9 research outputs found
Information-Preserving Markov Aggregation
We present a sufficient condition for a non-injective function of a Markov
chain to be a second-order Markov chain with the same entropy rate as the
original chain. This permits an information-preserving state space reduction by
merging states or, equivalently, lossless compression of a Markov source on a
sample-by-sample basis. The cardinality of the reduced state space is bounded
from below by the node degrees of the transition graph associated with the
original Markov chain.
We also present an algorithm listing all possible information-preserving
state space reductions, for a given transition graph. We illustrate our results
by applying the algorithm to a bi-gram letter model of an English text.Comment: 7 pages, 3 figures, 2 table
Optimal Kullback-Leibler Aggregation via Information Bottleneck
In this paper, we present a method for reducing a regular, discrete-time
Markov chain (DTMC) to another DTMC with a given, typically much smaller number
of states. The cost of reduction is defined as the Kullback-Leibler divergence
rate between a projection of the original process through a partition function
and a DTMC on the correspondingly partitioned state space. Finding the reduced
model with minimal cost is computationally expensive, as it requires an
exhaustive search among all state space partitions, and an exact evaluation of
the reduction cost for each candidate partition. Our approach deals with the
latter problem by minimizing an upper bound on the reduction cost instead of
minimizing the exact cost; The proposed upper bound is easy to compute and it
is tight if the original chain is lumpable with respect to the partition. Then,
we express the problem in the form of information bottleneck optimization, and
propose using the agglomerative information bottleneck algorithm for searching
a sub-optimal partition greedily, rather than exhaustively. The theory is
illustrated with examples and one application scenario in the context of
modeling bio-molecular interactions.Comment: 13 pages, 4 figure
Markov State Space Aggregation via the Information Bottleneck Method
Consider the problem of approximating a Markov chain by another Markov chain with a smaller state space that is obtained by partitioning the original state space. An information-theoretic cost function is proposed that is based on the relative entropy rate between the original Markov chain and a Markov chain defined by the partition. The state space aggregation problem can be sub-optimally solved by using the information bottleneck method
Метод и оценка сложности вычисления предельного распределения марковских функций, представляемых укрупненными цепями Маркова
Предложен метод вычисления предельного распределения марковских функций из класса укрупненных цепей Маркова, уменьшающий вычислительную сложность по сравнению с известным методом. Дана сравнительная оценка вычислительной сложности
An aggregation method for large-scale dynamic games
It is a well known fact that many dynamic games are subject to the curse of dimensionality, limiting the ability to use them in the study of real-world problems. I propose a new method to solve complex large-scale dynamic games using aggregation as an approximate solution. I obtain two fundamental characterization results. First, approximations with small within-state variation in the primitives have a smaller maximum error bound. I provide numerical results which compare the exact errors and the bound. Second, I find that for monotone games, order preserving aggregation is a necessary condition of any optimal aggregation. I suggest using quantiles as a straightforward implementation of an order preserving aggregation architecture for industry distributions. I conclude with an illustration, by solving and estimating a stylized dynamic reputation game for the hotel industry. Simulation results show maximal errors between the exact and approximated solutions below 6%, with average errors below 1%.publishersversionpublishe
Lumpings of Markov chains, entropy rate preservation, and higher-order lumpability
A lumping of a Markov chain is a coordinate-wise projection of the chain. We
characterise the entropy rate preservation of a lumping of an aperiodic and
irreducible Markov chain on a finite state space by the random growth rate of
the cardinality of the realisable preimage of a finite-length trajectory of the
lumped chain and by the information needed to reconstruct original trajectories
from their lumped images. Both are purely combinatorial criteria, depending
only on the transition graph of the Markov chain and the lumping function. A
lumping is strongly k-lumpable, iff the lumped process is a k-th order Markov
chain for each starting distribution of the original Markov chain. We
characterise strong k-lumpability via tightness of stationary entropic bounds.
In the sparse setting, we give sufficient conditions on the lumping to both
preserve the entropy rate and be strongly k-lumpable