Bayesian linear regression explained

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well as other parameters describing the distribution of the regressand) and ultimately allowing the out-of-sample prediction of the regressand (often labelled

y ) conditional on observed values of the regressors (usually X ). The simplest and most widely used version of this model is the normal linear model, in which y given X is distributed Gaussian. In this model, and under a particular choice of prior probabilities for the parameters—so-called conjugate priors—the posterior can be found analytically. With more arbitrarily chosen priors, the posteriors generally have to be approximated.

Model setup

Consider a standard linear regression problem, in which for

i = 1 , \ldots, n we specify the mean of the conditional distribution of y _{i} given a k x 1 predictor vector x _{i} :$$y\_\; =\; \backslash mathbf\_i^\backslash mathsf\; \backslash boldsymbol\backslash beta\; +\; \backslash varepsilon\_i,$$

where

\boldsymbol\beta is a k x 1 vector, and the \varepsilon _{i} are independent and identically normally distributed random variables:$$\backslash varepsilon\_\; \backslash sim\; N(0,\; \backslash sigma^2).$$

This corresponds to the following likelihood function:

$$\backslash rho(\backslash mathbf\backslash mid\backslash mathbf,\backslash boldsymbol\backslash beta,\backslash sigma^)\; \backslash propto\; (\backslash sigma^2)^\; \backslash exp\backslash left(-\backslash frac\; (\backslash mathbf-\; \backslash mathbf\; \backslash boldsymbol\backslash beta)^\backslash mathsf(\backslash mathbf-\; \backslash mathbf\; \backslash boldsymbol\backslash beta)\backslash right).$$

The ordinary least squares solution is used to estimate the coefficient vector using the Moore–Penrose pseudoinverse:$$\backslash hat\; =\; (\backslash mathbf^\backslash mathsf\backslash mathbf)^\backslash mathbf^\backslash mathsf\backslash mathbf$$

where

X is the n x k design matrix, each row of which is a predictor vector x ^{T} _{i} ; and y is the column n -vector [y _{1} … y ^{T} _{n]} .

This is a frequentist approach, and it assumes that there are enough measurements to say something meaningful about

\boldsymbol\beta . In the Bayesian approach, the data are supplemented with additional information in the form of a prior probability distribution. The prior belief about the parameters is combined with the data's likelihood function according to Bayes theorem to yield the posterior belief about the parameters \boldsymbol\beta and \sigma . The prior can take different functional forms depending on the domain and the information that is available a priori.

Since the data comprise both

y and X , the focus only on the distribution of y conditional on X needs justification. In fact, a "full" Bayesian analysis would require a joint likelihood \rho( y , X \mid\boldsymbol\beta,\sigma ^{2} ,\gamma) along with a prior \rho(\beta,\sigma ^{2} ,\gamma) , where \gamma symbolizes the parameters of the distribution for X . Only under the assumption of (weak) exogeneity can the joint likelihood be factored into \rho( y \mid\boldsymbol X ,\beta,\sigma ^{2} )\rho( X \mid\gamma) .^[1] The latter part is usually ignored under the assumption of disjoint parameter sets. More so, under classic assumptions X are considered chosen (for example, in a designed experiment) and therefore has a known probability without parameters.^[2]

With conjugate priors

Conjugate prior distribution

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

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

The likelihood is now re-written as$$\backslash rho(\backslash mathbf|\backslash mathbf,\backslash boldsymbol\backslash beta,\backslash sigma^)\; \backslash propto\; (\backslash sigma^2)^\; \backslash exp\backslash left(-\backslash frac\backslash right)(\backslash sigma^2)^\; \backslash exp\backslash left(-\backslash frac(\backslash boldsymbol\backslash beta\; -\; \backslash hat)^\backslash mathsf(\backslash mathbf^\backslash mathsf\backslash mathbf)(\backslash boldsymbol\backslash beta\; -\; \backslash hat)\backslash right),$$where$$vs^2\; =(\backslash mathbf-\; \backslash mathbf\; \backslash hat)^\backslash mathsf(\backslash mathbf-\; \backslash mathbf\; \backslash hat)\; \backslash quad\; \backslash text\; \backslash quad\; v\; =\; n-k,$$where

k is the number of regression coefficients.

This suggests a form for the prior:$$\backslash rho(\backslash boldsymbol\backslash beta,\backslash sigma^2)\; =\; \backslash rho(\backslash sigma^2)\backslash rho(\backslash boldsymbol\backslash beta\backslash mid\backslash sigma^2),$$where

\rho(\sigma ^{2)} is an inverse-gamma distribution$$\backslash rho(\backslash sigma^2)\; \backslash propto\; (\backslash sigma^2)^\; \backslash exp\backslash left(-\backslash frac\backslash right).$$

In the notation introduced in the inverse-gamma distribution article, this is the density of an

Inv-Gamma ( a _{0,} b _{0)} distribution with a _{0=\tfrac{v} _{0}{2}} and b _{0=\tfrac{1}{2}} v _{0s} ^{2} ^{0} with v _{0} and s ^{2} _{0} as the prior values of v and s ^{2} , respectively. Equivalently, it can also be described as a scaled inverse chi-squared distribution, Scale-inv- \chi ^{2 (v} _{0 ,} s ^{2 ).} _{0}

Further the conditional prior density

\rho(\boldsymbol\beta|\sigma ^{2} ) is a normal distribution,

$$\backslash rho(\backslash boldsymbol\backslash beta\backslash mid\backslash sigma^2)\; \backslash propto\; (\backslash sigma^2)^\; \backslash exp\backslash left(-\backslash frac(\backslash boldsymbol\backslash beta\; -\; \backslash boldsymbol\backslash mu\_0)^\backslash mathsf\; \backslash mathbf\_0\; (\backslash boldsymbol\backslash beta\; -\; \backslash boldsymbol\backslash mu\_0)\backslash right).$$

In the notation of the normal distribution, the conditional prior distribution is

l{N}\left(\boldsymbol\mu _{0,} \sigma ^{2} \boldsymbol Λ ^{-1} _{0} \right).

Posterior distribution

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

With some re-arrangement,^[3] the posterior can be re-written so that the posterior mean

\boldsymbol\mu _{n} of the parameter vector \boldsymbol\beta can be expressed in terms of the least squares estimator \hat{\boldsymbol\beta} and the prior mean \boldsymbol\mu _{0} , with the strength of the prior indicated by the prior precision matrix \boldsymbol Λ _{0}

$$\backslash boldsymbol\backslash mu\_n\; =\; (\backslash mathbf^\backslash mathsf\backslash mathbf+\backslash boldsymbol\backslash Lambda\_0)^(\backslash mathbf^\backslash mathsf\; \backslash mathbf\backslash hat+\backslash boldsymbol\backslash Lambda\_0\backslash boldsymbol\backslash mu\_0)\; .$$

To justify that

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

$$(\backslash mathbf-\; \backslash mathbf\; \backslash boldsymbol\backslash beta)^\backslash mathsf(\backslash mathbf-\; \backslash mathbf\; \backslash boldsymbol\backslash beta)\; +\; (\backslash boldsymbol\backslash beta\; -\; \backslash boldsymbol\backslash mu\_0)^\backslash mathsf\backslash boldsymbol\backslash Lambda\_0(\backslash boldsymbol\backslash beta\; -\; \backslash boldsymbol\backslash mu\_0)\; =(\backslash boldsymbol\backslash beta-\backslash boldsymbol\backslash mu\_n)^\backslash mathsf(\backslash mathbf^\backslash mathsf\backslash mathbf+\backslash boldsymbol\backslash Lambda\_0)(\backslash boldsymbol\backslash beta-\backslash boldsymbol\backslash mu\_n)+\backslash mathbf^\backslash mathsf\backslash mathbf-\backslash boldsymbol\backslash mu\_n^\backslash mathsf(\backslash mathbf^\backslash mathsf\backslash mathbf+\backslash boldsymbol\backslash Lambda\_0)\backslash boldsymbol\backslash mu\_n+\backslash boldsymbol\backslash mu\_0^\backslash mathsf\; \backslash boldsymbol\backslash Lambda\_0\backslash boldsymbol\backslash mu\_0\; .$$

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

$$\backslash rho(\backslash boldsymbol\backslash beta,\backslash sigma^2\backslash mid\backslash mathbf,\backslash mathbf)\; \backslash propto\; (\backslash sigma^2)^\; \backslash exp\backslash left(-\backslash frac(\backslash boldsymbol\backslash beta\; -\; \backslash boldsymbol\backslash mu\_n)^\backslash mathsf(\backslash mathbf^\backslash mathsf\; \backslash mathbf+\backslash mathbf\_0)(\backslash boldsymbol\backslash beta\; -\; \backslash boldsymbol\backslash mu\_n)\backslash right)\; (\backslash sigma^2)^\; \backslash exp\backslash left(-\backslash frac\backslash right)\; .$$

Therefore, the posterior distribution can be parametrized as follows.$$\backslash rho(\backslash boldsymbol\backslash beta,\backslash sigma^2\backslash mid\backslash mathbf,\backslash mathbf)\; \backslash propto\; \backslash rho(\backslash boldsymbol\backslash beta\; \backslash mid\; \backslash sigma^2,\backslash mathbf,\backslash mathbf)\; \backslash rho(\backslash sigma^2\backslash mid\backslash mathbf,\backslash mathbf),$$where the two factors correspond to the densities of

l{N}\left( \boldsymbol\mu _{n,} \sigma ^{-1} _{n} \right) and Inv-Gamma \left(a _{n,b} _{n} \right) distributions, with the parameters of these given by

$$\backslash boldsymbol\backslash Lambda\_n=(\backslash mathbf^\backslash mathsf\backslash mathbf+\backslash mathbf\_0),\; \backslash quad\; \backslash boldsymbol\backslash mu\_n\; =\; (\backslash boldsymbol\backslash Lambda\_n)^(\backslash mathbf^\backslash mathsf\; \backslash mathbf\; \backslash hat\; +\; \backslash boldsymbol\backslash Lambda\_0\; \backslash boldsymbol\backslash mu\_0),$$$$a\_n=\; a\_0\; +\; \backslash frac,\; \backslash qquad\; b\_n=b\_0+\backslash frac(\backslash mathbf^\backslash mathsf\; \backslash mathbf\; +\; \backslash boldsymbol\backslash mu\_0^\backslash mathsf\; \backslash boldsymbol\backslash Lambda\_0\backslash boldsymbol\backslash mu\_0-\backslash boldsymbol\backslash mu\_n^\backslash mathsf\; \backslash boldsymbol\backslash Lambda\_n\; \backslash boldsymbol\backslash mu\_n)\; .$$

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

Model evidence

p( y \mid m) is the probability of the data given the model m . It is also known as the marginal likelihood, and as the prior predictive density. Here, the model is defined by the likelihood function p( y \mid X ,\boldsymbol\beta,\sigma) and the prior distribution on the parameters, i.e. 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 p( y ,\boldsymbol\beta,\sigma\mid X ) over all possible values of \boldsymbol\beta and \sigma .$$p(\backslash mathbf|m)=\backslash int\; p(\backslash mathbf\backslash mid\backslash mathbf,\backslash boldsymbol\backslash beta,\backslash sigma)\backslash ,\; p(\backslash boldsymbol\backslash beta,\backslash sigma)\backslash ,\; d\backslash boldsymbol\backslash beta\backslash ,\; d\backslash sigma$$This integral can be computed analytically and the solution is given in the following equation.^[5] $$p(\backslash mathbf\backslash mid\; m)=\backslash frac\backslash sqrt\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash frac$$

Here

\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 \boldsymbol\beta and \sigma .$$p(\backslash mathbf\backslash mid\; m)=\backslash frac$$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.

Other cases

In general, it may be impossible or impractical to derive the posterior distribution analytically. However, it is possible to approximate the posterior by an approximate Bayesian inference method such as Monte Carlo sampling,^[6] INLA or variational Bayes.

The special case

\boldsymbol\mu _{0=0,} Λ _{0} = c I is called ridge regression.

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

See also

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

References

• Book: Box, G. E. P. . George E. P. Box . George Tiao. Tiao . G. C. . 1973 . Bayesian Inference in Statistical Analysis . Wiley . 0-471-57428-7 .
• Book: Carlin, Bradley P. . Louis, Thomas A.. Bayesian Methods for Data Analysis . Third . Boca Raton, FL: Chapman and Hall/CRC . 2008 . 1-58488-697-8.
• Book: Fahrmeir, L. . Kneib, T. . Lang, S.. Regression. Modelle, Methoden und Anwendungen . Second . Springer . Heidelberg . 2009 . 978-3-642-01836-7 . 10.1007/978-3-642-01837-4.
• Book: Gelman, Andrew . Andrew Gelman . Carlin . John B. . Stern . Hal S. . Dunson . David B. . Vehtari . Aki . Rubin . Donald B. . 1 . Introduction to regression models . 353–380 . Bayesian Data Analysis . Third . Boca Raton, FL: Chapman and Hall/CRC . 2013 . 978-1-4398-4095-5 .
• Book: Jackman, Simon . Regression models . 99–124 . Bayesian Analysis for the Social Sciences . Wiley . 2009 . 978-0-470-01154-6 .
• Book: Rossi, Peter E. . Greg M. . Allenby . Robert . McCulloch . Bayesian Statistics and Marketing . John Wiley & Sons . 2006 . 0470863676 .
• Book: O'Hagan, Anthony. Bayesian Inference. 2B . Kendall's Advanced Theory of Statistics. 1994 . First . 0-340-52922-9. Halsted.

External links

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

Notes and References

1. See Jackman (2009), p. 101.
2. See Gelman et al. (2013), p. 354.
3. The intermediate steps of this computation can be found in O'Hagan (1994) at the beginning of the chapter on Linear models.
4. The intermediate steps are in Fahrmeir et al. (2009) on page 188.
5. The intermediate steps of this computation can be found in O'Hagan (1994) on page 257.
6. Carlin and Louis (2008) and Gelman, et al. (2003) explain how to use sampling methods for Bayesian linear regression.