# For Prior definition in bayesian regression with R package MCMCglmm, how to convey different strength of believe via parameter nu?

I understand the strength of the Prior is set via parameter nu however, I can not find information what nu expresses in statistical terms, e.g. how strong would a prior that is similar to the number of variables x be in this example?

#Inverse Wishart (multivariate, variables=x)

prior.miw<-list(R=list(V=diag(x), nu=x),G=list(G1=list(V=diag(x),
nu=x)))


I also saw a lot of examples for weak priors with nu=0.01, does it mean we have a 1/100 degree of belief in the prior compared to the posterior?

From the course notes for MCMCglmm section 1.2 describing prior distributions.

For a single variance component the inverse Wishart takes two scalar parameters, V and nu. The distribution tends to a point mass on V as the degree of belief parameter, nu goes to infinity. The distribution tends to be right skewed when nu is not very large, with a mode of $$\frac{V∗nu}{nu+2}$$ but a mean of $$\frac{V∗nu}{nu−2}$$ (which is not defined for nu < 2).

So your answer would be that nu plays a large role in the determination of the prior likelihood of the estimated variance.

In the same section you see code for a log-prior created in this way:

logprior <- function(par, priorR, priorB) {
dnorm(par,
mean = priorB$mu, sd = sqrt(priorB$V),
log = TRUE) +
log(dinvgamma(par,
shape = priorR$nu/2, scale = (priorR$nu * priorR\$V)/2))
}


And we see that nu is used in the shape and scale parameter for the prior likelihood of the estimated variance, par.

For your question, "does it mean we have a 1/100 degree of belief in the prior compared to the posterior?", you must keep in mind that the posterior is composed of the prior likelihood and the data likelihood. That is, the posterior likelihood represents our final belief given the data and our prior belief. Working through the example from section 1.2 and 1.3 would give you an opportunity to adjust nu and see its effect on the posterior.