Robust, partially alive particle Metropolis-Hastings via the Frankenfilter

TL;DR

The paper tackles a key limitation of particle MCMC in zero-likelihood scenarios, proposing the Frankenfilter, an almost-alive particle filter that preserves unbiased likelihood estimation while bounding computational effort. It extends the alive filter by introducing minimum and maximum simulation budgets and allowing general nonnegative weights and success measures, making it suitable for PMMH and for incomplete/noisy observations. The authors provide rigorous unbiasedness results, tuning guidance to target a manageable relative variance, and practical recommendations for $m_+$ and $\,\mathfrak{s}$. Through simulations on Markov jump processes and a real SEIR example, the Frankenfilter demonstrates improved robustness to outliers and typically 2–3x efficiency gains over standard filters. This approach offers a practical, theoretically sound framework for reliable PMMH inference under challenging observation models and complex latent dynamics.

Abstract

When a hidden Markov model permits the conditional likelihood of an observation given the hidden process to be zero, all particle simulations from one observation time to the next could produce zeros. If so, the filtering distribution cannot be estimated and the estimated parameter likelihood is zero. The alive particle filter addresses this by simulating a random number of particles for each inter-observation interval, stopping after a target number of non-zero conditional likelihoods. For outlying observations or poor parameter values, a non-zero result can be extremely unlikely, and computational costs prohibitive. We introduce the Frankenfilter, a principled, partially alive particle filter that targets a user-defined amount of success whilst fixing lower and upper bounds on the number of simulations. The Frankenfilter produces unbiased estimators of the likelihood, suitable for pseudo-marginal Metropolis--Hastings (PMMH). We demonstrate that PMMH with the Frankenfilter is more robust to outliers and mis-specified initial parameter values than PMMH using standard particle filters, and is typically at least 2-3 times more efficient. We also provide advice for choosing the amount of success. In the case of n exact observations, this is particularly simple: target n successes.

Figures (4)

• Figure 1: Death model. Ratio of the expectation of the estimator of the likelihood and exact likelihood at $\theta=0.01$, based on the Frankenfilter (black lines) and alive particle filter (grey lines) using D50 (left panel) and D50mod (right panel).
• Figure 2: Protein dimerisation model. ESS/s versus required total success $\mathfrak{s}$, from the output of $50K$ iterations of PMMH, using data sets with 10 observations (left panel), 30 observations (middle panel), 50 observations (right panel). Solid lines correspond to data sets P10a, P30a and P50a whereas dotted lines correspond to data sets P10b, P30b and P50b (with ESS/s scaled to give the same maximum within observation regime). The value $\mathfrak{s}=T$ is indicated by a vertical line.
• Figure 3: Lotka-Volterra model. ESS/s versus required total success $\mathfrak{s}$, from the output of $50K$ iterations of PMMH, using data sets LV20 (top left), LV40 (top right), LV20prey (bottom left) and LV40prey (bottom right). Vertical lines show $\mathfrak{s}=T$ for the full observation data sets LV20 and LV40, and show the value of $\mathfrak{s}$ obtained by solving \ref{['eqn.Vrel.gen']} with $V_{rel}=1$ for the partial observation data sets LV20prey and LV40prey.
• Figure 4: SEIR model. Marginal posterior densities of $\beta$, $\mu$ and $\alpha$ based on the output of PMMH with FF (solid black lines) and BSPF (dashed black lines). The prior distribution is shown in grey.

Theorems & Definitions (8)

• Proposition 1
• Theorem 1
• Proposition 2
• Theorem 2
• Proposition 3
• Proposition 4
• Lemma 1
• Lemma 2