Computations of log-likelihoods

The full likelihood function for a sampled path $X$ given by:

\[\frac{ \color{forestgreen}{ \tilde{\rho}(0,X_0) } }{ \color{maroon}{ \rho(0,X_0) } }\exp\left\{ \color{royalblue}{ \int_0^T G(t, X_t) \dd t } \right\},\]

where

\[G(t,x):=\left[ (b-\tilde{b})^T\tilde{r} + 0.5 tr\left\{ (a-\tilde{a})(\tilde{r}\tilde{r}^T-H) \right\} \right](t,x),\]

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 — Function

loglikhd([::IntegrationRule=::LeftRule], P::GuidProp, X::Trajectory; skip=0)

Compute path contribution to the log-likelihood function, i.e.:

\[\int_0^T G(t, X_t) dt,\]

where

\[G(t,x):=\left[ (b-\tilde{b})^T\tilde{r} + 0.5 tr\left\{ (a-\tilde{a})(\tilde{r}\tilde{r}^T-H) \right\} \right](t,x),\]

and $\tilde{r}(t,x):=\nabla\log\tilde{\rho}(t,x)$.

loglikhd(
    [::IntegrationRule=::LeftRule],
    PP::AbstractArray{<:GuidProp}, XX::AbstractArray{<:Trajectory};
    skip=0
)

Compute path contribution to the log-likelihood function for a sequence of segments.

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and

GuidedProposals.loglikhd_obs — Function

loglikhd_obs(P::GuidProp, x0)

Compute the contribution of end-points to the log-likelihood function, i.e.: $\log\rho(t,x)$.

source

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! — Function

solve_and_ll!(X, W, P, y1)

Compute the trajectory under the law P for a given Wiener noise W and a starting point y1. Store the trajectory in XX. Compute the log-likelihood (path contribution) along the way. Return success_flag and log-likelihood. success_flag is set to false only if sampling was prematurely halted due to XX violating assumptions about state space.

source

solve_and_ll! computes only $\color{royalblue}{\int_0^T G(t, X_t) \dd t}$.

When rand! is called on a single GuidProp (i.e. a single interval) then only this path contribution is returned.
However, if rand! is called on a list of GuidProp, then apart from summing over the results from solve_and_ll! an additional end-point contribution is added, i.e.

\[\color{royalblue}{\int_0^T G(t, X_t) \dd t}+\log\left(\color{forestgreen}{\tilde{\rho}(0,X_0)}\right)\]

is returned.