Approximate Bayesian Computation for the inference of multivariate partially observed stochastic processes with application to neuroscience
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6th Austrian Stochastics Days
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In many signal-processing applications, it is of primary interest to decode/reconstruct the unobserved signal based on some partially observed information. Some examples are automatic speech, face, gesture and handwriting recognition, and neuroscience (ion channels modeling). From a mathematical point of view, this corresponds to estimate model parameters of an unknown coordinate based on discrete observations of one or more other coordinates. Here we consider a partially observed bivariate stochastic process and discuss it in the framework of stochastic modelling of single neuron dynamics. None of the two components is directly observed: the available observations correspond to hitting times of the first component to the second component. Our goal is to provide statistical inference of the underlying model parameters. This is particularly difficult because: 1. the considered process does not fit into the well-known class of hidden Markov models; 2. the available data consists of a point process with not iid consecutive intertime intervals; 3. the likelihood of the model is intractable. We tackle this problem using Approximate Bayesian Computation, a likelihood-free method requiring the development of suitable distances to apply in an algorithm similar to the important-sampling. After presenting the method (proposing several possible distances), I illustrate how to use it on the considered model.