There is an obscure and useful function that makes it easy to fit stochastic differential equations to data insofar as the model can be linearized without causing too much trouble. The function discretizes the continuous-time (i.e., differential equation) state matrices A, the drift or state transition matrix, B, the input or covariate coefficient matrix, and Q, diffusion or noise covariance matrix. That means that the function essentially takes the differential equation in matrix form and solves it for a given time step. The discretized matrices function like those of an autoregressive process. Some details of this approach and what this does can be found here but not exactly a complete implementation, namely with matrix B. So this is one of those code blocks I just have backed up in several project folders in various languages.
Understanding MCMC and autodifferentiation
In putting together educational materials related to Stan and posterior sampling, I remembered two good ones.
The MCMC interactive gallery is the best set of MCMC visualizations I’ve found.
Stan uses NUTS, so it has to calculate numerous gradients for each new sample and does so by autodifferentiation. I recommend this video for understanding autodiff. It helps a lot to know what Stan is doing with the model code to avoid giving it more work than necessary.
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