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| {{Bayesian statistics}}
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| In [[Bayesian probability]] theory, if the [[posterior probability|posterior distributions]] ''p''(θ|''x'') are in the same family as the [[prior probability distribution]] ''p''(θ), the prior and posterior are then called '''conjugate distributions,''' and the prior is called a '''conjugate prior''' for the [[likelihood function]]. For example, the [[Normal distribution|Gaussian]] family is conjugate to itself (or ''self-conjugate'') with respect to a Gaussian likelihood function: if the likelihood function is Gaussian, choosing a Gaussian prior over the mean will ensure that the posterior distribution is also Gaussian. This means that the Gaussian distribution is a conjugate prior for the likelihood which is also Gaussian. The concept, as well as the term "conjugate prior", were introduced by [[Howard Raiffa]] and [[Robert Schlaifer]] in their work on [[Bayesian decision theory]].<ref name="raiffa_schlaifer">[[Howard Raiffa]] and [[Robert Schlaifer]]. ''Applied Statistical Decision Theory''. Division of Research, Graduate School of Business Administration, Harvard University, 1961.</ref> A similar concept had been discovered independently by [[George Alfred Barnard]].<ref name="miller">Jeff Miller et al. [http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics], [http://jeff560.tripod.com/c.html "conjugate prior distributions"]. Electronic document, revision of November 13, 2005, retrieved December 2, 2005.</ref>
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| Consider the general problem of inferring a distribution for a parameter θ given some datum or data ''x''. From [[Bayes' theorem]], the posterior distribution is equal to the product of the likelihood function <math>\theta \mapsto p(x\mid\theta)\!</math> and prior <math>p( \theta )\!</math>, normalized (divided) by the probability of the data <math>p( x )\!</math>:
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| :<math> p(\theta|x) = \frac{p(x|\theta) \, p(\theta)}
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| {\int p(x|\theta') \, p(\theta') \, d\theta'}. \!</math>
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| Let the likelihood function be considered fixed; the likelihood function is usually well-determined from a statement of the data-generating process. It is clear that different choices of the prior distribution ''p''(θ) may make the integral more or less difficult to calculate, and the product ''p''(''x''|θ) × ''p''(θ) may take one algebraic form or another. For certain choices of the prior, the posterior has the same algebraic form as the prior (generally with different parameter values). Such a choice is a ''conjugate prior''.
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| A conjugate prior is an algebraic convenience, giving a [[closed-form expression]]
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| for the posterior: otherwise a difficult [[numerical integration]] may be necessary. Further, conjugate priors may give intuition, by more transparently showing how a likelihood function updates a prior distribution. | |
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| All members of the [[exponential family]] have conjugate priors. See Gelman et al.<ref name="gelman_et_al">[[Andrew Gelman]], John B. Carlin, Hal S. Stern, and Donald B. Rubin. ''Bayesian Data Analysis'', 2nd edition. CRC Press, 2003. ISBN 1-58488-388-X.</ref> for a catalog.
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| ==Example==
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| The form of the conjugate prior can generally be determined by inspection of the [[probability density function|probability density]] or [[probability mass function]] of a distribution. For example, consider a [[random variable]] which consists of the number of successes in ''n'' [[Bernoulli trial]]s with unknown probability of success ''q'' in [0,1]. This random variable will follow the [[binomial distribution]], with a probability mass function of the form | |
| :<math>p(x) = {n \choose x}q^x (1-q)^{n-x}</math>
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| Expressed as a function of <math>q</math>, this has the form
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| :<math>f(q) \propto q^a (1-q)^b</math> | |
| for some constants <math>a</math> and <math>b</math>. Generally, this functional form will have an additional multiplicative factor (the [[normalizing constant]]) ensuring that the function is a [[probability distribution]], i.e. the integral over the entire range is 1. This factor will often be a function of <math>a</math> and <math>b</math>, but never of <math>q</math>.
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| In fact, the usual conjugate prior is the [[beta distribution]] with parameters (<math>\alpha</math>, <math>\beta</math>):
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| :<math>p(q) = {q^{\alpha-1}(1-q)^{\beta-1} \over \Beta(\alpha,\beta)}</math>
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| where <math>\alpha</math> and <math>\beta</math> are chosen to reflect any existing belief or information (<math>\alpha</math> = 1 and <math>\beta</math> = 1 would give a [[uniform distribution (continuous)|uniform distribution]]) and ''Β''(<math>\alpha</math>, <math>\beta</math>) is the [[Beta function]] acting as a [[normalising constant]].
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| In this context, <math>\alpha</math> and <math>\beta</math> are called ''[[hyperparameter]]s'' (parameters of the prior), to distinguish them from parameters of the underlying model (here ''q''). It is a typical characteristic of conjugate priors 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 this means that 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]], and consider also the [[Wishart distribution]], conjugate prior of the [[covariance matrix]] of a [[multivariate normal distribution]], for an example where a large dimensionality is involved.)
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| If we then sample this random variable and get ''s'' successes and ''f'' failures, we have
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| :<math>P(s,f|q=x) = {s+f \choose s} x^s(1-x)^f, </math>
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| :<math>\begin{align} P(q=x|s,f) & = {{{s+f \choose s} x^{s+\alpha-1}(1-x)^{f+\beta-1} / \Beta(\alpha,\beta)} \over \int_{y=0}^1 \left({s+f \choose s} y^{s+\alpha-1}(1-y)^{f+\beta-1} / \Beta(\alpha,\beta)\right) dy} \\
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| & = {x^{s+\alpha-1}(1-x)^{f+\beta-1} \over \Beta(s+\alpha,f+\beta)}, \\
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| \end{align}</math>
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| which is another Beta distribution with parameters (<math>\alpha</math> + ''s'', <math>\beta</math> + ''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.
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| ==Pseudo-observations==
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| It is often useful to think of the hyperparameters of a conjugate prior distribution as corresponding to having observed a certain number of ''pseudo-observations'' with properties specified by the parameters. For example, the values <math>\alpha</math> and <math>\beta</math> of a [[beta distribution]] can be thought of as corresponding to <math>\alpha-1</math> successes and <math>\beta-1</math> failures if the posterior mode is used to choose an optimal parameter setting, or <math>\alpha</math> successes and <math>\beta</math> 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 both in providing an intuition behind the often messy update equations, as well as to help choose reasonable hyperparameters for a prior.
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| == Interpretations ==
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| === Analogy with eigenfunctions ===
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| Conjugate priors are analogous to [[eigenfunctions]] in [[operator theory]], 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.
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| In both eigenfunctions and conjugate priors, there is a ''finite dimensional'' space which is preserved by the operator: 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).
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| However, the processes are only analogous, not identical:
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| conditioning is not linear, as the space of distributions is not closed under [[linear combination]], only [[convex combination]], and the posterior is only of the same ''form'' as the prior, not a scalar multiple.
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| Just as one can easily analyze how a linear combination of eigenfunctions evolves under application of an operator (because, with respect to these functions, the operator is [[diagonalized]]), one can easily analyze how a convex combination of conjugate priors evolves under conditioning; this is called using a ''[[hyperprior]],'' and corresponds to using a [[mixture density]] of conjugate priors, rather than a single conjugate prior.
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| === Dynamical system ===
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| One can think of conditioning on conjugate priors as defining a kind of (discrete time) [[dynamical system]]: 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 inference, this is not simply dependent on time, but rather on data over time. For related approaches, see [[Recursive Bayesian estimation]] and [[Data assimilation]].
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| == Table of conjugate distributions ==
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| Let ''n'' denote the number of observations.
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| If the likelihood function belongs to the [[exponential family]], then a conjugate prior exists, often also in the exponential family; see [[Exponential family#Bayesian estimation: conjugate distributions|Exponential family: Conjugate distributions]].
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| ===Discrete distributions===
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| {| class="wikitable"
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| ! Likelihood !! Model parameters !! Conjugate prior distribution !! Prior hyperparameters !! Posterior hyperparameters !! Interpretation of hyperparameters<ref group=note name="beta-interp"/> !! Posterior predictive<ref group=note name=postpred/>
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| | [[Bernoulli distribution|Bernoulli]] || ''p'' (probability) || [[Beta distribution|Beta]] || <math>\alpha,\, \beta\!</math> || <math>\alpha + \sum_{i=1}^n x_i,\, \beta + n - \sum_{i=1}^n x_i\!</math>
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| | <math>\alpha - 1</math> successes, <math>\beta - 1</math> failures<ref group=note name="beta-interp"/>
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| | <math>p(\tilde{x}=1) = \frac{\alpha'}{\alpha'+\beta'}</math>
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| | [[binomial distribution|Binomial]] || ''p'' (probability) || [[Beta distribution|Beta]] || <math>\alpha,\, \beta\!</math> || <math>\alpha + \sum_{i=1}^n x_i,\, \beta + \sum_{i=1}^nN_i - \sum_{i=1}^n x_i\!</math>
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| | <math>\alpha - 1</math> successes, <math>\beta - 1</math> failures<ref group=note name="beta-interp"/>
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| | <math>\operatorname{BetaBin}(\tilde{x}|\alpha',\beta')</math><br />([[beta-binomial distribution|beta-binomial]])
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| | [[negative binomial distribution|Negative Binomial]]<br />with known failure number ''r'' || ''p'' (probability) || [[Beta distribution|Beta]] || <math>\alpha,\, \beta\!</math> || <math>\alpha + \sum_{i=1}^n x_i,\, \beta + rn\!</math>
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| | <math>\alpha - 1</math> total successes, <math>\beta - 1</math> failures<ref group=note name="beta-interp"/> (i.e. <math>\frac{\beta - 1}{r}</math> experiments, assuming <math>r</math> stays fixed)
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| | [[Poisson distribution|Poisson]] || ''λ'' (rate) || [[Gamma distribution|Gamma]] || <math>k,\, \theta\!</math> || <math>k+ \sum_{i=1}^n x_i,\ \frac {\theta} {n \theta + 1}\!</math>
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| | <math>k</math> total occurrences in <math>1/\theta</math> intervals
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| | <math>\operatorname{NB}(\tilde{x}|k', \frac{\theta'}{1+\theta'})</math><br />([[negative binomial distribution|negative binomial]])
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| | [[Poisson distribution|Poisson]] || ''λ'' (rate) || [[Gamma distribution|Gamma]] || <math>\alpha,\, \beta\!</math> <ref group=note name="beta_rate"/>|| <math>\alpha + \sum_{i=1}^n x_i ,\ \beta + n\!</math>
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| | <math>\alpha</math> total occurrences in <math>\beta</math> intervals
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| | <math>\operatorname{NB}(\tilde{x}|\alpha', \frac{1}{1+\beta'})</math><br />([[negative binomial distribution|negative binomial]])
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| | [[categorical distribution|Categorical]] || '''''p''''' (probability vector), ''k'' (number of categories, i.e. size of '''''p''''') || [[Dirichlet distribution|Dirichlet]] || <math>\boldsymbol\alpha\!</math> || <math>\boldsymbol\alpha+(c_1,\ldots,c_k),</math> where <math>c_i</math> is the number of observations in category ''i''
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| | <math>\alpha_i - 1</math> occurrences of category <math>i</math><ref group=note name="beta-interp"/>
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| | <math>p(\tilde{x}=i) = \frac{{\alpha_i}'}{\sum_i {\alpha_i}'}</math><br/>
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| <math>= \frac{\alpha_i + c_i}{\sum_i \alpha_i + n}</math>
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| | [[multinomial distribution|Multinomial]] || '''''p''''' (probability vector), ''k'' (number of categories, i.e. size of '''''p''''') || [[Dirichlet distribution|Dirichlet]] || <math>\boldsymbol\alpha\!</math> || <math>\boldsymbol\alpha+\sum_{i=1}^n\mathbf{x}_i\!</math>
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| | <math>\alpha_i - 1</math> occurrences of category <math>i</math><ref group=note name="beta-interp"/>
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| | <math>\operatorname{DirMult}(\tilde{\mathbf{x}}|\boldsymbol\alpha')</math><br />([[Dirichlet-multinomial distribution|Dirichlet-multinomial]])
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| | [[hypergeometric distribution|Hypergeometric]]<br />with known total population size ''N'' || ''M'' (number of target members) || [[Beta-binomial distribution|Beta-binomial]]<ref name="Fink"/> || <math>n=N, \alpha,\, \beta\!</math> || <math>\alpha + \sum_{i=1}^n x_i,\, \beta + \sum_{i=1}^nN_i - \sum_{i=1}^n x_i\!</math>
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| | <math>\alpha - 1</math> successes, <math>\beta - 1</math> failures<ref group=note name="beta-interp"/>
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| | [[geometric distribution|Geometric]] || ''p<sub>0</sub>'' (probability) || [[Beta distribution|Beta]] || <math>\alpha,\, \beta\!</math> || <math>\alpha + n,\, \beta + \sum_{i=1}^n x_i\!</math>
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| | <math>\alpha - 1</math> experiments, <math>\beta - 1</math> total failures<ref group=note name="beta-interp"/>
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| |}
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| ===Continuous distributions===
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| '''Note''': In all cases below, the data is assumed to consist of ''n'' points <math>x_1,\ldots,x_n</math> (which will be [[random vector]]s in the multivariate cases).
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| {| class="wikitable"
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| ! Likelihood !! Model parameters !! Conjugate prior distribution !! Prior hyperparameters !! Posterior hyperparameters!!Interpretation of hyperparameters!!Posterior predictive<ref group=note name=ppredNt/>
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| | [[normal distribution|Normal]]<br>with known variance ''σ''<sup>2</sup> || ''μ'' (mean) || [[normal distribution|Normal]] || <math>\mu_0,\, \sigma_0^2\!</math> || <math>\left.\left(\frac{\mu_0}{\sigma_0^2} + \frac{\sum_{i=1}^n x_i}{\sigma^2}\right)\right/\left(\frac{1}{\sigma_0^2} + \frac{n}{\sigma^2}\right),</math><br/><math> \left(\frac{1}{\sigma_0^2} + \frac{n}{\sigma^2}\right)^{-1}</math>
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| | mean was estimated from observations with total precision (sum of all individual precisions)<math>1/\sigma_0^2</math> and with sample mean <math>\mu_0</math>
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| | <math>\mathcal{N}(\tilde{x}|\mu_0', {\sigma_0^2}' +\sigma^2)</math><ref name="murphy">Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [http://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf]</ref>
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| | [[normal distribution|Normal]]<br>with known precision ''τ'' || ''μ'' (mean) || [[normal distribution|Normal]] || <math>\mu_0,\, \tau_0\!</math> || <math> \left.\left(\tau_0 \mu_0 + \tau \sum_{i=1}^n x_i\right)\right/(\tau_0 + n \tau),\, \tau_0 + n \tau</math>
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| | mean was estimated from observations with total precision (sum of all individual precisions)<math>\tau_0</math> and with sample mean <math>\mu_0</math>
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| | <math>\mathcal{N}\left(\tilde{x}|\mu_0', \frac{1}{\tau_0'} +\frac{1}{\tau}\right)</math><ref name="murphy"/>
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| | [[Normal distribution|Normal]]<br>with known mean ''μ'' || ''σ''<sup>2</sup> (variance) || [[Inverse gamma distribution|Inverse gamma]] || <math> \mathbf{\alpha,\, \beta} </math> <ref group=note name="beta_scale"/> || <math> \mathbf{\alpha}+\frac{n}{2},\, \mathbf{\beta} + \frac{\sum_{i=1}^n{(x_i-\mu)^2}}{2} </math>
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| | variance was estimated from <math>2\alpha</math> observations with sample variance <math>\beta/\alpha</math> (i.e. with sum of [[squared deviations]] <math>2\beta</math>, where deviations are from known mean <math>\mu</math>)
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| | <math>t_{2\alpha'}(\tilde{x}|\mu,\sigma^2 = \beta'/\alpha')</math><ref name="murphy"/>
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| | [[normal distribution|Normal]]<br>with known mean ''μ'' || ''σ''<sup>2</sup> (variance) || [[Scaled inverse chi-squared distribution|Scaled inverse chi-squared]] || <math>\nu,\, \sigma_0^2\!</math> || <math>\nu+n,\, \frac{\nu\sigma_0^2 + \sum_{i=1}^n (x_i-\mu)^2}{\nu+n}\!</math>
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| | variance was estimated from <math>\nu</math> observations with sample variance <math>\sigma_0^2</math>
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| | <math>t_{\nu'}(\tilde{x}|\mu,{\sigma_0^2}')</math><ref name="murphy"/>
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| | [[normal distribution|Normal]]<br>with known mean ''μ'' || ''τ'' (precision) || [[Gamma distribution|Gamma]] || <math>\alpha,\, \beta\!</math><ref group=note name="beta_rate"/> || <math>\alpha + \frac{n}{2},\, \beta + \frac{\sum_{i=1}^n (x_i-\mu)^2}{2}\!</math>
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| | precision was estimated from <math>2\alpha</math> observations with sample variance <math>\beta/\alpha</math> (i.e. with sum of [[squared deviations]] <math>2\beta</math>, where deviations are from known mean <math>\mu</math>)
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| | <math>t_{2\alpha'}(\tilde{x}|\mu,\sigma^2 = \beta'/\alpha')</math><ref name="murphy"/>
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| | [[Normal distribution|Normal]]<ref group=note>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 distribution|generalized inverse Gaussian]] as conjugate mixing distribution.</ref> || ''μ'' and ''σ<sup>2</sup>''<br>Assuming [[Exchangeable random variables|exchangeability]]|| [[Normal-inverse gamma distribution|Normal-inverse gamma]]
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| | <math> \mu_0 ,\, \nu ,\, \alpha ,\, \beta</math> || <math>\frac{\nu\mu_0+n\bar{x}}{\nu+n} ,\, \nu+n,\, \alpha+\frac{n}{2} ,\, </math><br/><math>
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| \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} </math>
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| *<math> \bar{x} </math> is the sample mean
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| | mean was estimated from <math>\nu</math> observations with sample mean <math>\mu_0</math>; variance was estimated from <math>2\alpha</math> observations with sample mean <math>\mu_0</math> and sum of [[squared deviations]] <math>2\beta</math>
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| | <math>t_{2\alpha'}\left(\tilde{x}|\mu',\frac{\beta'(\nu'+1)}{\alpha'\nu'}\right)</math><ref name="murphy"/>
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| | [[Normal distribution|Normal]] || ''μ'' and ''τ''<br>Assuming [[Exchangeable random variables|exchangeability]]|| [[Normal-gamma distribution|Normal-gamma]]
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| | <math> \mu_0 ,\, \nu ,\, \alpha ,\, \beta</math> || <math>\frac{\nu\mu_0+n\bar{x}}{\nu+n} ,\, \nu+n,\, \alpha+\frac{n}{2} ,\, </math><br/><math>
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| \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} </math>
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| *<math> \bar{x} </math> is the sample mean
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| | mean was estimated from <math>\nu</math> observations with sample mean <math>\mu_0</math>, and precision was estimated from <math>2\alpha</math> observations with sample mean <math>\mu_0</math> and sum of [[squared deviations]] <math>2\beta</math>
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| | <math>t_{2\alpha'}\left(\tilde{x}|\mu',\frac{\beta'(\nu'+1)}{\alpha'\nu'}\right)</math><ref name="murphy"/>
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| | [[multivariate normal distribution|Multivariate normal]] with known covariance matrix '''''Σ''''' || '''''μ''''' (mean vector) || [[multivariate normal distribution|Multivariate normal]] || <math>\boldsymbol{\boldsymbol\mu}_0,\, \boldsymbol\Sigma_0</math> || <math>\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),</math><br/><math>\left(\boldsymbol\Sigma_0^{-1} + n\boldsymbol\Sigma^{-1}\right)^{-1}</math>
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| *<math>\mathbf{\bar{x}}</math> is the sample mean
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| | mean was estimated from observations with total precision (sum of all individual precisions)<math>\boldsymbol\Sigma_0^{-1}</math> and with sample mean <math>\boldsymbol\mu_0</math>
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| | <math>\mathcal{N}(\tilde{\mathbf{x}}|{\boldsymbol\mu_0}', {\boldsymbol\Sigma_0}' +\boldsymbol\Sigma)</math><ref name="murphy">Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [http://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf] Note that <math>\mathcal{N}()</math> is a [[normal distribution]] or [[multivariate normal distribution]]; <math>t_n()</math> is a [[Student's t-distribution]] or [[multivariate t-distribution]].</ref>
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| | [[multivariate normal distribution|Multivariate normal]] with known precision matrix '''''Λ''''' || '''''μ''''' (mean vector) || [[multivariate normal distribution|Multivariate normal]] || <math>\mathbf{\boldsymbol\mu}_0,\, \boldsymbol\Lambda_0</math> || <math>\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)</math>
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| *<math>\mathbf{\bar{x}}</math> is the sample mean
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| | mean was estimated from observations with total precision (sum of all individual precisions)<math>\boldsymbol\Lambda</math> and with sample mean <math>\boldsymbol\mu_0</math>
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| | <math>\mathcal{N}\left(\tilde{\mathbf{x}}|{\boldsymbol\mu_0}', ({{\boldsymbol\Lambda_0}'}^{-1} + \boldsymbol\Lambda^{-1})^{-1}\right)</math><ref name="murphy"/>
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| | [[multivariate normal distribution|Multivariate normal]] with known mean '''''μ''''' || '''''Σ''''' (covariance matrix) || [[Inverse-Wishart distribution|Inverse-Wishart]] || <math>\nu ,\, \boldsymbol\Psi</math> || <math>n+\nu ,\, \boldsymbol\Psi + \sum_{i=1}^n (\mathbf{x_i} - \boldsymbol\mu) (\mathbf{x_i} - \boldsymbol\mu)^T </math>
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| | covariance matrix was estimated from <math>\nu</math> observations with sum of pairwise deviation products <math>\boldsymbol\Psi</math>
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| | <math>t_{\nu'-p+1}\left(\tilde{\mathbf{x}}|\boldsymbol\mu,\frac{1}{\nu'-p+1}\boldsymbol\Psi'\right)</math><ref name="murphy"/>
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| | [[multivariate normal distribution|Multivariate normal]] with known mean '''''μ''''' || '''''Λ''''' (precision matrix) || [[Wishart distribution|Wishart]] || <math>\nu ,\, \mathbf{V}</math> || <math>n+\nu ,\, \left(\mathbf{V}^{-1} + \sum_{i=1}^n (\mathbf{x_i} - \boldsymbol\mu) (\mathbf{x_i} - \boldsymbol\mu)^T\right)^{-1} </math>
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| | covariance matrix was estimated from <math>\nu</math> observations with sum of pairwise deviation products <math>\mathbf{V}^{-1}</math>
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| | <math>t_{\nu'-p+1}\left(\tilde{\mathbf{x}}|\boldsymbol\mu,\frac{1}{\nu'-p+1}{\mathbf{V}'}^{-1}\right)</math><ref name="murphy"/>
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| | [[multivariate normal distribution|Multivariate normal]] || '''''μ''''' (mean vector) and '''''Σ''''' (covariance matrix) || [[normal-inverse-Wishart distribution|normal-inverse-Wishart]] || <math>\boldsymbol\mu_0 ,\, \kappa_0 ,\, \nu_0 ,\, \boldsymbol\Psi</math> || <math>\frac{\kappa_0\boldsymbol\mu_0+n\mathbf{\bar{x}}}{\kappa_0+n} ,\, \kappa_0+n,\, \nu_0+n ,\,</math><br/><math> \boldsymbol\Psi + \mathbf{C} + \frac{\kappa_0 n}{\kappa_0+n}(\mathbf{\bar{x}}-\boldsymbol\mu_0)(\mathbf{\bar{x}}-\boldsymbol\mu_0)^T </math>
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| *<math> \mathbf{\bar{x}} </math> is the sample mean
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| *<math>\mathbf{C} = \sum_{i=1}^n (\mathbf{x_i} - \mathbf{\bar{x}}) (\mathbf{x_i} - \mathbf{\bar{x}})^T</math>
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| | mean was estimated from <math>\kappa_0</math> observations with sample mean <math>\boldsymbol\mu_0</math>; covariance matrix was estimated from <math>\nu_0</math> observations with sample mean <math>\boldsymbol\mu_0</math> and with sum of pairwise deviation products <math>\boldsymbol\Psi</math>
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| | <math>t_{{\nu_0}'-p+1}\left(\tilde{\mathbf{x}}|{\boldsymbol\mu_0}',\frac{{\kappa_0}'+1}{{\kappa_0}'({\nu_0}'-p+1)}\boldsymbol\Psi'\right)</math><ref name="murphy"/>
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| | [[multivariate normal distribution|Multivariate normal]] || '''''μ''''' (mean vector) and '''''Λ''''' (precision matrix)|| [[normal-Wishart distribution|normal-Wishart]] || <math>\boldsymbol\mu_0 ,\, \kappa_0 ,\, \nu_0 ,\, \mathbf{V}</math> || <math>\frac{\kappa_0\boldsymbol\mu_0+n\mathbf{\bar{x}}}{\kappa_0+n} ,\, \kappa_0+n,\, \nu_0+n ,\,</math><br/><math> \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} </math>
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| *<math> \mathbf{\bar{x}} </math> is the sample mean
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| *<math>\mathbf{C} = \sum_{i=1}^n (\mathbf{x_i} - \mathbf{\bar{x}}) (\mathbf{x_i} - \mathbf{\bar{x}})^T</math>
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| | mean was estimated from <math>\kappa_0</math> observations with sample mean <math>\boldsymbol\mu_0</math>; covariance matrix was estimated from <math>\nu_0</math> observations with sample mean <math>\boldsymbol\mu_0</math> and with sum of pairwise deviation products <math>\mathbf{V}^{-1}</math>
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| | <math>t_{{\nu_0}'-p+1}\left(\tilde{\mathbf{x}}|{\boldsymbol\mu_0}',\frac{{\kappa_0}'+1}{{\kappa_0}'({\nu_0}'-p+1)}{\mathbf{V}'}^{-1}\right)</math><ref name="murphy"/>
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| | [[Uniform distribution (continuous)|Uniform]] || <math> U(0,\theta)\!</math> || [[Pareto distribution|Pareto]] || <math> x_{m},\, k\!</math> || <math> \max\{\,x_1,\ldots,x_n,x_\mathrm{m}\},\, k+n\!</math>
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| | <math>k</math> observations with maximum value <math>x_m</math>
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| | [[Pareto distribution|Pareto]] <br/>with known minimum ''x''<sub>''m''</sub> || ''k'' (shape) || [[Gamma distribution|Gamma]] || <math>\alpha,\, \beta\!</math> || <math>\alpha+n,\, \beta+\sum_{i=1}^n \ln\frac{x_i}{x_{\mathrm{m}}}\!</math>
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| | <math>\alpha</math> observations with sum <math>\beta</math> of the [[order of magnitude]] of each observation (i.e. the logarithm of the ratio of each observation to the minimum <math>x_m</math>)
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| | [[Weibull distribution|Weibull]] <br/>with known shape ''β'' || ''θ'' (scale) || [[inverse-gamma distribution|Inverse gamma]]<ref name="Fink"/> || <math>a, b\!</math> || <math>a+n,\, b+\sum_{i=1}^n x_i^{\beta}\!</math>
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| | <math>a</math> observations with sum <math>b</math> of the ''β'''th power of each observation
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| | [[log-normal distribution|Log-normal]]<br>with known precision ''τ'' || ''μ'' (mean) || [[normal distribution|Normal]]<ref name="Fink"/> || <math>\mu_0,\, \tau_0\!</math> || <math> \left.\left(\tau_0 \mu_0 + \tau \sum_{i=1}^n \ln x_i\right)\right/(\tau_0 + n \tau),\, \tau_0 + n \tau</math>
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| | "mean" was estimated from observations with total precision (sum of all individual precisions)<math>\tau_0</math> and with sample mean <math>\mu_0</math>
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| | [[log-normal distribution|Log-normal]]<br>with known mean ''μ'' || ''τ'' (precision) || [[Gamma distribution|Gamma]]<ref name="Fink"/> || <math>\alpha,\, \beta\!</math><ref group=note name="beta_rate"/> || <math>\alpha + \frac{n}{2},\, \beta + \frac{\sum_{i=1}^n (\ln x_i-\mu)^2}{2}\!</math>
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| | precision was estimated from <math>2\alpha</math> observations with sample variance <math>\frac{\beta}{\alpha}</math> (i.e. with sum of [[squared deviations|squared log deviations]] <math>2\beta</math> — i.e. deviations between the logs of the data points and the "mean")
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| | [[exponential distribution|Exponential]] || ''λ'' (rate) || [[Gamma distribution|Gamma]] || <math>\alpha,\, \beta\!</math> <ref group=note name="beta_rate"/> || <math>\alpha+n,\, \beta+\sum_{i=1}^n x_i\!</math>
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| | <math>\alpha</math> observations that sum to <math>\beta</math>
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| | <math>\operatorname{Lomax}(\tilde{x}|\beta',\alpha')</math><br />([[Lomax distribution]])
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| | [[Gamma Distribution|Gamma]] <br>with known shape ''α''|| ''β'' (rate) || [[Gamma Distribution|Gamma]] || <math>\alpha_0,\, \beta_0\!</math>||<math>\alpha_0+n\alpha,\, \beta_0+\sum_{i=1}^n x_i\!</math>
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| | <math>\alpha_0</math> observations with sum <math>\beta_0</math>
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| | <math>\operatorname{CG}(\tilde{\mathbf{x}}|\alpha,{\alpha_0}',{\beta_0}')=\operatorname{\beta'}(\tilde{\mathbf{x}}|\alpha,{\alpha_0}',1,{\beta_0}')</math> <ref group=note name=CG/>
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| | [[Inverse-gamma distribution|Inverse Gamma]] <br>with known shape ''α''|| ''β'' (inverse scale) || [[Gamma Distribution|Gamma]] || <math>\alpha_0,\, \beta_0\!</math>||<math>\alpha_0+n\alpha,\, \beta_0+\sum_{i=1}^n \frac{1}{x_i}\!</math>
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| | <math>\alpha_0</math> observations with sum <math>\beta_0</math>
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| | [[Gamma Distribution|Gamma]] <br>with known rate ''β''|| ''α'' (shape)
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| | <math>\propto \frac{a^{\alpha-1} \beta^{\alpha c}}{\Gamma(\alpha)^b}</math>
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| | <math>a,\, b,\, c\!</math>||<math>a \prod_{i=1}^n x_i,\, b + n,\, c + n\!</math>
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| | <math>b</math> or <math>c</math> observations (<math>b</math> for estimating <math>\alpha</math>, <math>c</math> for estimating <math>\beta</math>) with product <math>a</math>
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| | [[Gamma Distribution|Gamma]] <ref name="Fink"/>|| ''α'' (shape), ''β'' (inverse scale) || <math>\propto \frac{p^{\alpha-1} e^{-\beta q}}{\Gamma(\alpha)^r \beta^{-\alpha s}}</math> || <math>p,\, q,\, r,\, s \!</math> || <math>p \prod_{i=1}^n x_i,\, q + \sum_{i=1}^n x_i,\, r + n,\, s + n \!</math>
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| | <math>\alpha</math> was estimated from <math>r</math> observations with product <math>p</math>; <math>\beta</math> was estimated from <math>s</math> observations with sum <math>q</math>
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| |}
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| == See also==
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| [[Beta-binomial distribution]]
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| ==Notes==
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| {{Reflist | group=note| refs=
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| <ref group=note name="beta-interp">The exact interpretation of the parameters of a [[beta distribution]] in terms of number of successes and failures depends on what function is used to extract a point estimate from the distribution. The mode of a beta distribution is <math>\frac{\alpha - 1}{\alpha + \beta - 2},</math> which corresponds to <math>\alpha - 1</math> successes and <math>\beta - 1</math> failures; but the mean is <math>\frac{\alpha}{\alpha + \beta},</math> which corresponds to <math>\alpha</math> successes and <math>\beta</math> failures. The use of <math>\alpha - 1</math> and <math>\beta - 1</math> has the advantage that a uniform <math>{\rm Beta}(1,1)</math> prior corresponds to 0 successes and 0 failures, but the use of <math>\alpha</math> and <math>\beta</math> is somewhat more convenient mathematically and also corresponds well with the fact that Bayesians generally prefer to use the posterior mean rather than the posterior mode as a point estimate. The same issues apply to the [[Dirichlet distribution]].</ref>
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| <ref group=note name=postpred>This is the [[posterior predictive distribution]] of a new data point <math>\tilde{x}</math> given the observed data points, with the parameters [[marginal distribution|marginalized out]]. Variables with primes indicate the posterior values of the parameters.</ref>
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| <ref group=note name=ppredNt>This is the [[posterior predictive distribution]] of a new data point <math>\tilde{x}</math> given the observed data points, with the parameters [[marginal distribution|marginalized out]]. Variables with primes indicate the posterior values of the parameters. <math>\mathcal{N}</math> and <math>t_n</math> refer to the [[normal distribution]] and [[Student's t-distribution]], respectively, or to the [[multivariate normal distribution]] and [[multivariate t-distribution]] in the multivariate cases.</ref>
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| <ref group=note name="beta_rate">''β'' is rate or inverse scale. In parameterization of [[gamma distribution]],''θ'' = 1/''β'' and ''k'' = ''α''.</ref>
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| <ref group=note name="beta_scale">In terms of the [[inverse gamma distribution|inverse gamma]], <math>\beta</math> is a [[scale parameter]]</ref>
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| <ref group=note name=CG><math>\operatorname{CG}()</math> is a [[compound gamma distribution]]; <math>\operatorname{\beta'}()</math> here is a [[generalized beta prime distribution]].</ref>
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| }}
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| == References ==
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| {{Reflist|refs=
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| <ref name="Fink">{{cite paper | last = Fink | first = D. | year = 1997 | title = A Compendium of Conjugate Priors | id = {{citeseerx|10.1.1.157.5540}} | format = (Caution: Unreliable source) In progress report: Beware of some errors in multivariate normal and models and Arethya's prior (see [http://finmathblog.blogspot.com/2013/06/a-little-addendum-to-compendium-of.html addendum])| journal = DOE contract 95‑831 }}</ref>
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| }}
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| [[Category:Bayesian statistics]]
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| [[Category:Conjugate prior distributions]]
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