Bayesian Linear Regression Models: Priors Distributions

You can specify the following prior distribution settings for the regression parameters and the variance of the errors. The following options are available only when the Characterize Posterior Distribution option is selected for Bayesian Analysis.

Note: Many applied researchers may question the need to specify a prior. Reference priors minimize the concern where the prior is generally overwhelmed as the data increases. When informative prior information is specified, Bayesian methods can effectively use the information. The requirement to specify a prior should not be viewed as a deterrent to using Bayesian analysis.

Reference

When selected, reference analysis produces objective Bayesian inference. Inferential statements depend only on the assumed model and the available data, and the prior distribution that is used to make an inference is the least informative. This is the default setting.

Conjugate

Provides options for defining conjugate prior distributions. Conjugate priors assume the Normal-Inverse-Gamma joint distribution. Although conjugate priors are not required when performing Bayesian updates, they aid the calculation processes.

Note: In order to specify conjugate priors for a linear regression model, set your expected mean of regression parameters in the Priors on variance of errors table. You can also choose to use the Variance of covariance matrix settings to specify the prior variance-covariance.

Priors on variance of errors

Shape Parameter: Specify the shape parameter a₀ for Inverse-Gamma distribution. You must enter a single value that is greater than 0.
Scale Parameter: Specify the scale parameter b₀ for Inverse-Gamma distribution. You must enter a single value that is greater than 0. The larger the scale parameter, the more spread out the distribution.

The table lists the mean of regression parameters (including intercept), which specifies the mean vector θ₀ for the defined regression parameters. The number of values must meet the number of regression parameters, including the intercept term.

The first variable name is always INTERCEPT. From the second row, the Variables column is automatically populated with the variables that are specified by Factor(s) and Covariate(s). The Mean column does not include any default values.

Click Reset to clear the values.

Variance of covariance matrix: σ²x

Specify V₀ the values in the lower triangle in the variance-covariance matrix for the multivariate normal prior. Note that V₀ must be semi-positive definite. The last value of each row must be positive. The next row should have one more value than the previous row. No values are specified for reference categories (if any).

Click Reset to clear the values.

Use identity matrix: When selected, the scaled identity matrix is used. You cannot specify V₀ values in the lower triangle in the variance-covariance matrix for the multivariate normal prior.