursively over time using inco ...
and
Data assimilation Data assimilation is a mathematical discipline that seeks to optimally combine theory (usually in the form of a numerical model) with observations. There may be a number of different goals sought – for example, to determine the optimal state es ...
.


Practical example

Suppose a rental car service operates in your city. Drivers can drop off and pick up cars anywhere inside the city limits. You can find and rent cars using an app. Suppose you wish to find the probability that you can find a rental car within a short distance of your home address at any time of day. Over three days you look at the app and find the following number of cars within a short distance of your home address: \mathbf = ,4,1/math> Suppose we assume the data comes from a
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
. In that case, we can compute the
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
estimate of the parameters of the model, which is \lambda = \frac \approx 2.67. Using this maximum likelihood estimate, we can compute the probability that there will be at least one car available on a given day: p(x>0 , \lambda \approx 2.67) = 1 - p(x=0 , \lambda \approx 2.67) = 1-\frac \approx 0.93 This is the Poisson distribution that is ''the'' most likely to have generated the observed data \mathbf. But the data could also have come from another Poisson distribution, e.g., one with \lambda = 3, or \lambda = 2, etc. In fact, there is an infinite number of Poisson distributions that ''could'' have generated the observed data. With relatively few data points, we should be quite uncertain about which exact Poisson distribution generated this data. Intuitively we should instead take a weighted average of the probability of p(x>0, \lambda) for each of those Poisson distributions, weighted by how likely they each are, given the data we've observed \mathbf. Generally, this quantity is known as the
posterior predictive distribution Posterior may refer to: * Posterior (anatomy), the end of an organism opposite to its head ** Buttocks, as a euphemism * Posterior horn (disambiguation) * Posterior probability The posterior probability is a type of conditional probability that r ...
p(x, \mathbf) = \int_\theta p(x, \theta)p(\theta, \mathbf)d\theta\,, where x is a new data point, \mathbf is the observed data and \theta are the parameters of the model. Using Bayes' theorem we can expand p(\theta, \mathbf) = \frac\,, therefore p(x, \mathbf) = \int_\theta p(x, \theta)\fracd\theta\,. Generally, this integral is hard to compute. However, if you choose a conjugate prior distribution p(\theta), a closed-form expression can be derived. This is the posterior predictive column in the tables below. Returning to our example, if we pick the
Gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...
as our prior distribution over the rate of the Poisson distributions, then the posterior predictive is the negative binomial distribution, as can be seen from the table below. The Gamma distribution is parameterized by two hyperparameters \alpha, \beta, which we have to choose. By looking at plots of the gamma distribution, we pick \alpha = \beta = 2, which seems to be a reasonable prior for the average number of cars. The choice of prior hyperparameters is inherently subjective and based on prior knowledge. Given the prior hyperparameters \alpha and \beta we can compute the posterior hyperparameters \alpha' = \alpha + \sum_i x_i = 2 + 3+4+1 = 10 and \beta' = \beta + n = 2+3 = 5 Given the posterior hyperparameters, we can finally compute the posterior predictive of p(x>0, \mathbf) = 1-p(x=0, \mathbf) = 1 - NB\left(0\, , \, 10, \frac\right) \approx 0.84 This much more conservative estimate reflects the uncertainty in the model parameters, which the posterior predictive takes into account.


Table of conjugate distributions

Let ''n'' denote the number of observations. In all cases below, the data is assumed to consist of ''n'' points x_1,\ldots,x_n (which will be
random vector In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value ...
s in the multivariate cases). If the likelihood function belongs to the
exponential family In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...
, then a conjugate prior exists, often also in the exponential family; see Exponential family: Conjugate distributions.


When the likelihood function is a discrete distribution


When likelihood function is a continuous distribution

{, class="wikitable" ! Likelihood !! Model parameters !! Conjugate prior distribution !! Prior hyperparameters !! Posterior hyperparameters!!Interpretation of hyperparameters!!Posterior predictive , - ,
Normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

with known variance ''σ''2 , , ''μ'' (mean) , ,
Normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
, , \mu_0,\, \sigma_0^2\! , , \frac{1}{\frac{1}{\sigma_0^2} + \frac{n}{\sigma^2\left(\frac{\mu_0}{\sigma_0^2} + \frac{\sum_{i=1}^n x_i}{\sigma^2}\right), \left(\frac{1}{\sigma_0^2} + \frac{n}{\sigma^2}\right)^{-1} , mean was estimated from observations with total precision (sum of all individual precisions) 1/\sigma_0^2 and with sample mean \mu_0 , \mathcal{N}(\tilde{x}, \mu_0', {\sigma_0^2}' +\sigma^2) , - ,
Normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

with known precision ''τ'' , , ''μ'' (mean) , ,
Normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
, , \mu_0,\, \tau_0^{-1}\! , , \frac{\tau_0 \mu_0 + \tau \sum_{i=1}^n x_i}{\tau_0 + n \tau},\, \left(\tau_0 + n \tau\right)^{-1} , mean was estimated from observations with total precision (sum of all individual precisions)\tau_0 and with sample mean \mu_0 , \mathcal{N}\left(\tilde{x}\mid\mu_0', \frac{1}{\tau_0'} +\frac{1}{\tau}\right) , - ,
Normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

with known mean ''μ'' , , ''σ''2 (variance) , , Inverse gamma , , \mathbf{\alpha,\, \beta} , , \mathbf{\alpha}+\frac{n}{2},\, \mathbf{\beta} + \frac{\sum_{i=1}^n{(x_i-\mu)^2{2} , variance was estimated from 2\alpha observations with sample variance \beta/\alpha (i.e. with sum of
squared deviations Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of ''variance'' is either the expected value of the SDM (when considering a theoretical distribution) or its average valu ...
2\beta, where deviations are from known mean \mu) , t_{2\alpha'}(\tilde{x}, \mu,\sigma^2 = \beta'/\alpha') , - ,
Normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

with known mean ''μ'' , , ''σ''2 (variance) , , Scaled inverse chi-squared , , \nu,\, \sigma_0^2\! , , \nu+n,\, \frac{\nu\sigma_0^2 + \sum_{i=1}^n (x_i-\mu)^2}{\nu+n}\! , variance was estimated from \nu observations with sample variance \sigma_0^2 , t_{\nu'}(\tilde{x}, \mu,{\sigma_0^2}') , - ,
Normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

with known mean ''μ'' , , ''τ'' (precision) , ,
Gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
, , \alpha,\, \beta\! , , \alpha + \frac{n}{2},\, \beta + \frac{\sum_{i=1}^n (x_i-\mu)^2}{2}\! , precision was estimated from 2\alpha observations with sample variance \beta/\alpha (i.e. with sum of
squared deviations Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of ''variance'' is either the expected value of the SDM (when considering a theoretical distribution) or its average valu ...
2\beta, where deviations are from known mean \mu) , t_{2\alpha'}(\tilde{x}\mid\mu,\sigma^2 = \beta'/\alpha') , - ,
Normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
A different conjugate prior for unknown mean and variance, but with a fixed, linear relationship between them, is found in the normal variance-mean mixture, with the generalized inverse Gaussian as conjugate mixing distribution. , , ''μ'' and ''σ2''
Assuming
exchangeability In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence ''X''1, ''X''2, ''X''3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change whe ...
, , Normal-inverse gamma , \mu_0 ,\, \nu ,\, \alpha ,\, \beta , , \frac{\nu\mu_0+n\bar{x{\nu+n} ,\, \nu+n,\, \alpha+\frac{n}{2} ,\,
\beta + \tfrac{1}{2} \sum_{i=1}^n (x_i - \bar{x})^2 + \frac{n\nu}{\nu+n}\frac{(\bar{x}-\mu_0)^2}{2} * \bar{x} is the sample mean , mean was estimated from \nu observations with sample mean \mu_0; variance was estimated from 2\alpha observations with sample mean \mu_0 and sum of
squared deviations Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of ''variance'' is either the expected value of the SDM (when considering a theoretical distribution) or its average valu ...
2\beta , t_{2\alpha'}\left(\tilde{x}\mid\mu',\frac{\beta'(\nu'+1)}{\nu' \alpha'}\right) , - ,
Normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
, , ''μ'' and ''τ''
Assuming
exchangeability In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence ''X''1, ''X''2, ''X''3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change whe ...
, , Normal-gamma , \mu_0 ,\, \nu ,\, \alpha ,\, \beta , , \frac{\nu\mu_0+n\bar{x{\nu+n} ,\, \nu+n,\, \alpha+\frac{n}{2} ,\,
\beta + \tfrac{1}{2} \sum_{i=1}^n (x_i - \bar{x})^2 + \frac{n\nu}{\nu+n}\frac{(\bar{x}-\mu_0)^2}{2} * \bar{x} is the sample mean , mean was estimated from \nu observations with sample mean \mu_0, and precision was estimated from 2\alpha observations with sample mean \mu_0 and sum of
squared deviations Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of ''variance'' is either the expected value of the SDM (when considering a theoretical distribution) or its average valu ...
2\beta , t_{2\alpha'}\left(\tilde{x}\mid\mu',\frac{\beta'(\nu'+1)}{\alpha'\nu'}\right) , - ,
Multivariate normal In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
with known covariance matrix ''Σ'' , , ''μ'' (mean vector) , ,
Multivariate normal In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
, , \boldsymbol{\boldsymbol\mu}_0,\, \boldsymbol\Sigma_0 , , \left(\boldsymbol\Sigma_0^{-1} + n\boldsymbol\Sigma^{-1}\right)^{-1}\left( \boldsymbol\Sigma_0^{-1}\boldsymbol\mu_0 + n \boldsymbol\Sigma^{-1} \mathbf{\bar{x \right),
\left(\boldsymbol\Sigma_0^{-1} + n\boldsymbol\Sigma^{-1}\right)^{-1} *\mathbf{\bar{x is the sample mean , mean was estimated from observations with total precision (sum of all individual precisions)\boldsymbol\Sigma_0^{-1} and with sample mean \boldsymbol\mu_0 , \mathcal{N}(\tilde{\mathbf{x\mid{\boldsymbol\mu_0}', {\boldsymbol\Sigma_0}' +\boldsymbol\Sigma) , - ,
Multivariate normal In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
with known precision matrix ''Λ'' , , ''μ'' (mean vector) , ,
Multivariate normal In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
, , \mathbf{\boldsymbol\mu}_0,\, \boldsymbol\Lambda_0 , , \left(\boldsymbol\Lambda_0 + n\boldsymbol\Lambda\right)^{-1}\left( \boldsymbol\Lambda_0\boldsymbol\mu_0 + n \boldsymbol\Lambda \mathbf{\bar{x \right),\, \left(\boldsymbol\Lambda_0 + n\boldsymbol\Lambda\right) *\mathbf{\bar{x is the sample mean , mean was estimated from observations with total precision (sum of all individual precisions)\boldsymbol\Lambda_0 and with sample mean \boldsymbol\mu_0 , \mathcal{N}\left(\tilde{\mathbf{x\mid{\boldsymbol\mu_0}', , \boldsymbol\mu,\frac{1}{\nu'-p+1}\boldsymbol\Psi'\right) , - ,
Multivariate normal In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
with known mean ''μ'' , , ''Λ'' (precision matrix) , , Wishart , , \nu ,\, \mathbf{V} , , n+\nu ,\, \left(\mathbf{V}^{-1} + \sum_{i=1}^n (\mathbf{x_i} - \boldsymbol\mu) (\mathbf{x_i} - \boldsymbol\mu)^T\right)^{-1} , covariance matrix was estimated from \nu observations with sum of pairwise deviation products \mathbf{V}^{-1} , t_{\nu'-p+1}\left(\tilde{\mathbf{x\mid\boldsymbol\mu,\frac{1}{\nu'-p+1}{\mathbf{V}'}^{-1}\right) , - ,
Multivariate normal In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
, , ''μ'' (mean vector) and ''Σ'' (covariance matrix) , , normal-inverse-Wishart , , \boldsymbol\mu_0 ,\, \kappa_0 ,\, \nu_0 ,\, \boldsymbol\Psi , , \frac{\kappa_0\boldsymbol\mu_0+n\mathbf{\bar{x}{\kappa_0+n} ,\, \kappa_0+n,\, \nu_0+n ,\,
\boldsymbol\Psi + \mathbf{C} + \frac{\kappa_0 n}{\kappa_0+n}(\mathbf{\bar{x-\boldsymbol\mu_0)(\mathbf{\bar{x-\boldsymbol\mu_0)^T * \mathbf{\bar{x is the sample mean *\mathbf{C} = \sum_{i=1}^n (\mathbf{x_i} - \mathbf{\bar{x) (\mathbf{x_i} - \mathbf{\bar{x)^T , mean was estimated from \kappa_0 observations with sample mean \boldsymbol\mu_0; covariance matrix was estimated from \nu_0 observations with sample mean \boldsymbol\mu_0 and with sum of pairwise deviation products \boldsymbol\Psi=\nu_0\boldsymbol\Sigma_0 , t_, {\boldsymbol\mu_0}',\frac{\kappa_0+n} ,\, \kappa_0+n,\, \nu_0+n ,\,
\left(\mathbf{V}^{-1} + \mathbf{C} + \frac{\kappa_0 n}{\kappa_0+n}(\mathbf{\bar{x-\boldsymbol\mu_0)(\mathbf{\bar{x-\boldsymbol\mu_0)^T\right)^{-1} * \mathbf{\bar{x is the sample mean *\mathbf{C} = \sum_{i=1}^n (\mathbf{x_i} - \mathbf{\bar{x) (\mathbf{x_i} - \mathbf{\bar{x)^T , mean was estimated from \kappa_0 observations with sample mean \boldsymbol\mu_0; covariance matrix was estimated from \nu_0 observations with sample mean \boldsymbol\mu_0 and with sum of pairwise deviation products \mathbf{V}^{-1} , t_\mid {\boldsymbol\mu_0}', \frac\! , \alpha observations with sum \beta of the
order of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic di ...
of each observation (i.e. the logarithm of the ratio of each observation to the minimum x_m) , , - ,
Weibull Weibull is a Swedish locational surname. The Weibull family share the same roots as the Danish / Norwegian noble family of Falsenbr>They originated from and were named after the village of Weiböl in Widstedts parish, Jutland, but settled in Skà ...

with known shape ''β'' , , ''θ'' (scale) , , Inverse gamma , , a, b\! , , a+n,\, b+\sum_{i=1}^n x_i^{\beta}\! , a observations with sum b of the ''βth power of each observation , , - ,
Log-normal In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
, colspan="6" , Same as for the normal distribution after applying the natural logarithm to the data for the posterior hyperparameters. Please refer to to see the details. , , - ,
Exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...
, , ''λ'' (rate) , ,
Gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
, , \alpha,\, \beta\! , , \alpha+n,\, \beta+\sum_{i=1}^n x_i\! , \alpha-1 observations that sum to \beta , \operatorname{Lomax}(\tilde{x}\mid\beta',\alpha')
(
Lomax distribution The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K.  ...
) , - ,
Gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...

with known shape ''α'', , ''β'' (rate) , ,
Gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
, , \alpha_0,\, \beta_0\!, , \alpha_0+n\alpha,\, \beta_0+\sum_{i=1}^n x_i\! , \alpha_0/\alpha observations with sum \beta_0 , \operatorname{CG}(\tilde{\mathbf{x\mid\alpha,{\alpha_0}',{\beta_0}')=\operatorname{\beta'}(\tilde{\mathbf{x, \alpha,{\alpha_0}',1,{\beta_0}') , - , Inverse Gamma
with known shape ''α'', , ''β'' (inverse scale) , ,
Gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
, , \alpha_0,\, \beta_0\!, , \alpha_0+n\alpha,\, \beta_0+\sum_{i=1}^n \frac{1}{x_i}\! , \alpha_0/\alpha observations with sum \beta_0 , , - ,
Gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...

with known rate ''β'', , ''α'' (shape) , \propto \frac{a^{\alpha-1} \beta^{\alpha c{\Gamma(\alpha)^b} , a,\, b,\, c\!, , a \prod_{i=1}^n x_i,\, b + n,\, c + n\! , b or c observations (b for estimating \alpha, c for estimating \beta) with product a , , - ,
Gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
, , ''α'' (shape), ''β'' (inverse scale) , , \propto \frac{p^{\alpha-1} e^{-\beta q{\Gamma(\alpha)^r \beta^{-\alpha s , , p,\, q,\, r,\, s \! , , p \prod_{i=1}^n x_i,\, q + \sum_{i=1}^n x_i,\, r + n,\, s + n \! , \alpha was estimated from r observations with product p; \beta was estimated from s observations with sum q , , - ,
Beta Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labiod ...
, , ''α'', ''β'' , , \propto \frac{\Gamma(\alpha+\beta)^k \, p^\alpha \, q^\beta}{\Gamma(\alpha)^k\,\Gamma(\beta)^k} , , p,\, q,\, k \! , , p \prod_{i=1}^n x_i,\, q \prod_{i=1}^n (1-x_i),\, k + n \! , \alpha and \beta were estimated from k observations with product p and product of the complements q , , -


See also

*
Beta-binomial distribution In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of B ...


Notes


References

{{Reflist , refs = {{cite web , last = Fink , first = Daniel , date = 1997 , title = A Compendium of Conjugate Priors , url=https://courses.physics.ucsd.edu/2018/Fall/physics210b/REFERENCES/conjugate_priors.pdf , citeseerx = 10.1.1.157.5540 , archive-url=https://web.archive.org/web/20090529203101/http://www.people.cornell.edu/pages/df36/CONJINTRnew%20TEX.pdf , archive-date=May 29, 2009 Bayesian statistics