Doubly stochastic model

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In statistics, a doubly stochastic model is a type of model that can arise in many contexts, but in particular in modelling time-series and stochastic processes.

The basic idea for a doubly stochastic model is that an observed random variable in modelled in two stages. In one stage, the distribution of the observed outcome is represented in a fairly standard way using one or more parameters. At a second stage, some of these parameters (often only one) are treated as being themselves random variables. In a univariate context this is essentially the same as the well-known concept of compounded distributions. For the more general case of doubly stochastic models, there is the idea that many values in a time-series or stochastic model are simultaneously affected by the underlying parameters, either by using a single parameter affecting many outcome variates, or by treating the underlying parameter as a time-series or stochastic process in its own right.

An example of a doubly stochastic model is the following. The observed values in a point process might be modelled as a Poisson Process in which the rate (the relevant underlying parameter) is modelling as being the exponential of a Gaussian Process.