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Computational methods for birth‐death processes

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Many important stochastic counting models can be written as general birth‐death processes (BDPs). BDPs are continuous‐time Markov chains on the non‐negative integers in which only jumps to adjacent states are allowed. BDPs can be used to easily parameterize a rich variety of probability distributions on the non‐negative integers, and straightforward conditions guarantee that these distributions are proper. BDPs also provide a mechanistic interpretation—birth and death of actual particles or organisms—that has proven useful in evolution, ecology, physics, and chemistry. Although the theoretical properties of general BDPs are well understood, traditionally statistical work on BDPs has been limited to the simple linear (Kendall) process. Aside from a few simple cases, it remains impossible to find analytic expressions for the likelihood of a discretely‐observed BDP, and computational difficulties have hindered development of tools for statistical inference. But the gap between BDP theory and practical methods for estimation has narrowed in recent years. There are now robust methods for evaluating likelihoods for realizations of BDPs: finite‐time transition, first passage, equilibrium probabilities, and distributions of summary statistics that arise commonly in applications. Recent work has also exploited the connection between continuously‐ and discretely‐observed BDPs to derive EM algorithms for maximum likelihood estimation. Likelihood‐based inference for previously intractable BDPs is much easier than previously thought and regression approaches analogous to Poisson regression are straightforward to derive. In this review, we outline the basic mathematical theory for BDPs and demonstrate new tools for statistical inference using data from BDPs. WIREs Comput Stat 2018, 10:e1423. doi: 10.1002/wics.1423 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms Applications of Computational Statistics > Computational Chemistry
Stochastic simulation of a BDP starting at X(0) = 1 on the interval 0 < t < 2
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Probabilistic control of a stochastic SIS epidemic. At top, the distribution of total epidemic cost Ci for different values of a control parameter ɛ. The dashed gray vertical line is at w = 7, and we wish to keep Ci < 7 with high probability. At bottom, the probability that Ci < 7 as a function of the control parameter ɛ. The horizontal gray dashed line denotes 0.95, and the vertical dashed line is the smallest epsilon that achieves Pr(Ci < 7) > 0.95; this yields ɛ ≈ 3.4. In this way, we can easily find the smallest value of a control parameter that bounds the probability that the epidemic will exceed a certain threshold
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Illustration of the integral of a functional of a general birth‐death process (BDP). On the left, a BDP begins at X(0) = 1 and ends when the process reaches the absorbing state 0 just before time t = 2. On the right, is the area under the trajectory of g(X(t)), where g : → [0, ∞) is an arbitrary positive “reward” or “cost” function. The upper limit of integration τ1 is the first passage time to zero, beginning at X(0) = 1
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Applications of Computational Statistics > Computational Chemistry
Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms
Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory

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