Can one use the RTS (Rauch–Tung–Striebel) smoother to simulate latent factors (as opposed to Carter and Kohn procedure)?

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.

• 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. – Nick Cox May 20 '15 at 17:26
• I have inserted what I believe to be the relevant references. Please check that's what you meant. – Glen_b -Reinstate Monica Dec 17 '15 at 6:57

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 $p(\mathbf{x}_{0:n}\,|\,\mathbf{y}_{1:n})$.