varying lag $\Delta$ and a varying number of particles $N$. The results are obtained by $1 \thinspace 000$ Monte Carlo runs on a single data set generated from the previously discussed LGSS model with $T=100$. We conclude again that faPF is preferable when available. The results are largely robust to the lag, as long as this is chosen large enough when using the faPF. A lag of about $12$ seems to be a good choice for this model when $T=100$ and when using the faPF with systematic resampling.

Burn-in and scale-invariance
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![The trace plots of the first $50$ steps using PMH0 (black), PMH1 (red) and PMH2 (blue). The dotted lines show the *true* parameters of the LGSS model. The gray contours show the log-posterior.[]{data-label="fig:lgss-scaleinvariance"}](lgss-scaleinvariance-paper.pdf){width="\columnwidth"}

Consider the problem of inferring $\{\theta_1,\theta_2\}=\{\phi,\sigma_v\}$ in the LGSS model . We simulate a single data set with parameters $\theta^{(1)}$ (as defined in the previous section) of length $T=250$. We use an uniform parameter prior over $|\phi|<1,\sigma_v > 0$ and initialise in $\theta_0=\{0.1,2\}$. We use faPF with systematic resampling, $N=100$ and $\Delta=12$. Here, we use the standard version of Algorithm \[alg:SMCfull\] to adjust the estimate of the Hessian in the cases when it is not 