A conjugate prior for regression coefficients, conditional on residual variance, and sample size.
RegressionCoefficientConjugatePrior(
mean,
sample.size,
additional.prior.precision = numeric(0),
diagonal.weight = 0)
The mean of the prior distribution, denoted 'b' below. See Details.
The value denoted \(\kappa\) below.
This can be interpreted as a number of observations worth of weight
to be assigned to mean in the posterior distribution.
A vector of non-negative numbers
representing the diagonal matrix \(\Lambda^{-1}\)
below. Positive values for additional.prior.precision will
ensure the distribution is proper even if the regression model has
no data. If all columns of the design matrix have positive variance
then additional.prior.precision can safely be set to zero. A
zero-length numeric vector is a slightly more efficient equivalent
to a vector of all zeros.
The weight given to the diagonal when XTX is
averaged with its diagonal. The purpose of diagonal.weight
is to keep the prior distribution proper even if X is less than full
rank. If the design matrix is full rank then diagonal.weight
can be set to zero.
Steven L. Scott steve.the.bayesian@gmail.com
A conditional prior for the coefficients (beta) in a linear regression model. The prior is conditional on the residual variance \(\sigma^2\), the sample size n, and the design matrix X. The prior is
$$\beta | \sigma^2, X \sim % N(b, \sigma^2 (\Lambda^{-1} + V$$
where $$V^{-1} = \frac{\kappa}{n} ((1 - w) X^TX + w diag(X^TX)) .$$
The prior distribution also depends on the cross product matrix XTX and the sample size n, which are not arguments to this function. It is expected that the underlying C++ code will get those quantities elsewhere (presumably from the regression modeled by this prior).
Gelman, Carlin, Stern, Rubin (2003), "Bayesian Data Analysis", Chapman and Hall.