## Alternative evidence estimation in phylogenetics

An alternative marginal likelihood estimator for phylogenetic models is the title of the paper recently posted by Arima and Tardella on arXiv. While working on phylogenetic trees, it does not consider an ABC approach for computing the evidence but instead relies on harmonic mean estimators (since thermodynamic alternatives take too long). It is based upon the harmonic mean estimator of the marginal (already discussed in this post), namely which

$\mathbb{E}\left[ \dfrac{\varphi(\theta)}{\pi(\theta) L(\theta|x)} \mid x \right] = \dfrac{1}{m(x)}$

holds for any density φ. As Arima and Tardella point out, the special case when φ is the prior (Newton and Raftery, 1994, JRSS Series B) is still the one used in phylogenetic softwares like MrBayes, PHASE and BEAST, despite clear warnings about its unreliability! In An alternative marginal likelihood estimator for phylogenetic models, the authors propose a choice of φ that avoids the infinite variance problem of the original harmonic mean estimator. They perturbate the unormalised posterior distribution, of the kind (in one dimension, when the mode is near zero)

$\tilde\pi(\theta|x)$ into $\tilde\pi_P(\theta|x)=\begin{cases}\tilde\pi(\theta+\delta) &\text{if }\theta<-\delta\\ \tilde\pi(\theta-\delta) &\text{if }\theta>-\delta\\ \tilde\pi(0) &\text{otherwise}\end{cases}$

by pulling the density up by a controlled amount of $2\delta\tilde\pi(0|x)$. I find it interesting if only because it goes against my own intuition (and solution) of removing mass from the posterior by concentrating around the mode. The variance of the corresponding harmonic mean estimator is finite if the tails of $\tilde\pi(\theta|x)$ do not decrease too rapidly, I think. The remainder of the paper studies the impact of this approximation technique in the setting of phylogenic models for comparing trees.

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