Prior selection in BSiZer

In BSiZer software one must give prior parameters for prior distribution(s). The following prior distributions are used in BSiZer:

• Scaled inverse Chi-square distribution in standard BSiZer (independent observations and fixed design) for the random error variance.
• Inverse Wishart and Normal distribution in extended BiZer (possibly correlated observations and errors in predictors) for the random error covariance matrix and predictors, respectively.

In this page some advice for the elicitation of the prior parameters is given.

Scaled inverse Chi-square distribution

Scaled inverse Chi-square (SIC) distribution is a conjugate prior for the variance under normal likelihood. Some properties of the distribution can be found at

• http://en.wikipedia.org/wiki/Scale-inverse-chi-square_distribution
• Book: Gelman, Carlin, Stern and Rubin: Bayesian Data Analysis

Briefly: it is a nonsymmetric distribution for a continuous positive valued random variable with negative skewness parameterized by two parameters:v (>0) (degrees of dreedom) and σ (>0) (scale). Using SIC with normal likelihood results in a closed form expression of the posterior of the variance. The next figure shows SIC probability density functions with different parameters:

• Blue curve: v=10, σ=2
• Red curve: v=10, σ=3
• Green curve: v=10, σ=4
• Magenta curve: v=5, σ=2

A simple protocol for prior parameter elicitation could as follows:

1. Set the expected value of the variance to some value x. Set the mean (v/(v-2) σ²) of the distribution equal to x.
   
   
2. Set the variance of the variance, that describes the (un)certainty of the quantity, to some value y. Set the variance (2v² σ⁴/ ((v-2)² (v-4)) of the distribution equal to y.
   
   
3. Solve for v and σ.
   
   

Drawing the probability density function and investigating the posterior realizations of σ can also provide valuable information.

Inverse Wishart distribution

Inverse Wishart distribution can be seen as a multivariate generalization of univariate SIC distribution to the k-dimensional case. It has conjugate properties analogous to those of the univariate SIC distribution; it is a conjugate prior for covariance matrix under multivariate normal likelihood. In the literature it is usually parameterized by v (degrees of freedom) and S^-1 which is a k by k scale matrix. However, for technical reasons, in the code parameters v and W are used, with the connection W=S^-1.

One approach for quantifying the prior parameters is the following:

1. Find the baseline value b for the expected covariance matrix. For instance, for a (10 by 10) covariance matrix with the expected value diag([.1:.1:1]) the baseline value is set to b=.1. The baseline value is the smallest value of the elements of the expected mean.
   
   
2. Select the degree of freedom v and let v'= (v-k-1)b. Note: large (small) v decreases (increases) the variance of the elements in the covariance matrix.
   
   
3. Set W=v'Q, where bQ is the desired expected value of the covariance matrix. In our example Q is diag([1:10]) and the prior mean is .1*diag([1:10]).
   
   

Investigating the samples from the prior and posterior is also recommended.

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