# Sequential importance sampling for multi-resolution Kingman-Tajima coalescent counting

## TL;DR

This paper introduces a sequential importance sampling method to estimate the size of genealogical tree spaces at various resolutions, aiding the choice of simpler models for evolutionary inference.

## Contribution

It presents a novel algorithm to accurately estimate genealogical tree space cardinalities, facilitating better model selection in coalescent-based evolutionary studies.

## Key findings

- The method effectively estimates genealogical space sizes across different resolutions.
- Coarser resolutions are advantageous in certain data settings.
- Application to real genetic data demonstrates practical utility.

## Abstract

Statistical inference of evolutionary parameters from molecular sequence data relies on coalescent models to account for the shared genealogical ancestry of the samples. However, inferential algorithms do not scale to available data sets. A strategy to improve computational efficiency is to rely on simpler coalescent and mutation models, resulting in smaller hidden state spaces. An estimate of the cardinality of the state-space of genealogical trees at different resolutions is essential to decide the best modeling strategy for a given dataset. To our knowledge, there is neither an exact nor approximate method to determine these cardinalities. We propose a sequential importance sampling algorithm to estimate the cardinality of the space of genealogical trees under different coalescent resolutions. Our sampling scheme proceeds sequentially across the set of combinatorial constraints imposed by the data. We analyse the cardinality of different genealogical tree spaces on simulations to study the settings that favor coarser resolutions. We estimate the cardinality of genealogical tree spaces from mtDNA data from the 1000 genomes and a sample from a Melanesian population to illustrate the settings in which it is advantageous to employ coarser resolutions.