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Conjugate prior

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Title: Conjugate prior  
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Subject: Normal distribution, Posterior predictive distribution, Gamma distribution, Bayesian inference, Normal-inverse-gamma distribution
Collection: Bayesian Statistics, Conjugate Prior Distributions
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Hypergeometric
with known total population size N
M (number of target members) Beta-binomial[4] n=N, \alpha,\, \beta\! \alpha + \sum_{i=1}^n x_i,\, \beta + \sum_{i=1}^nN_i - \sum_{i=1}^n x_i\! \alpha - 1 successes, \beta - 1 failures[note 1]
Geometric p0 (probability) Beta \alpha,\, \beta\! \alpha + n,\, \beta + \sum_{i=1}^n x_i\! \alpha - 1 experiments, \beta - 1 total failures[note 1]
Likelihood Model parameters Conjugate prior distribution Prior hyperparameters Posterior hyperparameters Interpretation of hyperparameters Posterior predictive[note 4]
Normal
with known variance σ2
μ (mean) Normal \mu_0,\, \sigma_0^2\! \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),
\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)[5]
Normal
with known precision τ
μ (mean) Normal \mu_0,\, \tau_0\! \left.\left(\tau_0 \mu_0 + \tau \sum_{i=1}^n x_i\right)\right/(\tau_0 + n \tau),\, \tau_0 + n \tau 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}|\mu_0', \frac{1}{\tau_0'} +\frac{1}{\tau}\right)[5]
Normal
with known mean μ
σ2 (variance) Inverse gamma \mathbf{\alpha,\, \beta} [note 5] \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 2\beta, where deviations are from known mean \mu) t_{2\alpha'}(\tilde{x}|\mu,\sigma^2 = \beta'/\alpha')[5]
Normal
with known mean μ
σ2 (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}')[5]
Normal
with known mean μ
τ (precision) Gamma \alpha,\, \beta\![note 3] \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 2\beta, where deviations are from known mean \mu) t_{2\alpha'}(\tilde{x}|\mu,\sigma^2 = \beta'/\alpha')[5]
Normal[note 6] μ and σ2
Assuming exchangeability
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 2\beta t_{2\alpha'}\left(\tilde{x}|\mu',\frac{\beta'(\nu'+1)}{\nu' \alpha'}\right)[5]
Normal μ and τ
Assuming exchangeability
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 2\beta t_{2\alpha'}\left(\tilde{x}|\mu',\frac{\beta'(\nu'+1)}{\alpha'\nu'}\right)[5]
Multivariate normal with known covariance matrix Σ μ (mean vector) Multivariate normal \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}}|{\boldsymbol\mu_0}', {\boldsymbol\Sigma_0}' +\boldsymbol\Sigma)[5]
Multivariate normal with known precision matrix Λ μ (mean vector) Multivariate normal \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 and with sample mean \boldsymbol\mu_0 \mathcal{N}\left(\tilde{\mathbf{x}}|{\boldsymbol\mu_0}', (|\boldsymbol\mu,\frac{1}{\nu'-p+1}\boldsymbol\Psi'\right)[5]
Multivariate normal 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}}|\boldsymbol\mu,\frac{1}{\nu'-p+1}{\mathbf{V}'}^{-1}\right)[5]
Multivariate normal 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_|{\boldsymbol\mu_0}',\frac}\! \alpha observations with sum \beta of the order of magnitude of each observation (i.e. the logarithm of the ratio of each observation to the minimum x_m)
Weibull
with known shape β
θ (scale) Inverse gamma[4] a, b\! a+n,\, b+\sum_{i=1}^n x_i^{\beta}\! a observations with sum b of the β'th power of each observation
Log-normal
with known precision τ
μ (mean) Normal[4] \mu_0,\, \tau_0\! \left.\left(\tau_0 \mu_0 + \tau \sum_{i=1}^n \ln x_i\right)\right/(\tau_0 + n \tau),\, \tau_0 + n \tau "mean" was estimated from observations with total precision (sum of all individual precisions)\tau_0 and with sample mean \mu_0
Log-normal
with known mean μ
τ (precision) Gamma[4] \alpha,\, \beta\![note 3] \alpha + \frac{n}{2},\, \beta + \frac{\sum_{i=1}^n (\ln x_i-\mu)^2}{2}\! precision was estimated from 2\alpha observations with sample variance \frac{\beta}{\alpha} (i.e. with sum of squared log deviations 2\beta — i.e. deviations between the logs of the data points and the "mean")
Exponential λ (rate) Gamma \alpha,\, \beta\! [note 3] \alpha+n,\, \beta+\sum_{i=1}^n x_i\! \alpha observations that sum to \beta \operatorname{Lomax}(\tilde{x}|\beta',\alpha')
(Lomax distribution)
Gamma
with known shape α
β (rate) Gamma \alpha_0,\, \beta_0\! \alpha_0+n\alpha,\, \beta_0+\sum_{i=1}^n x_i\! \alpha_0 observations with sum \beta_0 \operatorname{CG}(\tilde{\mathbf{x}}|\alpha,{\alpha_0}',{\beta_0}')=\operatorname{\beta'}(\tilde{\mathbf{x}}|\alpha,{\alpha_0}',1,{\beta_0}') [note 7]
Inverse Gamma
with known shape α
β (inverse scale) Gamma \alpha_0,\, \beta_0\! \alpha_0+n\alpha,\, \beta_0+\sum_{i=1}^n \frac{1}{x_i}\! \alpha_0 observations with sum \beta_0
Gamma
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 [4] 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

See also

Notes

  1. ^ a b c d e f g h 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 \frac{\alpha - 1}{\alpha + \beta - 2}, which corresponds to \alpha - 1 successes and \beta - 1 failures; but the mean is \frac{\alpha}{\alpha + \beta}, which corresponds to \alpha successes and \beta failures. The use of \alpha - 1 and \beta - 1 has the advantage that a uniform {\rm Beta}(1,1) prior corresponds to 0 successes and 0 failures, but the use of \alpha and \beta 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.
  2. ^ This is the posterior predictive distribution of a new data point \tilde{x} given the observed data points, with the parameters marginalized out. Variables with primes indicate the posterior values of the parameters.
  3. ^ a b c d β is rate or inverse scale. In parameterization of gamma distribution,θ = 1/β and k = α.
  4. ^ This is the posterior predictive distribution of a new data point \tilde{x} given the observed data points, with the parameters marginalized out. Variables with primes indicate the posterior values of the parameters. \mathcal{N} and t_n 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.
  5. ^ In terms of the inverse gamma, \beta is a scale parameter
  6. ^ 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.
  7. ^ \operatorname{CG}() is a compound gamma distribution; \operatorname{\beta'}() here is a generalized beta prime distribution.

References

  1. ^ Howard Raiffa and Robert Schlaifer. Applied Statistical Decision Theory. Division of Research, Graduate School of Business Administration, Harvard University, 1961.
  2. ^ Jeff Miller et al. Earliest Known Uses of Some of the Words of Mathematics, "conjugate prior distributions". Electronic document, revision of November 13, 2005, retrieved December 2, 2005.
  3. ^ For a catalog, see  
  4. ^ a b c d e Fink, D. (1997). "A Compendium of Conjugate Priors". DOE contract 95‑831 ((Caution: Unreliable source) In progress report: Beware of some errors in multivariate normal and models and Arethya's prior (see addendum)).  
  5. ^ a b c d e f g h i j k l m Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution" (PDF). 
P(s,f|q=x) = {s+f \choose s} x^s(1-x)^f,

If we then sample this random variable and get s successes and f failures, we have

In this context, \alpha and \beta are called hyperparameters (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.)

where \alpha and \beta are chosen to reflect any existing belief or information (\alpha = 1 and \beta = 1 would give a uniform distribution) and Β(\alpha\beta) is the Beta function acting as a normalising constant.

p(q) = {q^{\alpha-1}(1-q)^{\beta-1} \over \Beta(\alpha,\beta)}

In fact, the usual conjugate prior is the beta distribution with parameters (\alpha, \beta):

for some constants a and b. 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 a and b, but never of q.

f(q) \propto q^a (1-q)^b

Expressed as a function of q, this has the form

p(x) = {n \choose x}q^x (1-q)^{n-x}

The form of the conjugate prior can generally be determined by inspection of the 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 trials 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

Example

Contents

  • Example 1
  • Pseudo-observations 2
  • Interpretations 3
    • Analogy with eigenfunctions 3.1
    • Dynamical system 3.2
  • Table of conjugate distributions 4
    • Discrete distributions 4.1
    • Continuous distributions 4.2
  • See also 5
  • Notes 6
  • References 7

All members of the exponential family have conjugate priors.[3]

A conjugate prior is an algebraic convenience, giving a closed-form expression 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.

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.

p(\theta|x) = \frac{p(x|\theta) \, p(\theta)} {\int p(x|\theta') \, p(\theta') \, d\theta'}. \!

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 \theta \mapsto p(x\mid\theta)\! and prior p( \theta )\!, normalized (divided) by the probability of the data p( x )\!:

[2]

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