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Bayesian linear regression is a type ofconditional modeling in which the mean of one variable is described by alinear combination of other variables, with the goal of obtaining theposterior probability of the regression coefficients (as well as other parameters describing thedistribution of the regressand) and ultimately allowing theout-of-sample prediction of theregressand (often labelled${\displaystyle y}$)conditional on observed values of the regressors (usually${\displaystyle X}$). The simplest and most widely used version of this model is thenormal linear model, in which${\displaystyle y}$ given${\displaystyle X}$ is distributedGaussian. In this model, and under a particular choice ofprior probabilities for the parameters—so-calledconjugate priors—the posterior can be found analytically. With more arbitrarily chosen priors, the posteriors generally have to be approximated.

Consider a standardlinear regression problem, in which for${\displaystyle i=1,\dots ,n}$ we specify the mean of theconditional distribution of${\displaystyle y_i}$ given a${\displaystyle k\times 1}$ predictor vector${\displaystyle {\mathbf{x}}_i}$:$${\displaystyle y_i={\mathbf{x}}_i^{\mathsf{T}}\mathit{\beta }+{\epsilon }_i,}$$

where${\displaystyle \mathit{\beta }}$ is a${\displaystyle k\times 1}$ vector, and the${\displaystyle {\epsilon }_i}$ areindependent and identicallynormally distributed random variables:$${\displaystyle {\epsilon }_i\sim N(0,{\sigma }^2).}$$

This corresponds to the followinglikelihood function:

$${\displaystyle \rho (\mathbf{y}\mid \mathbf{X},\mathit{\beta },{\sigma }^2)\propto ({\sigma }^2{)}^{-n/2}\exp \left(-\frac{1}{2{\sigma }^2}(\mathbf{y}-\mathbf{X}\mathit{\beta }{)}^{\mathsf{T}}(\mathbf{y}-\mathbf{X}\mathit{\beta })\right).}$$

Theordinary least squares solution is used to estimate the coefficient vector using theMoore–Penrose pseudoinverse:$${\displaystyle \hat{\mathit{\beta }}=({\mathbf{X}}^{\mathsf{T}}\mathbf{X}{)}^{-1}{\mathbf{X}}^{\mathsf{T}}\mathbf{y}}$$

where${\displaystyle \mathbf{X}}$ is the${\displaystyle n\times k}$design matrix, each row of which is a predictor vector${\displaystyle {\mathbf{x}}_i^{\mathsf{T}}}$; and${\displaystyle \mathbf{y}}$ is the column${\displaystyle n}$-vector${\displaystyle [y_1\cdots y_n{]}^{\mathsf{T}}}$.

This is afrequentist approach, and it assumes that there are enough measurements to say something meaningful about${\displaystyle \mathit{\beta }}$. In theBayesian approach,^[1] the data is supplemented with additional information in the form of aprior probability distribution. The prior belief about the parameters is combined with the data's likelihood function according toBayes' theorem to yield theposterior belief about the parameters${\displaystyle \mathit{\beta }}$ and${\displaystyle \sigma }$. The prior can take different functional forms depending on the domain and the information that is availablea priori.

Since the data comprises both${\displaystyle \mathbf{y}}$ and${\displaystyle \mathbf{X}}$, the focus only on the distribution of${\displaystyle \mathbf{y}}$ conditional on${\displaystyle \mathbf{X}}$ needs justification. In fact, a "full" Bayesian analysis would require a joint likelihood${\displaystyle \rho (\mathbf{y},\mathbf{X}\mid \mathit{\beta },{\sigma }^2,\gamma )}$ along with a prior${\displaystyle \rho (\beta ,{\sigma }^2,\gamma )}$, where${\displaystyle \gamma }$ symbolizes the parameters of the distribution for${\displaystyle \mathbf{X}}$. Only under the assumption of (weak) exogeneity can the joint likelihood be factored into${\displaystyle \rho (\mathbf{y}\mid \mathbf{X},\beta ,{\sigma }^2)\rho (\mathbf{X}\mid \gamma )}$.^[2] The latter part is usually ignored under the assumption of disjoint parameter sets. More so, under classic assumptions${\displaystyle \mathbf{X}}$ is considered chosen (for example, in a designed experiment) and therefore has a known probability without parameters.^[3]

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

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

The likelihood is now re-written as$${\displaystyle \rho (\mathbf{y}|\mathbf{X},\mathit{\beta },{\sigma }^2)\propto ({\sigma }^2{)}^{-\frac{v}{2}}\exp \left(-\frac{v{s}^2}{2{\sigma }^2}\right)({\sigma }^2{)}^{-\frac{n-v}{2}}\exp \left(-\frac{1}{2{\sigma }^2}(\mathit{\beta }-\hat{\mathit{\beta }}{)}^{\mathsf{T}}({\mathbf{X}}^{\mathsf{T}}\mathbf{X})(\mathit{\beta }-\hat{\mathit{\beta }})\right),}$$where$${\displaystyle v{s}^2=(\mathbf{y}-\mathbf{X}\hat{\mathit{\beta }}{)}^{\mathsf{T}}(\mathbf{y}-\mathbf{X}\hat{\mathit{\beta }})\text{ and }v=n-k,}$$where${\displaystyle k}$ is the number of regression coefficients.

This suggests a form for the prior:$${\displaystyle \rho (\mathit{\beta },{\sigma }^2)=\rho ({\sigma }^2)\rho (\mathit{\beta }\mid {\sigma }^2),}$$where${\displaystyle \rho ({\sigma }^2)}$ is aninverse-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 theinverse-gamma distribution article, this is the density of an${\displaystyle \text{Inv-Gamma}(a_0,b_0)}$ distribution with${\displaystyle a_0=\frac{v_0}{2}}$ and${\displaystyle b_0=\frac{1}{2}{v}_0{s}_0^2}$ 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 ascaled inverse chi-squared distribution,${\displaystyle \text{Scale-inv-}{\chi }^2(v_0,s_0^2).}$

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

$${\displaystyle \rho (\mathit{\beta }\mid {\sigma }^2)\propto ({\sigma }^2{)}^{-k/2}\exp \left(-\frac{1}{2{\sigma }^2}(\mathit{\beta }-{\mathit{\mu }}_0{)}^{\mathsf{T}}{\mathbf{\Lambda }}_0(\mathit{\beta }-{\mathit{\mu }}_0)\right).}$$

In the notation of thenormal distribution, the conditional prior distribution is${\displaystyle \mathcal{N}\left({\mathit{\mu }}_0,{\sigma }^2{\mathbf{\Lambda }}_0^{-1}\right).}$

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

With some re-arrangement,^[4] the posterior can be re-written so that the posterior mean${\displaystyle {\mathit{\mu }}_n}$ of the parameter vector${\displaystyle \mathit{\beta }}$ can be expressed in terms of the least squares estimator${\displaystyle \hat{\mathit{\beta }}}$ and the prior mean${\displaystyle {\mathit{\mu }}_0}$, with the strength of the prior indicated by the prior precision matrix${\displaystyle {\mathbf{\Lambda }}_0}$

$${\displaystyle {\mathit{\mu }}_n=({\mathbf{X}}^{\mathsf{T}}\mathbf{X}+{\mathbf{\Lambda }}_0{)}^{-1}({\mathbf{X}}^{\mathsf{T}}\mathbf{X}\hat{\mathit{\beta }}+{\mathbf{\Lambda }}_0{\mathit{\mu }}_0).}$$

To justify that${\displaystyle {\mathit{\mu }}_n}$ is indeed the posterior mean, the quadratic terms in the exponential can be re-arranged as aquadratic form in${\displaystyle \mathit{\beta }-{\mathit{\mu }}_n}$.^[5]

$${\displaystyle (\mathbf{y}-\mathbf{X}\mathit{\beta }{)}^{\mathsf{T}}(\mathbf{y}-\mathbf{X}\mathit{\beta })+(\mathit{\beta }-{\mathit{\mu }}_0{)}^{\mathsf{T}}{\mathbf{\Lambda }}_0(\mathit{\beta }-{\mathit{\mu }}_0)=(\mathit{\beta }-{\mathit{\mu }}_n{)}^{\mathsf{T}}({\mathbf{X}}^{\mathsf{T}}\mathbf{X}+{\mathbf{\Lambda }}_0)(\mathit{\beta }-{\mathit{\mu }}_n)+{\mathbf{y}}^{\mathsf{T}}\mathbf{y}-{\mathit{\mu }}_n^{\mathsf{T}}({\mathbf{X}}^{\mathsf{T}}\mathbf{X}+{\mathbf{\Lambda }}_0){\mathit{\mu }}_n+{\mathit{\mu }}_0^{\mathsf{T}}{\mathbf{\Lambda }}_0{\mathit{\mu }}_0.}$$

Now the posterior can be expressed as anormal distribution times aninverse-gamma distribution:

$${\displaystyle \rho (\mathit{\beta },{\sigma }^2\mid \mathbf{y},\mathbf{X})\propto ({\sigma }^2{)}^{-k/2}\exp \left(-\frac{1}{2{\sigma }^2}(\mathit{\beta }-{\mathit{\mu }}_n{)}^{\mathsf{T}}({\mathbf{X}}^{\mathsf{T}}\mathbf{X}+{\mathbf{\Lambda }}_0)(\mathit{\beta }-{\mathit{\mu }}_n)\right)({\sigma }^2{)}^{-\frac{n+2a_0}{2}-1}\exp \left(-\frac{2b_0+{\mathbf{y}}^{\mathsf{T}}\mathbf{y}-{\mathit{\mu }}_n^{\mathsf{T}}({\mathbf{X}}^{\mathsf{T}}\mathbf{X}+{\mathbf{\Lambda }}_0){\mathit{\mu }}_n+{\mathit{\mu }}_0^{\mathsf{T}}{\mathbf{\Lambda }}_0{\mathit{\mu }}_0}{2{\sigma }^2}\right).}$$

Therefore, the posterior distribution can be parametrized as follows.$${\displaystyle \rho (\mathit{\beta },{\sigma }^2\mid \mathbf{y},\mathbf{X})\propto \rho (\mathit{\beta }\mid {\sigma }^2,\mathbf{y},\mathbf{X})\rho ({\sigma }^2\mid \mathbf{y},\mathbf{X}),}$$where the two factors correspond to the densities of${\displaystyle \mathcal{N}\left({\mathit{\mu }}_n,{\sigma }^2{\mathbf{\Lambda }}_n^{-1}\right)}$ and${\displaystyle \text{Inv-Gamma}\left(a_n,b_n\right)}$ distributions, with the parameters of these given by

$${\displaystyle {\mathbf{\Lambda }}_n=({\mathbf{X}}^{\mathsf{T}}\mathbf{X}+{\mathbf{\Lambda }}_0),{\mathit{\mu }}_n=({\mathbf{\Lambda }}_n{)}^{-1}({\mathbf{X}}^{\mathsf{T}}\mathbf{X}\hat{\mathit{\beta }}+{\mathbf{\Lambda }}_0{\mathit{\mu }}_0),}$$$${\displaystyle a_n=a_0+\frac{n}{2},b_n=b_0+\frac{1}{2}({\mathbf{y}}^{\mathsf{T}}\mathbf{y}+{\mathit{\mu }}_0^{\mathsf{T}}{\mathbf{\Lambda }}_0{\mathit{\mu }}_0-{\mathit{\mu }}_n^{\mathsf{T}}{\mathbf{\Lambda }}_n{\mathit{\mu }}_n).}$$

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

Themodel evidence${\displaystyle p(\mathbf{y}\mid m)}$ is the probability of the data given the model${\displaystyle m}$. It is also known as themarginal likelihood, and as theprior predictive density. Here, the model is defined by the likelihood function${\displaystyle p(\mathbf{y}\mid \mathbf{X},\mathit{\beta },\sigma )}$ and the prior distribution on the parameters, i.e.${\displaystyle p(\mathit{\beta },\sigma )}$. 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 byBayes factors. 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${\displaystyle p(\mathbf{y},\mathit{\beta },\sigma \mid \mathbf{X})}$ over all possible values of${\displaystyle \mathit{\beta }}$ and${\displaystyle \sigma }$.$${\displaystyle p(\mathbf{y}|m)=\int p(\mathbf{y}\mid \mathbf{X},\mathit{\beta },\sigma )p(\mathit{\beta },\sigma )d\mathit{\beta }d\sigma }$$This integral can be computed analytically and the solution is given in the following equation.^[6]$${\displaystyle p(\mathbf{y}\mid m)=\frac{1}{(2\pi {)}^{n/2}}\sqrt{\frac{det({\mathbf{\Lambda }}_0)}{det({\mathbf{\Lambda }}_n)}}\cdot \frac{b_0^{a_0}}{b_n^{a_n}}\cdot \frac{\mathrm{\Gamma }(a_n)}{\mathrm{\Gamma }(a_0)}}$$

Here${\displaystyle \mathrm{\Gamma }}$ denotes thegamma 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${\displaystyle \mathit{\beta }}$ and${\displaystyle \sigma }$.^[7]$${\displaystyle p(\mathbf{y}\mid m)=\frac{p(\mathit{\beta },\sigma |m)p(\mathbf{y}\mid \mathbf{X},\mathit{\beta },\sigma ,m)}{p(\mathit{\beta },\sigma \mid \mathbf{y},\mathbf{X},m)}}$$Note that this equation follows from a re-arrangement ofBayes' theorem. Inserting the formulas for the prior, the likelihood, and the posterior and simplifying the resulting expression leads to the analytic expression given above.

In general, it may be impossible or impractical to derive the posterior distribution analytically. However, it is possible to approximate the posterior by anapproximate Bayesian inference method such asMonte Carlo sampling,^[8]INLA orvariational Bayes.

The special case${\displaystyle {\mathit{\mu }}_0=0,{\mathbf{\Lambda }}_0=c\mathbf{I}}$ is calledridge regression.

A similar analysis can be performed for the general case of the multivariate regression and part of this provides for Bayesianestimation of covariance matrices: seeBayesian multivariate linear regression.

• Bayes linear statistics
• Constrained least squares
• Regularized least squares
• Tikhonov regularization
• Spike and slab variable selection
• Bayesian interpretation of kernel regularization

This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(August 2011) (Learn how and when to remove this message)

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• Bayesian estimation of linear models (R programming wikibook). Bayesian linear regression as implemented inR.

• Bayesian inference
• Single-equation methods (econometrics)

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