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Sequential learning theory for Markov genealogy processes

David J Pascall

Abstract

We introduce a filtration-based framework for studying when and why adding taxa improves phylodynamic inference, by constructing a natural ordering of observed tips and applying sequential Bayesian analysis to the resulting filtration. We decompose the expected variance reduction on taxa addition into learning, mismatch, and covariance components, classify estimands into learning classes based on the pathwise behaviour of the mismatch, and show that for absorbing estimands an oracle who knows the latent absorption status obtains event-wise learning guarantees unavailable to the analyst. The gap between oracle and analyst is irreducible assumptions that are likely to hold for many real phylodynamic estimands, establishing a fundamental limit on what sequence data alone can reveal about the latent genealogy.

Sequential learning theory for Markov genealogy processes

Abstract

We introduce a filtration-based framework for studying when and why adding taxa improves phylodynamic inference, by constructing a natural ordering of observed tips and applying sequential Bayesian analysis to the resulting filtration. We decompose the expected variance reduction on taxa addition into learning, mismatch, and covariance components, classify estimands into learning classes based on the pathwise behaviour of the mismatch, and show that for absorbing estimands an oracle who knows the latent absorption status obtains event-wise learning guarantees unavailable to the analyst. The gap between oracle and analyst is irreducible assumptions that are likely to hold for many real phylodynamic estimands, establishing a fundamental limit on what sequence data alone can reveal about the latent genealogy.
Paper Structure (4 sections, 8 theorems, 9 equations)

This paper contains 4 sections, 8 theorems, 9 equations.

Table of Contents

  1. Abstract
  2. Introduction
  3. Mathematical setup
  4. Results

Key Result

Proposition 1

Under the construction above, $(\mathcal{F}_n)_{n\leq f(\mathcal{G})}$ is a filtration on $(\Omega,\mathcal{F},\mathbb{P}^*)$, and for any square-integrable permutation-invariant estimand, $K$, $\mathbb{E}_{\mathbb{P}^*}\left[\mathrm{Var}(K|\mathcal{F}_n)\right]\geq \mathbb{E}_{\mathbb{P}^*}\left[\m

Theorems & Definitions (16)

  • Proposition 1: Standard learning for permutation-invariant estimands
  • proof
  • Lemma 1: Variance decomposition for sequential estimands
  • proof
  • Theorem 1: Sequential learning decomposition
  • proof
  • Lemma 2: Partitioned decomposition conditional on absorption status
  • proof
  • Corollary 1: Event-wise in expectation variance reduction for the oracle
  • proof
  • ...and 6 more