data [@jacquier-johannes-polson; @lele2010estimability]. Also, for $K\rightarrow \infty$ and independently of the chosen prior, $\pi(\theta|y^{(K)})$ is degenerate at $\theta=\hat{\theta}$, where $\hat{\theta}$ is the MLE of $\theta$.

Notice the simplification occurring in the expression for $\alpha$, due to taking $v(X|\theta)\equiv p(X|\theta)$, this resulting in . The simplification above solves the typically difficult problem of not having a ready expression for the transition densities of $\{X_t\}_{t\geq t_0}$. In fact, here all we need is the ability to (somehow) simulate the process $\{X_t\}_{t\geq t_0}$, and having access to transition densities is not strictly required. For example, in section \[sec:gompertz\] we know the solution of the considered stochastic differential equation (SDE) model, so we can simulate from it. When the exact solution to an SDE is not available, a numerical discretization with stepsize $h$ (e.g. the Euler-Maruyama scheme) generates an approximate solution, converging to the exact one as $h\rightarrow 0$. [@beskos2006exact] even devised a numerical scheme resulting in exact simulation of the SDE solution (i.e. without discretization error), though this is of not so general applicability.

It is important to realize that dealing with a “powered-up” posterior as in results in a surface having increasingly peaked modes for increasing $K$ and deeper “valleys” in-between modes 