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PARAMETER ESTIMATION OF BILINEAR PROCESSES USING ABC 141
of data is naturally done sequentially. In these cases, decisions have to be taken immediately as data becomes available and conditionally on the existing infor- mation. The decisions cannot be postponed until the entire data set becomes available (Doucet et al. [13]). In this framework of bayesian inference, and con- sidering that the focus is on the posterior and/or the predictive distributions, SMC methods simulate values sequentially from the intermediate distributions, in the sense that only the available data (up to the moment) are used.
ABC algorithms appear in the literature as a way to overcome computational problems associated to models that have analytically complicated or intractable likelihoods. In this framework, using MCMC methods can be very hard or even impossible to accomplish. The ABC algorithms are based on simulated values from the model of interest for a given set of parameter values and the information contained in the sample simulated is resumed in a set of summary statistics which is compared to the same summary statistics obtained from the fixed observed sample. The set of parameter values is then kept or discarded according to how these sets of summary statistics relate. The success of the ABC algorithm that is implemented depends on several factors, such as how easy it is to simulate data from the model of interest and how accurately do the summary statistics represent the data in hand. In this bilinear framework, there is an obvious lack of summary statistics. Although the Bayesian approaches via ABC method were developed in a population genetics context, they are nowadays used in several other areas such as archeology, epidemiology and ecology (see, for instance, Beaumont et al. [1], Csill´ery et al. [9]) and Marin et al. [27] for further references).
3.1. The ABC algorithm. On what follows, let ✓ be a parameter, or a vector of parameters, associated to a sampling model M, p(✓) the prior distribution of ✓ and Dt = (x1,x2,...,xt) the observations available at instant t (t 1). The posterior distribution of ✓ is referred as p(✓ | Dt) / L(✓ | Dt)p(✓), where L(·) is the likelihood.
Let us denote by Dt, the fixed observed data set. Similarly, denote by S, a vector of summary statistics which give a good representative of the model. Ideally, such vector of summary statistics should be su cient for the model, but when the likelihood is not known or is intractable, then su cient statistics other than the ordered sample will not exist. Let S0 = S(Dt) and S⇤ = S(Dt⇤) be respectively be the value of the summary statistics calculated at the fixed observed and simulated samples Dt and Dt⇤. The ABC algorithms are based on the widely know rejection method (Paulino et al. [29]). Essentially, they consist