Look at the following please :)
Assuming an uninformative prior for $N$ then $$ \begin{aligned} \log p(N \mid y) &=\text { const }+\log \mathrm{p}(\mathbf{y} \mid N) \\ & \approx \text { const }-\frac{1}{2} \mathrm{BIC},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (17) \end{aligned} $$ using (16) and substituting for the BIC. Generally, the BIC would give an overprecise approximation to the posterior $p(N \mid y)$.
I'm kind of lost on how they got to (17) and the Bayesian information criterion.
(16) if it may be related
$$ \log \mathrm{p}(\mathbf{y} \mid N) \approx \log p(\mathbf{y} \mid \hat{\Theta}, N)-\frac{p_{N}}{2} \log (T),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (16) $$
