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Register a new userEfficient Computation of Updated Lower Expectations for Imprecise Continuous-Time Hidden Markov Chains
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We consider the problem of performing inference with imprecise continuous-time hidden Markov chains, that is, imprecise continuous-time Markov chains that are augmented with random output variables whose distribution depends on the hidden state of the chain. The prefix `imprecise refers to the fact that we do not consider a classical continuous-time Markov chain, but replace it with a robust extension that allows us to represent various types of model uncertainty, using the theory of imprecise probabilities. The inference problem amounts to computing lower expectations of functions on the state-space of the chain, given observations of the output variables. We develop and investigate this problem with very few assumptions on the output variables; in particular, they can be chosen to be either discrete or continuous random variables. Our main result is a polynomial runtime algorithm to compute the lower expectation of functions on the state-space at any given time-point, given a collection of observations of the output variables.
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Continuous-time Markov chains are mathematical models that are used to describe the state-evolution of dynamical systems under stochastic uncertainty, and have found widespread applications in various fields. In order to make these models computationally tractable, they rely on a number of assumptions that may not be realistic for the domain of application; in particular, the ability to provide exact numerical parameter assessments, and the applicability of time-homogeneity and the eponymous Markov property. In this work, we extend these models to imprecise continuous-time Markov chains (ICTMCs), which are a robust generalisation that relaxes these assumptions while remaining computationally tractable. More technically, an ICTMC is a set of precise continuous-time finite-state stochastic processes, and rather than computing expected values of functions, we seek to compute lower expectations, which are tight lower bounds on the expectations that correspond to such a set of precise models. Note that, in contrast to e.g. Bayesian methods, all the elements of such a set are treated on equal grounds; we do not consider a distribution over this set. The first part of this paper develops a formalism for describing continuous-time finite-state stochastic processes that does not require the aforementioned simplifying assumptions. Next, this formalism is used to characterise ICTMCs and to investigate their properties. The concept of lower expectation is then given an alternative operator-theoretic characterisation, by means of a lower transition operator, and the properties of this operator are investigated as well. Finally, we use this lower transition operator to derive tractable algorithms (with polynomial runtime complexity w.r.t. the maximum numerical error) for computing the lower expectation of functions that depend on the state at any finite number of time points.
This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices $N_0^d$, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes are abundant in applications, in particular in biology. We characterize the structure of the state space of general CTMCs on $N_0^d$ in terms of the set of jump vectors and their corresponding transition rate functions. For CTMCs on $N_0$ with polynomial transition rate functions, we provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, certain absorption, positive/null recurrence, implosivity, and existence and non-existence of moments of hitting times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained, and the few gap cases are well-illustrated by examples. Subtle differences in conditions for different dynamical properties are revealed in terms of examples. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes which are not birth-death processes.
We study certain properties of the function space of autocorrelation functions of Unit Continuous Time Markov Chains (CTMCs). It is shown that under particular conditions, the $L^p$ norm of the autocorrelation function of arbitrary finite state space CTMCs is infinite. Several interesting inferences are made for point processes associated with CTMCs/ Discrete Time Markov Chains (DTMCs).
This paper is motivated by demands in stochastic reaction network theory. The $Q$-matrix of a stochastic reaction network can be derived from the reaction graph, an edge-labelled directed graph encoding the jump vectors of an associated continuous time Markov chain on an ambient space $mathbb{N}^d_0$. An open question is whether the ambient space $mathbb{N}^d_0$ can be decomposed into neutral, trapping, and escaping states, and open and closed communicating classes, based on the reaction graph alone. We characterize the structure of the state space of a $Q$-matrix on $mathbb{N}^d_0$ that generates continuous time Markov chains taking values in $mathbb{N}_0^d$, in terms of the set of jump vectors and their corresponding transition rate functions. We also define structural equivalence of two $Q$-matrices, and provide sufficient conditions for structural equivalence. Such stochastic processes are abundant in applications. We apply our results to stochastic reaction networks, a Lotka-Volterra model in ecology, the EnvZ-OmpR system in systems biology, and a class of extended branching processes, none of which are birth-death processes.
This paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions of continuous-time Markov chains on a subset of the non-negative integers. A new identity for stationary measures is established. In particular, for continuous-time Markov chains with asymptotic power-law transition rates, tail asymptotics for stationary distributions are classified into three types by three easily computable parameters: (i) Conley-Maxwell-Poisson distributions (light-tailed), (ii) exponential-tailed distributions, and (iii) heavy-tailed distributions. Similar results are derived for quasi-stationary distributions. The approach to establish tail asymptotics is different from the classical semimartingale approach. We apply our results to biochemical reaction networks (modeled as continuous-time Markov chains), a general single-cell stochastic gene expression model, an extended class of branching processes, and stochastic population processes with bursty reproduction, none of which are birth-death processes.
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