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,...,2m−1}. Are these to be counted as m parameters or one parameter?

**Answer**

It is partly because of this imprecision in the “number of parameters” in BIC that DIC (the deviance information criterion) introduced an effective number of parameters as

pD(x)=E[D(θ)|x]−D(E[θ|x])

where

D(θ)=−2logf(x|θ)

and

DIC(x)=pD(x)+E[D(θ)|x]

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

**Attribution***Source : Link , Question Author : highBandWidth , Answer Author : Xi’an*