How to interpret this pmf and likelihood? From 'The Bayesian Choice by Christian Robert I'm having some difficulties in understanding the definitions in the following example:

How do we interpret the pmf and the likelihood? is it always 1/3? If so, then why the branching out for values of theta? Berger and Wolpert (1988) is compilation of lecture notes, and I have no access to it. Also what's the meaning of $\mathbb{N}^*$? I've never seen this notation. This example is from 'The Bayesian Choice' by Christian Robert.
Any help would be appreciated.
 A: Some answers to your questions about that admittedly rather artificial example:

How do we interpret the pmf and the likelihood?

The pmf is the probability density of $X$ against the counting measure over the positive integers. Depending on the parity of $\theta$, i.e., whether it is odd or even, $X$ has a different support with three possible values, all with equal probabilities. 
The likelihood is a function of the parameter $\theta$ indexed by the value of the realisation $x$ of $X$. It is zero except for three values of $\theta$ depending on $x$. This can be interpreted as a case where there are three and only three values of $\theta$ which are compatible with this realisation or equivalently which can lead to this realisation.

is it always 1/3?

The functions are equal to $1/3$ for three entries and zero everywhere else.

If so, then why the branching out for values of theta? 

The density is defined on a different support of three integers for each value of $\theta$ and this support is described differently for odd and even values of $\theta$.

Berger and Wolpert (1988) is compilation of lecture notes, and I have no access to it. 

The entire monograph is available for free on Project Euclid.

Also what's the meaning of $ℕ^∗$?

This is a standard notation for the set of positive integers.
