Dear Bugs users,

I wish to model a multivariate normal response (X1, X2, X3, X4)   and to use a prior distribution which includes information on the parameters of (X1, X2) only (the prior should be "objective" for other parameters)

Does anybody know if there is a known way to do this ?

I had an idea that I will detail below. I encounter failures of the MCMC algorithm's convergence.

My idea is to decompose the distribution of (X1, X2, X3, X4) into the product of the marginal distribution of (X1, X2) and the conditional distribution of (X3, X4) given (X1, X2), and then to include informative priors on the parameters of the marginal distribution, and non-informative priors on the parameters of the conditional distribution.

I have implemented this but the MCMC algorithm's convergence fails. However, convergence is successful and estimations are good when there are only two components.

Now I give details of the two-components case for improving understanding.

Suppose that (X,Y) is distributed as a bivariate normal, and you have some realizations (X[1], Y[1]), (X[2], Y[2]), ..., (X[n], Y[n]). You have some prior beliefs about X but not about Y.

It is well-known that the distribution of Y given X=x is a normal distribution with mean a*X+b and with a variance which does not depend on x.

Then it is possible to estimate the parameters of the bivariate distribution of (X,Y) by using your prior beliefs about X with the following model:

model{
muX ~ dnorm(3, 1)    # subjective prior for mean(X)
tauX ~ dgamma(10, 200)   # subjective prior for precision(X)
a ~ dnorm(0, 0.0001)
b ~ dnorm(0, 0.0001)
tauY.given.X ~ dgamma(0.0001, 0.0001)
for(i in 1:n){
   X[i] ~ dnorm(muX, tauX)
     muY.given.X[i] <- a*X[i]+b    # conditional mean of Y given X
     Y[i] ~ dnorm(muY.given.X[i], tauY.given.X)   # conditional of Y given X
}
muY <- a*muX+b    # this is the (marginal) mean of Y
sigmaY <- 1/tauY.given.X + a*a/tauX    # this is the (marginal) variance of Y
}

That seems to work well, in this bivariate case.

What do you think about this ?

Thanks,

Stéphane.

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