# Accounting for discrete or binary parameters in Bayesian information criterion

BIC penalizes based on the number of parameters. What if some of the parameters are some sort of binary indicator variables? Do these count as full parameters? But I can combine $m$ binary parameters into one discrete variable that takes values in $\{0,1,...,2^m-1\}$. Are these to be counted as $m$ parameters or one parameter?

Note that $p_D(x)$ is then data-dependent. (As discussed there, DIC also has problems of its own!)