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# Conjugate prior

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 Title: Conjugate prior Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Conjugate prior

In

\begin{align} P(q=x|s,f) &= \frac{P(s,f|x)P(x)}{\int P(s,f|x)P(x)dx}\\ & = |\boldsymbol\alpha')
(Dirichlet-multinomial)
Hypergeometric
with known total population size N
M (number of target members) Beta-binomial 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)
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)
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')
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}')
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')
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)
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)
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)
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)
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)
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 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 \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 \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  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