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mcmc_setup.md

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Definition of the MCMC sampler

To define the MCMC sampler there are two objects that need to be defined. First, we must specify a range of possible transition kernels that the MCMC sampler can use (these transition kernels can change and adapt as the MCMC sampler progresses if certain settings are turned on). These will be stored in the object MCMCSetup. We then have to specify the schedule i.e. what steps need to be taken by the MCMC sampler at each of its step.

MCMC Setup

As in the previous section, we define transition steps for the problem of inference for diffusion processes. We will need two types of updates: imputations and parameter updates. Suppose that we don't use any blocking, to see how to modify the code below to include blocking see [this page](@ref blocking_header). Then imputation can be defined via:

precond_Crank_Nicolson = ...
ODE_solver = ... # possible choices Ralston3, RK4, Tsit5, Vern7
impute_step = Imputation(NoBlocking(), precond_Crank_Nicolson, ODE_solver)
Imputation

To define parameter updates more parameters need to be defined:

pu1 = ParamUpdate(MetropolisHastingsUpdt(), # the type of parameter update
                  5,                        # which coordinate is updated
                  θ_init,                   # needed just for the dimension of the parameter
                  UniformRandomWalk(0.5, true), # transition kernel
                  ImproperPosPrior(),       # prior
                  UpdtAuxiliary(            # auxiliary information
                      Vern7(),              # ODE solver
                      true))                # whether the update prompts for re-computing H,Hnu,c
pu2 = ...

Then the setup can be defined as:

mcmc_setup = MCMCSetup(impute_step, pu1, pu2)

There are currently two different ways of updating parameters:

ConjugateUpdt
MetropolisHastingsUpdt

And two generic transition kernels:

UniformRandomWalk
GaussianRandomWalk

MCMC schedule

It is now necessary to tell the mcmc sampler what to do at each iteration. To this end let's define the MCMCSchedule

MCMCSchedule

An example of how to define it is given here:

MCMCSchedule(111, [[1,2], [1,3]],
              (save=5, verbose=10, warm_up=7,
                readjust=(x->x%20==0), fuse=(x->false)))

It specifies the total number of MCMC steps to be 111, and the array [[1,2], [1,3]] makes the sampler alternate between performing imputation and parameter update pu1 on each odd step and imputation and parameter update pu2 on each even step. It saves the imputed path on every 5th iteration, prints out useful progress message to the console every 10th step, ignores parameter update steps for the first 7 iterations readjusts the proposals once in every 20th step (see [this page](@ref adaptive_header) for more information on adaptive schemes). It also never fuses transition kernels (see [this page](@ref fusion_header) for more information about fusion).