Objective Bayesian analysis for the multivariate skew-t model

Abstract

We propose a novel Bayesian analysis of the p-variate skew-t model, providing a new parameterization, a set of non-informative priors and a sampler specifically designed to explore the posterior density of the model parameters. Extensions, such as the multivariate regression model with skewed errors and the stochastic frontiers model, are easily accommodated. A novelty introduced in the paper is given by the extension of the bivariate skew-normal model given in Liseo and Parisi (2013) to a more realistic p-variate skew-t model. We also introduce the R package mvst, which produces a posterior sample for the parameters of a multivariate skew-t model.

Appendices

Proof of Proposition 2.1

• (a) From one of the possible definitions of a multivariate ST r.v., it is known that $U\sim \textit{SN}_p( 0, \alpha , \varOmega , \nu )$; since Y is a simple transformation of U, its distribution is readily obtained.

• (b) Start from $f(y,z,v) = f(v) f(z) f(y\mid z, v)$. By assumption, f(z) is a standard Gaussian density, and

  $$\begin{aligned} \left( Y \mid Z=z, V=v\right) = \left( \xi + \omega U \mid Z=z, V=v\right) = {\left\{ \begin{array}{ll} \xi + \omega X v^{-1/2} &{} z \ge 0 \\ \xi - \omega X v^{-1/2} &{} z < 0 \\ \end{array}\right. }. \end{aligned}$$

  Then, by using simple results on conditional Gaussian densities, one gets

  $$\begin{aligned} \left( Y \mid Z=z, V=v\right) \sim \left\{ \begin{array}{ll} N_p\left( \xi + \omega \delta \displaystyle \frac{z}{\sqrt{v}}, \frac{1}{v}\omega (\varOmega - \delta \delta ^\prime ) \omega \right) &{} z \ge 0\\ N_p\left( \xi - \omega \delta \displaystyle \frac{z}{\sqrt{v}}, \frac{1}{v}\omega (\varOmega - \delta \delta ^\prime ) \omega \right) &{} z < 0\\ \end{array} \right. \end{aligned}$$

  Hence the result in (4).

Proposal distributions

We use the full conditional distributions as proposals for the latent variables Z and $\xi$: each $Z_i$ has the following full conditional distribution

$$\begin{aligned} \pi (z_i|\cdots ) = \frac{\phi (z^+_i|m_i, v_\theta )}{2(1-\varPhi (z_i|m_i,v_\theta ))} \end{aligned}$$

(10)

where

$$\begin{aligned} v_\theta= & {} (1+\psi 'G^{-1}\psi )^{-1}\\ m_i= & {} v_\theta \sqrt{v_i} (\psi ' G^{-1}(y_i-\xi )) \end{aligned}$$

The variables $Z_i$ can be drawn as the product of $Z^+_i$, a normal r.v. with parameters $m_i$ and $v_\theta$ truncated in 0 and the sign $S_i$, uniform on $\{-1, 1\}$. To generate values $Z^+$ a rejection sampler has been employed (see Robert 1995).

The parameter $\xi$ has the following full conditional density:

$$\begin{aligned} (\xi |\cdots ) \sim N_p\left( \displaystyle \frac{1}{\sum _{i=1}^n v_i} \left( \sum _{i=1}^n(v_i y_i) - \psi \sum _{i=1}^n \sqrt{v_i}|z_i|\right) ,\displaystyle \frac{1}{\sum _{i=1}^n v_i}\,G\right) \end{aligned}$$

The parameters $\psi$ and G have untractable full conditional distributions. To obtain a proposal distribution, they are approximated using only the contribution of the likelihood to the full conditional density. The parameter $\psi$ has the following full conditional distribution

$$\begin{aligned}&\pi (\psi \mid \cdots ) \propto \prod _{j=1}^p \left[ (G_{jj} + \psi _j^2)^{-1/2}\right] \mathbb {1}_{{\delta }}(\varDelta _{\varSigma })\\&\quad \exp \Bigg \{-\displaystyle \frac{1}{2} \sum _{i=1}^n v_i \left( y_i-\xi -\psi \frac{|z_i|}{\sqrt{v_i}}\right) ' G^{-1} \left( y_i-\xi -\psi \frac{|z_i|}{\sqrt{v_i}}\right) \Bigg \}, \end{aligned}$$

where $\mathbb {1}_{x}(\cdot )$ denotes the indicator function. By ignoring the first two factors, we obtain the following proposal distribution

$$\begin{aligned} q(\psi ) = \phi _p \left( \psi \, \Big | \frac{1}{\sum _{i=1}^n z_i^2}\sum _{i=1}^n |z_i|\sqrt{v_i}(y_i-\xi ),\frac{1}{\sum _{i=1}^n z_i^2} G\right) \end{aligned}$$

The proposal distribution has a positive density on $\mathbb R^p$, while the full conditional is bounded on $\varDelta _\varSigma$. This feature improves the ability of the sampler to explore the parameter space; moreover, particles which don’t respect the constraint (7) will be automatically discarded, as they have null prior (and posterior) probability density, hence a null importance weight.

The parameter G has the following full conditional density

$$\begin{aligned} \pi (G|\cdots ) \propto \pi (\varSigma )|J| |G|^{-n/2}\exp \left\{ -\frac{1}{2} {\text {tr}}(G^{-1}\varXi )\right\} \end{aligned}$$

where

$$\begin{aligned} \varXi = \sum _{i=1}^n v_i \left( y_i-\xi -\psi \frac{|z_i|}{\sqrt{v_i}}\right) \left( y_i-\xi -\psi \frac{|z_i|}{\sqrt{v_i}}\right) '. \end{aligned}$$

Ignoring the prior term we obtain

$$\begin{aligned} q(G) = \textit{IW}(n-p-1, \varXi ). \end{aligned}$$

Details about the rejection sampler

For a generic latent variable $V_i$, the Kullback–Leibler divergence $\textit{KL}(f||\pi _v)$ is given by

$$\begin{aligned} \textit{KL}(f||\pi _v) = \int _{\mathbb R^+} f(v_i)\log \left( \frac{k_v\beta _v^{\alpha _v}}{2\varGamma (\alpha _v)} v_i^{\alpha _v/2-C} \exp \{A_i v_i + (B_i-\beta _v)\sqrt{v_i}\}\right) dv_i \end{aligned}$$

which has an analytical solution for $\alpha _v^\star = 2C$:

$$\begin{aligned} \textit{KL}(f||\pi _v) = \log \left( \frac{k_v\beta _v^{2C}}{2\varGamma (2C)}\right) + \frac{2C(2C+1)A_i}{\beta _v^2} + 2C\log (\beta _v) + \frac{2\textit{CB}_i}{\beta } - 2C. \end{aligned}$$

This divergence has always one (and only one) minimum in $\mathbb R^+$, given by

$$\begin{aligned} \beta _v^\star = \frac{1}{2} \left( B_i + \sqrt{B_i^2 + 8 A_i (2C+1)}\right) . \end{aligned}$$