Lumpability In Markov Chains Research Articles

Information geometry of Markov Kernels: a survey

Information geometry and Markov chains are two powerful tools used in modern fields such as finance, physics, computer science, and epidemiology. In this survey, we explore their intersection, focusing on the theoretical framework. We attempt to provide a self-contained treatment of the foundations without requiring a solid background in differential geometry. We present the core concepts of information geometry of Markov chains, including information projections and the pivotal information geometric construction of Nagaoka. We then delve into recent advances in the field, such as geometric structures arising from time reversibility, lumpability of Markov chains, or tree models. Finally, we highlight practical applications of this framework, such as parameter estimation, hypothesis testing, large deviation theory, and the maximum entropy principle.

Markov chain lumpability and applications to credit risk modelling in compliance with the International Financial Reporting Standard 9 framework

The aim of this paper is threefold. Firstly, we define the necessary quantities associated to the lumpability of a Markov chain and study their fundamental properties. Secondly, we examine the case of approximate lumpability of a non-lumpable Markov and an efficient method of minimizing the error in the approximation. Finally, we introduce a family of general minimization problems that can be approached using this method and examine applications in credit risk modelling, particularly under recent regulatory changes related to loan classification and provision calculations under IFRS 9.

Step-by-step community detection in volume-regular graphs

Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph).Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations. While the results of Becchetti et al. apply to perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, we extend their approach to graphs containing a hidden k partition and characterized by a milder form of volume-regularity. We show that the class of k-volume regular graphs is the largest class of undirected (possibly weighted) graphs whose transition matrix admits k “stepwise” eigenvectors (i.e., vectors that have constant entries over the components corresponding to the same set of the hidden partition). To obtain this result, we highlight a connection between volume regularity and lumpability of Markov chains. Moreover, we prove that if the stepwise eigenvectors are those associated to the first k largest eigenvalues of the transition matrix of a random walk on the graph and the gap between the k-th and the (k+1)-th eigenvalues is sufficiently large, the Averaging dynamics of Becchetti et al. recovers the underlying community structure of the graph in logarithmic time, with high probability.

On the relations between Markov chain lumpability and reversibility

In the literature, the notions of lumpability and time reversibility for large Markov chains have been widely used to efficiently study the functional and non-functional properties of computer systems. In this paper we explore the relations among different definitions of lumpability (strong, exact and strict) and the notion of time-reversed Markov chain. Specifically, we prove that an exact lumping induces a strong lumping on the reversed Markov chain and a strict lumping holds both for the forward and the reversed processes. Based on these results we introduce the class of \(\lambda \rho \)-reversible Markov chains which combines the notions of lumping and time reversibility modulo state renaming. We show that the class of autoreversible processes, previously introduced in Marin and Rossi (Proceedings of the IEEE 21st international symposium on modeling, analysis and simulation of computer and telecommunication systems MASCOTS, pp. 151–160, 2013), is strictly contained in the class of \(\lambda \rho \)-reversible chains.

Testing lumpability in Markov chains

The chi-squared test of Markov chain lumpability is shown to operate reliably under a corrected derivation of the degrees of freedom. The test is used to screen out lumping schemes that corrupt the Markov property and give rise to higher order dependence.

Markovian couplings staying in arbitrary subsets of the state space

Let (X t ) and (Y t ) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (X t ) and (Y t ) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

Markovian couplings staying in arbitrary subsets of the state space

Let (Xt) and (Yt) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (Xt) and (Yt) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

On weak lumpability in Markov chains

We analyse the conditions under which the aggregated process constructed from an homogeneous Markov chain over a given partition of the state space is also Markov homogeneous. The past work on the subject is revised and new properties are obtained.

On weak lumpability in Markov chains

We analyse the conditions under which the aggregated process constructed from an homogeneous Markov chain over a given partition of the state space is also Markov homogeneous. The past work on the subject is revised and new properties are obtained.

Technical Note—An Eigenvector Condition for Markov Chain Lumpability

Under certain conditions the state space of a discrete parameter Markov chain may be partitioned to form a smaller “lumped” chain that retains the Markov property. Existing statements of conditions on the transition matrix P of the original chain characterizing such lumpability are difficult to verify in practice, especially when the dimension of P is large. An alternate approach, based on the eigenvectors of P, is presented and illustrated with examples.

An Approximate Test of Markov Chain Lumpability

Abstract Under certain conditions the state space of a discrete parameter Markov chain may be partitioned to form a smaller lumped chain that retains the Markov property. The problem of formulating lump-ability hypotheses when the transition probability matrix P is not known and is possibly of large dimension is discussed. An approximate test of these hypotheses is described, based on well-known nonparametric methods. The procedure is illustrated with an application to a Markov manpower model.

An Approximate Test of Markov Chain Lumpability

Abstract Under certain conditions the state space of a discrete parameter Markov chain may be partitioned to form a smaller lumped chain that retains the Markov property. The problem of formulating lump-ability hypotheses when the transition probability matrix P is not known and is possibly of large dimension is discussed. An approximate test of these hypotheses is described, based on well-known nonparametric methods. The procedure is illustrated with an application to a Markov manpower model.