Bayesian Analysis of Dependent Non-Gaussian Data

The Conjugate Multivariate Distribution:


Suppose Z is distributed according to the natural exponential family, then

f(Z|Y) = exp{ZY - b ψ(Y) + c(Z)}, 

where f denotes a generic probability density function/probability mass function (pdf/pmf). The function bψ(Y) is often called the log partition function, and exp{c(Z)} is a normalizing constant. It follows from Diaconis and Ylvisaker (1979) that the conjugate prior distribution for Y is given by,

f(Y|a,k) = K(a,k) exp{a Y - k ψ(Y)},

where K(a, k) is a normalizing constant. Let DY(a;k; ψ) denote a shorthand for the pdf in (2). Here “DY” stands for “Diaconis-Ylvisaker.” It is immediate from the previous two equations that Y|Z ~ DY(a+Z;k +b; ψ). This conjugacy motivates the development of a multivariate version of the DY random variable to model dependent non-Gaussian data from the natural exponential family. Specifically, we let,

Z = mu + Vw,

where Z, mu, and w are n-dimensional random vectors, mu is unknown, w consists of iid DY random variables, and V is an n by n real-valued matrix representing the Cholesky decomposition of a covariance matrix. In our work we show that conditional conjugacy exists in this multivariate setting aswell. This allows one to improve, in terms of computation and predictive performance, upon the typical latent Gaussian process modeling strategy.



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Bayesian Models with Unknown Transformations:

 

Let Zij denote the observed data for i = 1,..., nj and j = 1, 2, 3. We consider the setting where for each i, Zi1 is continuous-valued, Zi2 is integer-valued ranging from 0,..., bi, and Zi3 is binary. It is assumed that Zi1, Zi2, and Zi3 are distributed according to different classes of probability density functions/probability mass functions (hence referred to as multiple-type responses). One classical strategy to model data of this type is to impose a transformation,

 

 

hj(Zij) | Yij, θ ~Dist (Yij, θ ),  i = 1,…, nj, j = 1,2,3,

 

where hj is a transformation of the datum Zij, “Dist” is a short-hand used for a probability density function (pdf), gj{E(Zij} = Yij, θ is a real-valued parameter vector, and gj is known as a link function. Additionally, Yij is defined for  i = 1,…, n, j = 1,2,3. Here, “Dist (Yij, θ )'' represents any preferred model for continuous data, and inference on Yij and θ is the primary goal.

 

Drop the functional notation for hj and write hij = hj(Zij). These transformations convert a multiple response type data set (e.g., { Zij }) to a single response type data set (e.g., { hij }), since hij follows a single distribution with a continuous support.

 

We introduce a Bayesian solution to the problem of an unknown transformation. In particular, we define pdfs and probability mass functions (pmf), f(Zij | hij) and f(hij).

 

 

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