Computational methods for birth-death processes.

Area of Science:

• Stochastic processes
• Mathematical biology
• Statistical inference

Background:

• Birth-death processes (BDPs) are continuous-time Markov chains used in various scientific fields.
• Traditional statistical inference for BDPs was limited due to computational challenges with likelihood calculations.
• General BDPs offer rich probability distributions and mechanistic interpretations for phenomena like particle birth and death.

Purpose of the Study:

• To review the mathematical theory of general birth-death processes (BDPs).
• To demonstrate novel computational tools for statistical inference with BDPs.
• To bridge the gap between theoretical BDP properties and practical estimation methods.

Main Methods:

• Utilizing robust methods for evaluating likelihoods of BDP realizations (e.g., transition, passage, equilibrium probabilities).
• Employing EM algorithms derived from the connection between continuous- and discrete-time BDPs for maximum likelihood estimation.
• Developing regression approaches analogous to Poisson regression for BDP data.

Main Results:

• Analytic expressions for likelihoods of discretely-observed BDPs are now feasible.
• EM algorithms facilitate maximum likelihood estimation for previously intractable BDPs.
• Regression techniques simplify likelihood-based inference for BDPs.

Conclusions:

• Statistical inference for general birth-death processes is more accessible than previously assumed.
• New computational tools significantly enhance the practical application of BDPs in science.
• This work provides a foundation for applying advanced BDP models across diverse research areas.