MorphZ: Enhancing evidence estimation through the Morph approximation

TL;DR

The Morph approximation, a class of product approximations of probability densities that selects low-order disjoint parameter blocks by maximizing the sum of their total correlations, is introduced and used as the importance distribution in optimal bridge sampling.

Abstract

We introduce the Morph approximation, a class of product approximations of probability densities that selects low-order disjoint parameter blocks by maximizing the sum of their total correlations. We use the posterior approximation via Morph as the importance distribution in optimal bridge sampling. We denote this procedure by MorphZ, which serves as a post-processing estimator of the marginal likelihood. The MorphZ estimator requires only posterior samples together with the prior and likelihood, and is fully agnostic to the choice of sampler. We evaluate MorphZ's performance across statistical benchmarks, pulsar timing array (PTA) models, compact binary coalescence (CBC) gravitational-wave (GW) simulations and the GW150914 event. Across these applications, spanning low to high dimensionalities, MorphZ yields accurate evidence at substantially reduced computational cost relative to standard approaches, and can improve these estimates even when posterior coverage is incomplete. Its bridge sampling relative error diagnostic provides conservative uncertainty estimates. Because MorphZ operates directly on posterior draws, it complements exploration-oriented samplers by enabling fast and reliable evidence estimation, while it can be seamlessly integrated into existing inference workflows.

Figures (5)

• Figure 1: Convergence and accuracy of Gaussian-shells$\log(z)$ estimates versus $n_{\rm live}$ the number of live points. The posterior samples from one NS run with $n_{\rm live}$ is used for MorphZ estimates plotted at $n_{\rm live}$, with 3000 likelihood calls per each MorphZ estimate. Top: Estimated $\log(z)$, NS (black) and the MorphZ with Morph approximations of second order ($\mathcal{M}_2$, gray) and third order ($\mathcal{M}_3$, red). Symbols denote means over 100 run; error bars indicate $1\sigma$ uncertainties. The horizontal dotted line marks the true $\log(z)$. Bottom: Root-mean-square error (RMSE) of $\log(z)$ relative to the true value .
• Figure 2: Comparison of bridge sampling RE and empirical errors for $\log(z)$ across models in Table \ref{['tab:pta']}. For each model, the red dot marks and the faint red bar shows respectively the mean and $\pm 1 \sigma$ of 100 RE, and the black square marks the empirical error as the standard deviation of $100$ estimates of $\log(z)$. The horizontal red segment connects the approximate and empirical values; indicating the degree of under/overestimation.
• Figure 3: Comparison of $\log(z)$ estimates for the EPTA DR2FULL PSR+CURN model using GSS and MorphZ. The box plots show the distribution of $\log(z)$ for each Morph approximation order. The horizontal dashed line and shaded band indicate the mean and $\pm1\sigma$ range of the reference GSS estimates with 64 temperatures. Red diamonds (right axis) denote the forward Kullback-Leibler divergence $D_{\mathrm{KL}}(P|\mathcal{M}_{\mathcal{B}_{L}})$ between the posterior and each corresponding order of the Morph approximation.
• Figure 4: Comparison of dynesty, MorphZ$_{\text{(PT)}}$, and MorphZ$_{\text{(NS)}}$ across 100 independent BBH simulation (see Table \ref{['tab:ensemble-deltas']}). The mean and standard deviation of 100 $\log(z)$ estimate per simulation is used to compute $\Delta \log(z)= \log(z)_{\mathrm{method}} - \log(z)_{\mathrm{NS}}$ for both independent MorphZ estimates. Note that $\log(z)_{\mathrm{NS}}$ is used as a reference estimate because the true $\log(z)$ is unknown.
• Figure 5: Relative difference of the total sum of simulated weights and runtime comparison of SGM and ILP across different orders of the Morph approximation for 100 independent sum of weight maximization simulation. Each simulation assign a weight drawn from $\text{Beta}(\alpha,2)$ with $\alpha\sim\text{Uniform}(5,9)$ for each possible group from $\binom{50}{L}$. Top: Relative difference of the total sum of chosen weights, $((\mathrm{SGM}-\mathrm{ILP})/\mathrm{ILP})\times 100\%)$, shown as box-and-whisker plots for $L=2-5$ with the corresponding $\text{N}_b$ number of blocks of length $L$ ; the horizontal dashed line marks zero difference. Bottom: Wall-clock time in seconds for SGM (black) and ILP (red) on the same instances; markers denote mean and ($\pm1\sigma$). The dashed line marks $1\mathrm{s}$.

Theorems & Definitions (2)

• Definition 1
• Definition 2