The analysis of population genetics and the evolution of allele frequencies over time can be framed as a filtering problem within the Hidden Markov Model (HMM) framework. To find a computable filter, we follow the work of Papaspiliopoulos and Ruggiero (2014), who established duality as a sufficient condition for filtering. We focus on the K-allele model and identify its dual, building on the work of Barbour et al. (2000). Through this duality, the core problem reduces to simulating a birth-and-death process and calculating its transition rates. When selection is introduced into the model, the tractability of these rates diminishes, as they depend on the ratio of multivariate density functions. These densities are the product of a normal distribution and a Dirichlet distribution, both defined over an n-dimensional simplex. We propose various methods to compute these ratios and compare their performances. First, we compute the normalizing constants for the numerator and denominator separately using Monte Carlo integration with importance sampling. We compare our approximations and their computational costs to the analytical method of nested integration proposed by Genz and Joyce (2000). To address the bias introduced by the first approach, we turn to direct approximations of the ratio using Annealed Importance Sampling (AIS) and Linked Importance Sampling (LIS), as described by Neal (2005). Finally, we evaluate all methods based on accuracy and computational time, ultimately defining the optimal approach for the K-allele model.
The work has been supervised by Professor Matteo Ruggiero
project.py: contains the full thesis pdf.
methods.py: contains the AIS, LIS methodologies, the necessary markov chains (MH) and the respective computations corrections
simulations.py: contains the MC integrations with Importance sampling
stoch_int_to_run.ipynb: runs AIS and LIS, evaluating and comparing their accucary and computational cost
nested_int_to_run.ipynb: defines and runs Nested Analytical Integration