An adaptive mixture view of particle filters

Figure 1.  Toy Example. In this experiment we show that the proposals of AM-PF-based algorithms (IAPF and OAPF) are closer to true posteriors compared to the competitors. In the first row, we show the transition kernels $ f({\bf x}_t | {\bf x}_{t-1}^{(m)}) $ and the likelihood associated with the true filtering distribution. We calculated $ \chi^2 $-divergence for these examples in Table 1. Note that here OAPF uses transition kernels for the proposal and their means $ \boldsymbol{\mu}_{t | t-1}^{(m)} $ as evaluation points, with $ K = M $

Figure 2.  (Linear Gaussian SSM.) Normalized (divided by the true value) MSE for the mean of a posterior distribution in the linear Gaussian state-space model, with 100 Monte Carlo replications. In this experiment, $ K $ was set to $ 5 $ for all numbers of particles, retaining performance. We found similar qualitative results for other values of the SSM parameters

Figure 3.  (Lorenz 63 system.) The plot shows effective sample size (ESS) over timesteps for $ 1000 $ Monte Carlo replications. In the legend we show the value of $ K $, which was set based on $ M = 1000 $, the number of particles. In this example, we show that unlike in the linear Gaussian system, unfortunately reducing too much the number of kernels $ K $ impacts performance for these settings of the system parameters. When $ K = M $, OAPF outperforms the other algorithms, which suggests that indeed the proposal constructed by the optimization of the mixture weights better matches the posterior distribution

Figure 4.  Here, we show for the Lorenz 63 experiment, (ⅰ) runtime (in seconds) per iteration in $ t $ (ⅱ) ESS and (ⅲ) ESS/runtime, vs the number of particles $ M \in \{ 100,200,300,400,500,600,700,800,900, 1000 \} $. The AM-PF algorithms (IAPF, OAPF) clearly suffer from the $ M^2 $ computational cost as discussed in Section 3. Recall that ESS is bounded by $ M $. Also, notice how the ESS for OAPF grows almost linearly as a function of $ M $.