Doubly stochastic model
Encyclopedia
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 is 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 distribution
s. 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.
The basic idea here is essentially similar to that broadly used in latent variable model
s except that here the quantities playing the role of latent variable
s usually have an underlying dependence structure related to the time-series or spatial context.
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 treated as being the exponential of a Gaussian process
.
The basic idea for a doubly stochastic model is that an observed random variable is 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 distribution
Compound probability distribution
In probability theory, a compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution F with an unknown parameter θ that is distributed according to some other distribution G, and then...
s. 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.
The basic idea here is essentially similar to that broadly used in latent variable model
Latent variable model
A latent variable model is a statistical model that relates a set of variables to a set of latent variables.It is assumed that 1) the responses on the indicators or manifest variables are the result of...
s except that here the quantities playing the role of latent variable
Latent variable
In statistics, latent variables , are variables that are not directly observed but are rather inferred from other variables that are observed . Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models...
s usually have an underlying dependence structure related to the time-series or spatial context.
An example of a doubly stochastic model is the following. The observed values in a point process might be modelled as a Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
in which the rate (the relevant underlying parameter) is treated as being the exponential of a Gaussian process
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...
.