What is the relation between the effective sample size $n$ and the model dimension (the effective size of parameters) $p$ in Bayesian model selection? Or is there any articles talking about this?
I guess $p$ is a function of $n$, intuitively since if we have more data points, larger $p$ will give a better model fit. But in terms of the information criteria, like AIC, BIC, WAIC, WBIC, DIC etc., I only found papers that talking about the model dimension $p$ related with these citeria, so how does the effective sample size effect these criteria?
Many thanks!
Effective sample size relates to the data; model dimension relates to the number of parameters. The two are unrelated.
In Spiegelhalter et al’s (2002) well known paper “Bayesian measures of model complexity and fit”, the connection is reiterated:
“...definition of p * that was derived by Moody (1992) and termed the ‘effective number of parameters’. This is the measure of dimensionality that is used in NIC ( Murata et al., 1994 ).”
To clarify, I say model dimension and [effective] sample size are unrelated because you could theoretically calculate the model dimension a priori (before you collect any data) using a point estimate of the prior distribution. Of course a posterior point estimate is more desirable, which is conditioned on the observance of data, and therefore you would justified in saying $p$ is a function of $n$.