Multivariate Non-central t-distribution fit

In this section we describe how the multivariate non-central t-distribution fit is performed in BIDET.

Assume that we have $P \times N_J$ matrix, $x$, with elements, $(x_{jp})$, where $j=1\ldots N_J$ indexes samples and $p=1\ldots P$ indexes parameters. The task is to fit to these samples a multivariate non-central t-distribution, $t(\mu,\sigma^2 \Sigma,\nu)$ (as described in appendix 10.3).

In BIDET we constrain the mean of the multivariate non-central t-distribution, $\mu_{\beta_g}$, to be equal to that from the fast posterior approximation for $\mu_{\beta_g}$ described in section 3.5. If we are not using this constraint then we can set the mean $\mu$ to the sample mean, i.e:

$$\mu_p=\frac{1}{N_J}\sum_j x_{jp}$$ (27)

We can also directly estimate the normalised covariance $\Sigma$ using the sample covariance, $\widehat{\Sigma}$:

$$\tilde{\Sigma} = \widehat{\Sigma}/ \vert\widehat{\Sigma}\vert^{1/P}$$

$$\widehat{\Sigma} = (x-M)(x-M)^T/(N_J-1)$$ (28)

where $M=\{\mu,\mu \ldots, \mu\}^T$.

We still need to estimate $\sigma^2$ and $\nu$. Fortunately, we can represent a multivariate non-central t-distribution using a two-parameter gamma distribution and a multivariate Normal distribution in a Bayesian framework, by introducing hidden variables $\tau_i$ (see appendix 10.3). With hidden variables we can use the Expection-Maximisation (EM) algorithm. In the E-step we obtain the expected value of the hidden variables, $\tau_j$:

$$E_{\tau_j\vert\nu_{(t)},\sigma^2_{(t)},x}[\tau_j] = \frac{\sigma^2_{(t)}(\nu_{(t)}+P)}{\nu_{(t)}\sigma^2_{(t)}+s_j}$$ (29)

where:

$$s_j = (x_j-\mu_j)^T \tilde{\Sigma}^{-1} (x_j-\mu_j)$$ (30)

and then in the M-step we can minimise the joint posterior over $\nu,\sigma^2$ given $\tau_{j(t)}=E_{\tau_j\vert\nu_{(t)},\sigma_{(t)}^2,x}[\tau_j]$ to get updates for $\nu,\sigma^2$ as:

$$\sigma^2_{(t+1)} = \frac{1}{N_J P}\sum_j \tau_{j(t)} s_j$$

$$\nu_{(t+1)} = \frac{2}{1-\sigma^2_{(t)}/(\frac{1}{N_J-1}\sum_j s_j)}$$ (31)

Convergence normally occurs after about $10$ iterations. To be conservative we therefore use $50$ iterations.