Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Bayesian linear regression with known covariance ▷ Conjugate prior distribution

Theorem: Let

\[\label{eq:GLM} y = X \beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \Sigma)\]

be a linear regression model with measured $n \times 1$ data vector $y$, known $n \times p$ design matrix $X$ and known $n \times n$ covariance matrix $\Sigma$ as well as unknown $p \times 1$ regression coefficients $\beta$.

Then, the conjugate prior for this model is a multivariate normal distribution

\[\label{eq:GLM-N-prior} p(\beta) = \mathcal{N}(\beta; \mu_0, \Sigma_0) \; .\]

Proof: By definition, a conjugate prior is a prior distribution that, when combined with the likelihood function, leads to a posterior distribution that belongs to the same family of probability distributions. This is fulfilled when the prior density and the likelihood function are proportional to the model parameters in the same way, i.e. the model parameters appear in the same functional form in both.

Equation \eqref{eq:GLM} implies the following likelihood function:

\[\label{eq:GLM-LF} p(y|\beta) = \mathcal{N}(y; X \beta, \Sigma) = \sqrt{\frac{1}{(2 \pi)^n |\Sigma|}} \, \exp\left[ -\frac{1}{2} (y-X\beta)^\mathrm{T} \Sigma^{-1} (y-X\beta) \right] \; .\]

Expanding the product in the exponent, we have:

\[\label{eq:GLM-LF-s1} p(y|\beta) = \sqrt{\frac{1}{(2 \pi)^n |\Sigma|}} \cdot \exp\left[ -\frac{1}{2} \left( y^\mathrm{T} \Sigma^{-1} y - y^\mathrm{T} \Sigma^{-1} X \beta - \beta^\mathrm{T} X^\mathrm{T} \Sigma^{-1} y + \beta^\mathrm{T} X^\mathrm{T} \Sigma^{-1} X \beta \right) \right] \; .\]

Completing the square over $\beta$, one obtains

\[\label{eq:GLM-LF-s2} p(y|\beta) = \sqrt{\frac{1}{(2 \pi)^n |\Sigma|}} \cdot \exp\left[ -\frac{1}{2} \left( (\beta - \tilde{X}y)^\mathrm{T} X^\mathrm{T} \Sigma^{-1} X (\beta - \tilde{X}y) - y^\mathrm{T} Q y + y^\mathrm{T} \Sigma^{-1} y \right) \right]\]

where $\tilde{X} = \left( X^\mathrm{T} \Sigma^{-1} X \right)^{-1} X^\mathrm{T} \Sigma^{-1}$ and $Q = \tilde{X}^\mathrm{T} \left( X^\mathrm{T} \Sigma^{-1} X \right) \tilde{X}$.

Separating constant and variable terms, we get:

\[\label{eq:GLM-LF-s3} p(y|\beta) = \sqrt{\frac{1}{(2 \pi)^n |\Sigma|}} \cdot \exp\left[ -\frac{1}{2} \left( y^\mathrm{T} Q y + y^\mathrm{T} \Sigma^{-1} y \right) \right] \cdot \exp\left[ -\frac{1}{2} (\beta - \tilde{X}y)^\mathrm{T} X^\mathrm{T} \Sigma^{-1} X (\beta - \tilde{X}y) \right] \; .\]

In other words, the likelihood function is proportional to an exponential of a squared form of $\beta$:

\[\label{eq:GLM-LF-s4} p(y|\beta) \propto \exp\left[ -\frac{1}{2} (\beta - \tilde{X}y)^\mathrm{T} X^\mathrm{T} \Sigma^{-1} X (\beta - \tilde{X}y) \right] \; .\]

The same is true for a multivariate normal distribution over $\beta$

\[\label{eq:GLM-N-prior-s1} p(\beta) = \mathcal{N}(\beta; \mu_0, \Sigma_0)\]

the probability density function of which

\[\label{eq:GLM-N-prior-s2} p(\beta) = \sqrt{\frac{1}{(2 \pi)^p |\Sigma_0|}} \cdot \exp\left[ -\frac{1}{2} (\beta-\mu_0)^\mathrm{T} \Sigma_0^{-1} (\beta-\mu_0) \right]\]

exhibits the same proportionality

\[\label{eq:GLM-N-prior-s3} p(\beta) \propto \exp\left[ -\frac{1}{2} (\beta-\mu_0)^\mathrm{T} \Sigma_0^{-1} (\beta-\mu_0) \right]\]

and is therefore conjugate relative to the likelihood.


Metadata: ID: P432 | shortcut: blrkc-prior | author: JoramSoch | date: 2024-01-19, 08:42.