It is a little surprising that I have not found anything online in the literature which clarified this.

I am working on an MCMC Gibbs sampling procedure to calibrate a "dynamic factor model". One of the steps involves simulating latent factors through a simulation smoother such as the Carter and Kohn (1994)[1] algorithm.

That basically entails running a Kalman filter forward in time (say $1:T$), then drawing randomly from the posterior at time $T$, followed by backward recursions and random draws from $T-1 : 1$.

In my specific problem, the matrices are quite sparse and I found that in the backward recursions I run into some numerical problems in the variances (a.k.a. smoothed matrix $P$) of my drawn latent factors. The simulated latent factors are sensible, however.

I found, however, that when I use the RTS fixed-interval smoother [2] at each step (very similar equations to Carter and Kohn equations except for variance propagation) I get good results for the variances.

Is what I am doing in violation of any assumptions I am not aware of? Is it a valid procedure?

[1] Carter C. K. and Kohn R.(1994),
"On Gibbs Sampling for State Space Models"
Biometrika 81:3 (Aug.), pp. 541-553

[2] Rauch H.E. Striebel, C.T., and Tungh F. (1965),
"Maximum likelihood estimates of linear dynamic systems",
AIAA Journal, 3:8, pp. 1445-1450.

  • $\begingroup$ It's good practice here as elsewhere not to assume that name (date) references are sufficient for interested readers, but to give full references or stable web links. $\endgroup$ – Nick Cox May 20 '15 at 17:26
  • $\begingroup$ I have inserted what I believe to be the relevant references. Please check that's what you meant. $\endgroup$ – Glen_b -Reinstate Monica Dec 17 '15 at 6:57

In a linear Gaussian system the RTS smoother provides MLE parameter estimates for marginal smoothing densities. In contrast, the Carter & Kohn algorithm is a "simulation" smoother which requires repeated draws on the backwards pass to estimate marginal smoothing density parameters.

Lindsten and Schon (2013) compare the RTS smoother vs. a simulation smoother on a linear Gaussian system in Ex. 1.4 and Fig. 1.3, and in Section 3.2 they discuss convergence properties of simulation smoothers on nonlinear/non-Gaussian systems.

The additional noise from the simulation smoother variance estimates may be causing numerical instabilities when calculating inverses in your recursive backwards steps. Regardless, if your system is linear Gaussian, then why are you using a simulation smoother instead of the RTS smoother?


It depends on what you mean by "...I use the RTS fixed-interval smoother..."

Off-the-shelf, the RTS fixed-interval smoother (as stated here for example) is recursively computing the means and covariance matrices of the marginal distributions $p(\mathbf{x}_k\,|\, \mathbf{y}_{1:n})$, $k < n$ (the past states given all of the data). So, if you're using RTS to compute those means and covariances, and then naively drawing the states out of those marginals (or worse, taking those means to be your draws), that's incorrect. You're drawing the states as if they were independent, which they are not.

But if what you're doing is drawing recursively backwards, treating your draw of $\mathbf{x}_{k+1}$ as the value of $\hat{\mathbf{x}}_{k+1|n}$ and setting $\mathbf{P}_{k+1|n}$ to $\mathbf{0}$ when you apply the RTS recursion, then that should be equivalent to Carter-Kohn. Both are valid algorithms for generating direct, iid draws from the full posterior $p(\mathbf{x}_{0:n}\,|\,\mathbf{y}_{1:n})$.


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