Bayesian linear regression

For an arbitrary prior distribution, there may be no analytical solution for the posterior distribution. In this section, we will consider a so-called conjugate prior for which the posterior distribution can be derived analytically.

A prior ${\displaystyle \rho ({\boldsymbol {\beta }},\sigma ^{2})}$ is conjugate to this likelihood function if it has the same functional form with respect to ${\displaystyle {\boldsymbol {\beta }}}$ and ${\displaystyle \sigma }$. Since the log-likelihood is quadratic in ${\displaystyle {\boldsymbol {\beta }}}$, the log-likelihood is re-written such that the likelihood becomes normal in ${\displaystyle ({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})}$. Write

$${\displaystyle {\begin{aligned}(\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }})^{\mathsf {T}}(\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }})&=[(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})+(\mathbf {X} {\hat {\boldsymbol {\beta }}}-\mathbf {X} {\boldsymbol {\beta }})]^{\mathsf {T}}[(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})+(\mathbf {X} {\hat {\boldsymbol {\beta }}}-\mathbf {X} {\boldsymbol {\beta }})]\\&=(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})^{\mathsf {T}}(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})+({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})^{\mathsf {T}}(\mathbf {X} ^{\mathsf {T}}\mathbf {X} )({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})+\underbrace {2(\mathbf {X} {\hat {\boldsymbol {\beta }}}-\mathbf {X} {\boldsymbol {\beta }})^{\mathsf {T}}(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})} _{=\ 0}\\&=(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})^{\mathsf {T}}(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})+({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})^{\mathsf {T}}(\mathbf {X} ^{\mathsf {T}}\mathbf {X} )({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})\,.\end{aligned}}}$$

The likelihood is now re-written as

$${\displaystyle \rho (\mathbf {y} |\mathbf {X} ,{\boldsymbol {\beta }},\sigma ^{2})\propto (\sigma ^{2})^{-{\frac {v}{2}}}\exp \left(-{\frac {vs^{2}}{2{\sigma }^{2}}}\right)(\sigma ^{2})^{-{\frac {n-v}{2}}}\exp \left(-{\frac {1}{2{\sigma }^{2}}}({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})^{\mathsf {T}}(\mathbf {X} ^{\mathsf {T}}\mathbf {X} )({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})\right),}$$

where

$${\displaystyle vs^{2}=(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})^{\mathsf {T}}(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})\quad {\text{ and }}\quad v=n-k,}$$

where ${\displaystyle k}$ is the number of regression coefficients.

This suggests a form for the prior:

$${\displaystyle \rho ({\boldsymbol {\beta }},\sigma ^{2})=\rho (\sigma ^{2})\rho ({\boldsymbol {\beta }}\mid \sigma ^{2}),}$$

where ${\displaystyle \rho (\sigma ^{2})}$ is an inverse-gamma distribution

$${\displaystyle \rho (\sigma ^{2})\propto (\sigma ^{2})^{-{\frac {v_{0}}{2}}-1}\exp \left(-{\frac {v_{0}s_{0}^{2}}{2\sigma ^{2}}}\right).}$$

In the notation introduced in the inverse-gamma distribution article, this is the density of an ${\displaystyle {\text{Inv-Gamma}}(a_{0},b_{0})}$ distribution with ${\displaystyle a_{0}={\tfrac {v_{0}}{2}}}$ and ${\displaystyle b_{0}={\tfrac {1}{2}}v_{0}s_{0}^{2}}$ with ${\displaystyle v_{0}}$ and ${\displaystyle s_{0}^{2}}$ as the prior values of ${\displaystyle v}$ and ${\displaystyle s^{2}}$, respectively. Equivalently, it can also be described as a scaled inverse chi-squared distribution, ${\displaystyle {\text{Scale-inv-}}\chi ^{2}(v_{0},s_{0}^{2}).}$

Further the conditional prior density ${\displaystyle \rho ({\boldsymbol {\beta }}|\sigma ^{2})}$ is a normal distribution,

$${\displaystyle \rho ({\boldsymbol {\beta }}\mid \sigma ^{2})\propto (\sigma ^{2})^{-k/2}\exp \left(-{\frac {1}{2\sigma ^{2}}}({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{0})^{\mathsf {T}}\mathbf {\Lambda } _{0}({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{0})\right).}$$

In the notation of the normal distribution, the conditional prior distribution is ${\displaystyle {\mathcal {N}}\left({\boldsymbol {\mu }}_{0},\sigma ^{2}{\boldsymbol {\Lambda }}_{0}^{-1}\right).}$

With the prior now specified, the posterior distribution can be expressed as

$${\displaystyle {\begin{aligned}\rho ({\boldsymbol {\beta }},\sigma ^{2}\mid \mathbf {y} ,\mathbf {X} )&\propto \rho (\mathbf {y} \mid \mathbf {X} ,{\boldsymbol {\beta }},\sigma ^{2})\rho ({\boldsymbol {\beta }}\mid \sigma ^{2})\rho (\sigma ^{2})\\&\propto (\sigma ^{2})^{-n/2}\exp \left(-{\frac {1}{2{\sigma }^{2}}}(\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }})^{\mathsf {T}}(\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }})\right)(\sigma ^{2})^{-k/2}\exp \left(-{\frac {1}{2\sigma ^{2}}}({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{0})^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{0})\right)(\sigma ^{2})^{-(a_{0}+1)}\exp \left(-{\frac {b_{0}}{\sigma ^{2}}}\right)\end{aligned}}}$$

With some re-arrangement,[3] the posterior can be re-written so that the posterior mean ${\displaystyle {\boldsymbol {\mu }}_{n}}$ of the parameter vector ${\displaystyle {\boldsymbol {\beta }}}$ can be expressed in terms of the least squares estimator ${\displaystyle {\hat {\boldsymbol {\beta }}}}$ and the prior mean ${\displaystyle {\boldsymbol {\mu }}_{0}}$, with the strength of the prior indicated by the prior precision matrix ${\displaystyle {\boldsymbol {\Lambda }}_{0}}$

$${\displaystyle {\boldsymbol {\mu }}_{n}=(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0})^{-1}(\mathbf {X} ^{\mathsf {T}}\mathbf {X} {\hat {\boldsymbol {\beta }}}+{\boldsymbol {\Lambda }}_{0}{\boldsymbol {\mu }}_{0}).}$$

To justify that ${\displaystyle {\boldsymbol {\mu }}_{n}}$ is indeed the posterior mean, the quadratic terms in the exponential can be re-arranged as a quadratic form in ${\displaystyle {\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{n}}$.[4]

$${\displaystyle (\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }})^{\mathsf {T}}(\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }})+({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{0})^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{0})=({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{n})^{\mathsf {T}}(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0})({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{n})+\mathbf {y} ^{\mathsf {T}}\mathbf {y} -{\boldsymbol {\mu }}_{n}^{\mathsf {T}}(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0}){\boldsymbol {\mu }}_{n}+{\boldsymbol {\mu }}_{0}^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}{\boldsymbol {\mu }}_{0}.}$$

Now the posterior can be expressed as a normal distribution times an inverse-gamma distribution:

$${\displaystyle \rho ({\boldsymbol {\beta }},\sigma ^{2}\mid \mathbf {y} ,\mathbf {X} )\propto (\sigma ^{2})^{-k/2}\exp \left(-{\frac {1}{2{\sigma }^{2}}}({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{n})^{\mathsf {T}}(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +\mathbf {\Lambda } _{0})({\boldsymbol {\beta }}-{\boldsymbol {\mu }}_{n})\right)(\sigma ^{2})^{-{\frac {n+2a_{0}}{2}}-1}\exp \left(-{\frac {2b_{0}+\mathbf {y} ^{\mathsf {T}}\mathbf {y} -{\boldsymbol {\mu }}_{n}^{\mathsf {T}}(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0}){\boldsymbol {\mu }}_{n}+{\boldsymbol {\mu }}_{0}^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}{\boldsymbol {\mu }}_{0}}{2\sigma ^{2}}}\right).}$$

Therefore, the posterior distribution can be parametrized as follows.

$${\displaystyle \rho ({\boldsymbol {\beta }},\sigma ^{2}\mid \mathbf {y} ,\mathbf {X} )\propto \rho ({\boldsymbol {\beta }}\mid \sigma ^{2},\mathbf {y} ,\mathbf {X} )\rho (\sigma ^{2}\mid \mathbf {y} ,\mathbf {X} ),}$$

where the two factors correspond to the densities of ${\displaystyle {\mathcal {N}}\left({\boldsymbol {\mu }}_{n},\sigma ^{2}{\boldsymbol {\Lambda }}_{n}^{-1}\right)\,}$ and ${\displaystyle {\text{Inv-Gamma}}\left(a_{n},b_{n}\right)}$ distributions, with the parameters of these given by

$${\displaystyle {\boldsymbol {\Lambda }}_{n}=(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +\mathbf {\Lambda } _{0}),\quad {\boldsymbol {\mu }}_{n}=({\boldsymbol {\Lambda }}_{n})^{-1}(\mathbf {X} ^{\mathsf {T}}\mathbf {X} {\hat {\boldsymbol {\beta }}}+{\boldsymbol {\Lambda }}_{0}{\boldsymbol {\mu }}_{0}),}$$

$${\displaystyle a_{n}=a_{0}+{\frac {n}{2}},\qquad b_{n}=b_{0}+{\frac {1}{2}}(\mathbf {y} ^{\mathsf {T}}\mathbf {y} +{\boldsymbol {\mu }}_{0}^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}{\boldsymbol {\mu }}_{0}-{\boldsymbol {\mu }}_{n}^{\mathsf {T}}{\boldsymbol {\Lambda }}_{n}{\boldsymbol {\mu }}_{n}).}$$

which illustrates Bayesian inference being a compromise between the information contained in the prior and the information contained in the sample.

The model evidence ${\displaystyle p(\mathbf {y} \mid m)}$ is the probability of the data given the model ${\displaystyle m}$. It is also known as the marginal likelihood, and as the prior predictive density. Here, the model is defined by the likelihood function ${\displaystyle p(\mathbf {y} \mid \mathbf {X} ,{\boldsymbol {\beta }},\sigma )}$ and the prior distribution on the parameters, i.e. ${\displaystyle p({\boldsymbol {\beta }},\sigma )}$. The model evidence captures in a single number how well such a model explains the observations. The model evidence of the Bayesian linear regression model presented in this section can be used to compare competing linear models by Bayesian model comparison. These models may differ in the number and values of the predictor variables as well as in their priors on the model parameters. Model complexity is already taken into account by the model evidence, because it marginalizes out the parameters by integrating ${\displaystyle p(\mathbf {y} ,{\boldsymbol {\beta }},\sigma \mid \mathbf {X} )}$ over all possible values of ${\displaystyle {\boldsymbol {\beta }}}$ and ${\displaystyle \sigma }$.

$${\displaystyle p(\mathbf {y} |m)=\int p(\mathbf {y} \mid \mathbf {X} ,{\boldsymbol {\beta }},\sigma )\,p({\boldsymbol {\beta }},\sigma )\,d{\boldsymbol {\beta }}\,d\sigma }$$

This integral can be computed analytically and the solution is given in the following equation.[5]

$${\displaystyle p(\mathbf {y} \mid m)={\frac {1}{(2\pi )^{n/2}}}{\sqrt {\frac {\det({\boldsymbol {\Lambda }}_{0})}{\det({\boldsymbol {\Lambda }}_{n})}}}\cdot {\frac {b_{0}^{a_{0}}}{b_{n}^{a_{n}}}}\cdot {\frac {\Gamma (a_{n})}{\Gamma (a_{0})}}}$$

Here ${\displaystyle \Gamma }$ denotes the gamma function. Because we have chosen a conjugate prior, the marginal likelihood can also be easily computed by evaluating the following equality for arbitrary values of ${\displaystyle {\boldsymbol {\beta }}}$ and ${\displaystyle \sigma }$.

$${\displaystyle p(\mathbf {y} \mid m)={\frac {p({\boldsymbol {\beta }},\sigma |m)\,p(\mathbf {y} \mid \mathbf {X} ,{\boldsymbol {\beta }},\sigma ,m)}{p({\boldsymbol {\beta }},\sigma \mid \mathbf {y} ,\mathbf {X} ,m)}}}$$

Note that this equation is nothing but a re-arrangement of Bayes theorem. Inserting the formulas for the prior, the likelihood, and the posterior and simplifying the resulting expression leads to the analytic expression given above.