Adaptive kernel density estimation proposal in gravitational wave data analysis

The Markov Chain Monte Carlo approach is frequently used within a Bayesian framework to sample the target posterior distribution. Its efficiency strongly depends on the proposal distribution used to build the chain. The best jump proposal is the one that closely resembles the unknown target distribution; therefore, we suggest an adaptive proposal distribution based on kernel density estimation (KDE). We group the model’s parameters according to their correlation and build a KDE based on the already accepted points for each group. We update the KDE-based proposal until it stabilizes. We argue that such a proposal distribution could be efficient in applications where the data volume is increasing. We tested it on several astrophysical datasets (IPTA and LISA) and have shown that, in some cases, the KDE-based proposal also helps to reduce the chains’ autocorrelation length. The efficiency of this proposal distribution is reduced in case of strong correlations between a large group of parameters.

Figure 1

We plot three examples of datasets where we have on the left no correlation, in the middle linear correlation and on the right more elaborate features. On the top panels, we have the corresponding values of the correlation coefficient based on a simple evaluation of the covariance matrix. We see that it excels at finding the linear correlation but completely fails with the right panel features. In the bottom panels, we have the same three datasets in red with their corresponding shuffled version that destroys correlations in blue. While the left panel remains unchanged, the others are affected, and the JS divergence captures it.

Figure 2

Illustration of the parameter grouping process using a JSD matrix. For this example, the JSD threshold is set to 0.25; hence the parameter subgroups will be [0, 2, 3, 4] and [1]. Starting from parameter 0, we find that JSD for parameters 2 and 3 are above the threshold, so they are both correlated with parameter 0. Then we check for parameters 2 and 3 and find that 4 is correlated with 3 while 2 sees no additional correlation. The last step would have been to check for 4 and find no additional correlations. Therefore, 0, 2, 3, and 4 will form a subgroup of correlated parameters. The parameter “1” is the last parameter that does not correlate with others (according to the adopted threshold) and will be a subgroup on its own.

Figure 3

For a chain of current length $N$, we get rid of the burn-in, then we extract $n_s$ linearly-spaced samples from the remaining fraction of the chain. These $n_s$ points are used to build the KDE.

Figure 4

Distribution of the KL divergence for all sub-KDEs. 1D sub-KDEs are of best quality.

Figure 5

Two-dimensional corner plot of binned original dataset against smooth KDE plot. KDE is blue, original is orange. The contour levels for the original dataset in the bottom left panel are [0.1, 0.95]. For this sub-KDE, $\mathrm{KL}\simeq 0.11$.

Figure 6

Acceptance rate for several values of $n_{\mathrm{kde}}$.

Figure 7

Mean, maximum, and minimum autocorrelation lengths for several values of $n_{\mathrm{kde}}$.

Figure 8

Evolution of KL and $\mathrm{\Delta }\mathrm{KL}$. Left panel, the evolution of KL before fixing parameter groups. Right panel, the evolution of KL after fixing parameter groups, we start computing $\mathrm{\Delta }\mathrm{KL}$ 5 updates after the grouping was fixed. The thick red line indicates the $\mathrm{\Delta }\mathrm{KL}$ threshold level of 5% below which we reach convergence.

Figure 9

Acceptance rate of all proposals for $n_{\mathrm{kde}}=1$. KDE and SCAM are on top panel, SLICE on bottom panel. Red plot and black dashed line shows where KDE finally converged and stopped updating.

Figure 10

Acceptance rate of all proposals for $n_{\mathrm{kde}}=5$. KDE and SCAM are on top panel, SLICE on bottom panel. Red plot and black-dashed line shows where KDE finally converged and stopped updating.

Figure 11

Maximum and mean autocorrelation lengths for $n_{\mathrm{kde}}=1$, comparing a run with KDE against a run without KDE. Black dashed line shows where KDE finally converged and stopped updating.