The HPFMM Procedure

Prior Distributions

The following list displays the parameterization of prior distributions for situations in which the HPFMM procedure uses a conjugate sampler in mixture models without model effects and certain basic distributions (binary, binomial, exponential, Poisson, normal, and t). You specify the parameters a and b in the formulas below in the MUPRIORPARMS= and PHIPRIORPARMS= options in the BAYES statement in these models.

Betaleft-parenthesis a comma b right-parenthesis
f left-parenthesis y right-parenthesis equals StartFraction normal upper Gamma left-parenthesis a plus b right-parenthesis Over normal upper Gamma left-parenthesis a right-parenthesis normal upper Gamma left-parenthesis b right-parenthesis EndFraction y Superscript a minus 1 Baseline left-parenthesis 1 minus y right-parenthesis Superscript b minus 1

where a greater-than 0, b greater-than 0. In this parameterization, the mean and variance of the distribution are mu equals a slash left-parenthesis a plus b right-parenthesis and mu left-parenthesis 1 minus mu right-parenthesis slash left-parenthesis a plus b plus 1 right-parenthesis, respectively. The beta distribution is the prior distribution for the success probability in binary and binomial distributions when conjugate sampling is used.

Dirichletleft-parenthesis a 1 comma ellipsis comma a Subscript k Baseline right-parenthesis
f left-parenthesis bold y right-parenthesis equals StartFraction normal upper Gamma left-parenthesis sigma-summation Underscript i equals 1 Overscript k Endscripts a Subscript i Baseline right-parenthesis Over product Underscript i equals 1 Overscript k Endscripts normal upper Gamma left-parenthesis a Subscript i Baseline right-parenthesis EndFraction y 1 Superscript a 1 minus 1 Baseline midline-horizontal-ellipsis y Subscript k Superscript a Super Subscript k Superscript minus 1

where sigma-summation Underscript i equals 1 Overscript k Endscripts y Subscript i Baseline equals 1 and the parameters a Subscript i Baseline greater-than 0. If any a Subscript i were zero, an improper density would result. The Dirichlet density is the prior distribution for the mixture probabilities. You can affect the choice of the a Subscript i through the MIXPRIORPARMS option in the BAYES statement. If k=2, the Dirichlet is the same as the betaleft-parenthesis a comma b right-parenthesis distribution.

Gammaleft-parenthesis a comma b)
f left-parenthesis y right-parenthesis equals StartFraction b Superscript a Baseline Over normal upper Gamma left-parenthesis a right-parenthesis EndFraction y Superscript a minus 1 Baseline exp left-brace minus b y right-brace

where a greater-than 0, b greater-than 0. In this parameterization, the mean and variance of the distribution are mu equals a slash b and mu slash b, respectively. The gamma distribution is the prior distribution for the mean parameter of the Poisson distribution when conjugate sampling is used.

Inverse gammaleft-parenthesis a comma b right-parenthesis
f left-parenthesis y right-parenthesis equals StartFraction b Superscript a Baseline Over normal upper Gamma left-parenthesis a right-parenthesis EndFraction y Superscript negative a minus 1 Baseline exp left-brace negative b slash y right-brace

where a greater-than 0, b greater-than 0. In this parameterization, the mean and variance of the distribution are mu equals b slash left-parenthesis a minus 1 right-parenthesis if a greater-than 1 and mu squared slash left-parenthesis a minus 2 right-parenthesis if a greater-than 2, respectively. The inverse gamma distribution is the prior distribution for the mean parameter of the exponential distribution when conjugate sampling is used. It is also the prior distribution for the scale parameter phi in all models.

Multinomialleft-parenthesis 1 comma pi 1 comma ellipsis comma pi Subscript k Baseline right-parenthesis
f left-parenthesis bold y right-parenthesis equals StartFraction 1 Over y 1 factorial midline-horizontal-ellipsis y Subscript k Baseline factorial EndFraction pi 1 Superscript y 1 Baseline midline-horizontal-ellipsis pi Subscript k Superscript y Super Subscript k

where sigma-summation Underscript j equals 1 Overscript k Endscripts y Subscript j Baseline equals n, y Subscript j Baseline greater-than-or-equal-to 0, sigma-summation Underscript j equals 1 Overscript k Endscripts pi Subscript j Baseline equals 1, and n is the number of observations included in the analysis. The multinomial density is the prior distribution for the mixture proportions. The mean and variance of upper Y Subscript j are mu Subscript j Baseline equals pi Subscript j and mu Subscript j Baseline left-parenthesis 1 minus mu Subscript j Baseline right-parenthesis, respectively.

Normalleft-parenthesis a comma b right-parenthesis
f left-parenthesis y right-parenthesis equals StartFraction a Over StartRoot 2 pi b EndRoot EndFraction exp left-brace minus one-half StartFraction left-parenthesis y minus a right-parenthesis squared Over b EndFraction right-brace

where b greater-than 0. The mean and variance of the distribution are mu equals a and b, respectively. The normal distribution is the prior distribution for the mean parameter of the normal and t distribution when conjugate sampling is used.

When a MODEL statement contains effects or if you specify the METROPOLIS option, the prior distribution for the regression parameters is multivariate normal, and you can specify the means and variances of the parameters in the BETAPRIORPARMS= option in the BAYES statement.

Last updated: December 09, 2022