There is a slight mistake in the Gibbs sampling step of the dynamics parameters of the LinearGaussianConjugateSSM model.
Briefly, defining the dynamics parameters as $W_{z} \equiv [F, B, b]^\intercal $, for this model the prior is given by:
$$Q, W_{z} \sim \mathcal{MN}\mathcal{W}^{-1} (M_{z0}, V_{z0}, \nu_{q0}, \psi_{q0})$$
and states are subsequently sampled according to
$$\begin{aligned}
z^{(i)}_t \sim \mathcal{N}(F z^{(i)}_{t-1} + B u^{(i)}_t + b, Q) \equiv \mathcal{N}(W_{z} x^{(i)}_{zt}, Q)\\\
\end{aligned}$$
where we defined $x^{(i)}_{zt} = [z^{(i)}_{t-1}, u^{(i)}_t, 1]^\intercal$. Using the conjugacy of the matrix normal inverse-Wishart distribution, the dynamics parameters and variances, $Q, W_{z} \sim \mathcal{MN}\mathcal{W}^{-1} (M_{z}, V_{z}, \nu_{q}, \psi_{q})$, given the states are derived from the summary statistics:
$$\begin{aligned}
S_{z}^{xx} = V_{z0} + \sum_{i=1}^{m} \sum_{t=2}^T x^{(i)}_{zt} {x^{(i)}_{zt}}^\intercal\\\
S_{z}^{yy} = M_{z0} V_{z0} M_{z0}^\intercal + \sum_{i=1}^{m} \sum_{t=2}^T z^{(i)}_{t} {z^{(i)}_{t}}^\intercal\\\
S_{z}^{yx} = M_{z0} V_{z0} + \sum_{i=1}^{m} \sum_{t=2}^Tz^{(i)}_{t} {x^{(i)}_{t}}^\intercal \\\
S_{z}^{y|x} = S_{z}^{yx} - S_{z}^{yx} (S_{z}^{xx})^{-1} {S_{z}^{yx}}^\intercal
\end{aligned}$$
However, in the code the summary statistics are calculated based on $[z^{(i)}_{t}, u^{(i)}_t, 1]^\intercal $ instead of $x^{(i)}_{zt} = [z^{(i)}_{t-1}, u^{(i)}_t, 1]^\intercal $.
I had already corrected this mistake in PR #403. But if you want, I can make a separate PR with just this fix instead.
Kind regards,
Hylke
There is a slight mistake in the Gibbs sampling step of the dynamics parameters of the
LinearGaussianConjugateSSMmodel.Briefly, defining the dynamics parameters as$W_{z} \equiv [F, B, b]^\intercal $ , for this model the prior is given by:
and states are subsequently sampled according to
where we defined$x^{(i)}_{zt} = [z^{(i)}_{t-1}, u^{(i)}_t, 1]^\intercal$ . Using the conjugacy of the matrix normal inverse-Wishart distribution, the dynamics parameters and variances, $Q, W_{z} \sim \mathcal{MN}\mathcal{W}^{-1} (M_{z}, V_{z}, \nu_{q}, \psi_{q})$ , given the states are derived from the summary statistics:
However, in the code the summary statistics are calculated based on$[z^{(i)}_{t}, u^{(i)}_t, 1]^\intercal $ instead of $x^{(i)}_{zt} = [z^{(i)}_{t-1}, u^{(i)}_t, 1]^\intercal $ .
I had already corrected this mistake in PR #403. But if you want, I can make a separate PR with just this fix instead.
Kind regards,
Hylke