72,211 research outputs found
Exponential families of mixed Poisson distributions
If I=(I1,…,Id) is a random variable on [0,∞)d with distribution μ(dλ1,…,dλd), the mixed Poisson distribution MP(μ) on View the MathML source is the distribution of (N1(I1),…,Nd(Id)) where N1,…,Nd are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v0+v1Y1+cdots, three dots, centered+vqYq for some qless-than-or-equals, slantd, for some vectors v0,…,vq of [0,∞)d with disjoint supports and for independent standard real gamma random variables Y1,…,Yq
Reduction of Markov chains with two-time-scale state transitions
In this paper, we consider a general class of two-time-scale Markov chains
whose transition rate matrix depends on a parameter . We assume that
some transition rates of the Markov chain will tend to infinity as
. We divide the state space of the Markov chain
into a fast state space and a slow state space and define a reduced chain
on the slow state space. Our main result is that the distribution of the
original chain will converge in total variation distance to that of the
reduced chain uniformly in time as .Comment: 30 pages, 3 figures; Stochastics: An International Journal of
Probability and Stochastic Processes, 201
Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification
Gaussian processes are a natural way of defining prior distributions over
functions of one or more input variables. In a simple nonparametric regression
problem, where such a function gives the mean of a Gaussian distribution for an
observed response, a Gaussian process model can easily be implemented using
matrix computations that are feasible for datasets of up to about a thousand
cases. Hyperparameters that define the covariance function of the Gaussian
process can be sampled using Markov chain methods. Regression models where the
noise has a t distribution and logistic or probit models for classification
applications can be implemented by sampling as well for latent values
underlying the observations. Software is now available that implements these
methods using covariance functions with hierarchical parameterizations. Models
defined in this way can discover high-level properties of the data, such as
which inputs are relevant to predicting the response
- …