A Decision-Theoretic Approach to Summarising Posterior Information in a Sample of step Functions

Abstract

An important task in analysing time series of the current passed by a single ion channel is the restoration of the true signal from the noisy recording. Hodgson [1] devised a reversible jump Markov chain Monte Carlo (MCMC) algorithm to implement fully Bayesian single channel analysis. Noiseless ion channel signals are typically modelled as step functions with a discrete number of levels. As the number of steps is unknown, the joint posterior distribution under the Bayesian paradigm exhibits variable dimensionality, necessitating the use of reversible jump MCMC (Green [2]). This paper addresses the task of summarising posterior knowledge about the step function representing the noiseless channel signal, using Bayesian decision theory and the ideas of Rue [3]. We compute Bayes estimates of the true step function under a selection of loss functions from the same family.