RWHM_chain function

Simulating MCMC single chains using MCMC algorithms

These functions are used to define the elements $chain of the MCMC algorithms that are (and must be) implemented as lists in EntropyMCMC. These functions are usually only called by higher-level functions, see details below.

RWHM_chain(theta0, it = 100, target, f_param, q_param, q_pdf = gaussian_pdf, 
                q_proposal = gaussian_proposal) 
HMIS_norm_chain(theta0, it = 100, target, f_param, q_param, q_pdf = q_pdf_ISnorm, 
                q_proposal = q_proposal_ISnorm)
AMHaario_chain(theta0, it = 100, target, f_param, q_param, q_pdf = gaussian_pdf,
                q_proposal = gaussian_proposal)
IID_chain(theta0 = NULL, it = 100, target, f_param, q_param = NULL, q_pdf = NULL,
                q_proposal = NULL)

Arguments

• it: the number of iterations to simulate
• theta0: the initial position of the chain, a $d$-dim vector
• target: the user-defined target density
• f_param: the parameters (hyperparameters, data) of the user-defined target density
• q_param: the parameters of the proposal density, which structure depends on the algorithm and the proposal density chosen by the user. Defaults are for RWHM: a list with the mean and covariance matrix of the proposal. For AMHaario: a list that must contain three elements: v the initial covariance matrix, t0 the iteration of the end of initial stage with that matrix, and epsi the epsilon parameter (for the nondegenerate matrix part), see Haario et. al.(2001).
• q_pdf: the proposal density
• q_proposal: the function simulating the proposal for the next move

Details

Each MCMC algorithm is defined as a list with five elements, see the object RWHM for an example. The element $chain must provide the name of the function performing simulation of a single chain and returning that chain, with arguments that must follow the definition above. Each user can define its own MCMC starting with the algorithms provided (see also section below). These functions are thus usually called by higher-level functions like MCMCcopies, EntropyParallel, or their multicore versions, for simulating copies of MCMC chains in an automatic manner.

• RWHM_chain is used in RWHM, a standard Randow-Walk Hastings-Metropolis algorithm.
• HMIS_norm_chain is used in HMIS_norm, an Independence Sampler HM with gaussian proposal
• AMHaario_chain is used in AMHaario, the Haario Adaptive Hastings-Metropolis algorithm (Haario 2001), and is provided as an example of a benchmark AMCMC.
• IID_chain is used in IID_norm, a fake MCMC that is just a gaussian IID sampler.

Returns

A list with elements:

• theta: the simulated chain in an array of $it$ rows and $d$ columns (the dimension)

• paccept: the empirical acceptance rate

• finalcov: the last covariance matrix

• algo: the name of the algorithm (for plot methods)

References

H. Haario, E. Saksman, and J. Tamminen (2001), An adaptive Metropolis algorithm. Bernoulli 7, 223--242.

Author(s)

Didier Chauveau.

See Also

The algorithm already implemented, listed in EntropyMCMC-package.

The higher level functions that use these functions for simulation: MCMCcopies, EntropyParallel and their multicore versions.