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and $\tilde{r}(t,x):=\nabla\log\tilde{\rho}(t,x)$. The terms $\color{royalblue}{\int_0^T G(t, X_t) \dd t}$, and $\log\left(\color{forestgreen}{\tilde{\rho}(0,X_0)}\right)$ may be computed with functions:
GuidedProposals.loglikhd
and
GuidedProposals.loglikhd_obs
respectively. In general, deriving the term $\color{maroon}{\rho(0,X_0)}$ explicitly is impossible. Thankfully though, in an MCMC or an importance sampling setting this term always cancels out and so never needs to be computed.
Log-likelihood computation whilst sampling
Function rand!—when called with a parameter Val(:ll)—computes the "log-likelihood" at the time of sampling. Internally the following function is called after the Wiener process is sampled.
GuidedProposals.solve_and_ll!
solve_and_ll! computes only $\color{royalblue}{\int_0^T G(t, X_t) \dd t}$.
When rand! is called on a singleGuidProp (i.e. a single interval) then only this path contribution is returned.
However, if rand! is called on a list ofGuidProp, then apart from summing over the results from solve_and_ll! an additional end-point contribution is added, i.e.