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<p>In this paper, the spread of an epidemiological disease over time is modeled using a Bienaymé-Galton-Watson (BGW) process. A discrete random variable, referred to as contagion, models the number of infections caused by each infector and it is the key element in understanding the dynamics of this branching process. The objective is to study the inverse problem of Bayesian inference for the parameters of the contagion. Two distinct strategies are presented: one where data are obtained from different realizations of the same generation’s random variable in the BGW process, and another where data are derived from a single realization of the process observed over multiple generations. In each strategy, once a prior distribution is selected, the main challenge is to determine an appropriate likelihood function. This paper introduces new methodologies for constructing these likelihood functions, representing a novel contribution to the literature. For the first strategy, probability generating functions and Markov chain are proposed as tools for constructing appropriate likelihood functions. For the second strategy, one-step transition probabilities, aided by probability generating functions, are employed. The <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-Wasserstein distance is introduced as a convergence criterion to stop the Bayesian update process. Since the data are generated stochastically, the number of update steps required to achieve convergence becomes a discrete random variable, and is therefore analyzed within a probabilistic framework.</p>

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Bayesian parametric inference problem applied on branching processes

  • 期刊论文

  • 发表日期：

    2025年12月22日

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  • João P. Freitas,
  • Roberta Lima,
  • Rubens Sampaio

摘要

In this paper, the spread of an epidemiological disease over time is modeled using a Bienaymé-Galton-Watson (BGW) process. A discrete random variable, referred to as contagion, models the number of infections caused by each infector and it is the key element in understanding the dynamics of this branching process. The objective is to study the inverse problem of Bayesian inference for the parameters of the contagion. Two distinct strategies are presented: one where data are obtained from different realizations of the same generation’s random variable in the BGW process, and another where data are derived from a single realization of the process observed over multiple generations. In each strategy, once a prior distribution is selected, the main challenge is to determine an appropriate likelihood function. This paper introduces new methodologies for constructing these likelihood functions, representing a novel contribution to the literature. For the first strategy, probability generating functions and Markov chain are proposed as tools for constructing appropriate likelihood functions. For the second strategy, one-step transition probabilities, aided by probability generating functions, are employed. The \(L_{2}\) L 2 -Wasserstein distance is introduced as a convergence criterion to stop the Bayesian update process. Since the data are generated stochastically, the number of update steps required to achieve convergence becomes a discrete random variable, and is therefore analyzed within a probabilistic framework.