Vancouver, June 4-6 2009

Standard MCMC uses a Markov Chain where a new state is proposed, then with some probability, the proposed state is accepted or the previous state is maintained. After a long time of continuing this process, (under some conditions) states visited by the Markov Chain approximate a sample from the posterior density of model parameters given the data.

When the posterior of interest is the posterior probability of a model given the data, marginalizing over the distribution of the models associated parameters, the Markov chain must be able to visit the state space associated with the potential models. This process of sampling values from different models, called Reversible Jump MCMC (RJMCMC), requires altering the dimension of the state space of the Markov chain. For instance, consider proposing a move that transitions the Markov chain between model 1 with parameter dimension d1 and model 2 with parameter dimension d2 > d1. In order to determine if the move is acceptable, the proposed parameter space dimensions must be augmented or reduced in such a way as to provide model parameters a reasonable set of parameter values from the candidate model to be accepted at rates coinciding with the relative posterior probabilities of the models given the data. When the models in question are non-nested dynamic systems, the model and dimension jumping moves are further complicated by the variety of dynamics that a system of differential equations can produce.

In this talk I propose a new algorithm to improve the model changing and dimension altering process of RJMCMC in dynamic system models. The new method transitions parameters between candidate models by stepping through the function spaces of the dynamic systems. When moving from model 1 to model 2, proposed values of parameters from model 2 are initially determined by this transformation through the function spaces of models 1 and 2. The proposed parameters from model 2 are then augmented in a direction orthogonal to the space spanned by model 1 but within the space spanned by model 2 before deciding whether or not to accept the proposed values. A similar strategy of proposing parameters through a transformation across function spaces and parameter space reduction through a move in orthogonal dimensions spanning the non-overlapping parameter space between candidate models is used to propose a change of models from m2 to m1.

I will describe the process of Reversible Jumps through Function Spaces MCMC and show it's performance in some examples.

Dave Campbell

Assistant Professor

Department of Statistics and

Actuarial Science

Simon Fraser University

Surrey, British Columbia