88 research outputs found
Model checking for imprecise Markov chains.
We extend probabilistic computational tree logic for expressing properties of Markov chains to imprecise Markov chains, and provide an efficient algorithm for model checking of imprecise Markov chains. Thereby, we provide a formal framework to answer a very wide range of questions about imprecise Markov chains, in a systematic and computationally efficient way
Hitting Times and Probabilities for Imprecise Markov Chains
We consider the problem of characterising expected hitting times and hitting
probabilities for imprecise Markov chains. To this end, we consider three
distinct ways in which imprecise Markov chains have been defined in the
literature: as sets of homogeneous Markov chains, as sets of more general
stochastic processes, and as game-theoretic probability models. Our first
contribution is that all these different types of imprecise Markov chains have
the same lower and upper expected hitting times, and similarly the hitting
probabilities are the same for these three types. Moreover, we provide a
characterisation of these quantities that directly generalises a similar
characterisation for precise, homogeneous Markov chains
Hitting times and probabilities for imprecise Markov chains
We consider the problem of characterising expected hitting times and hitting probabilities for imprecise Markov chains. To this end, we consider three distinct ways in which imprecise Markov chains have been defined in the literature: as sets of homogeneous Markov chains, as sets of more general stochastic processes, and as game-theoretic probability models. Our first contribution is that all these different types of imprecise Markov chains have the same lower and upper expected hitting times, and similarly the hitting probabilities are the same for these three types. Moreover, we provide a characterisation of these quantities that directly generalises a similar characterisation for precise, homogeneous Markov chains
Model Checking for Imprecise Markov Chains
We extend probabilistic computational tree logic for expressing properties of Markov chains to imprecise Markov chains, and provide an efficient algorithm for model checking of imprecise Markov chains. Thereby, we provide a formal framework to answer a very wide range of questions about imprecise Markov chains, in a systematic and computationally efficient way
Limit behaviour for imprecise Markov Chains
An imprecise Markov chain is defined by a closed convex set of transition matrices instead of a unique one for a classical precise Markov chain. These imprecise Markov chains allow us to model situations where we do not have enough information to specify a unique transition matrix, or to approximate the behaviour of nonâstationary Markov chains. We show that there are efficient, dynamic programmingâ like ways to work and reason with these imprecise Markov chains; e.g. to calculate the resulting distribution over the states at any time instant. We prove that this distribution converges in time, similarly to the precise case and under very mild conditions. We thus effectively prove a PerronâFrobenius theorem for a special class of nonâlinear systems
A Recursive Algorithm for Computing Inferences in Imprecise Markov Chains
We present an algorithm that can efficiently compute a broad class of
inferences for discrete-time imprecise Markov chains, a generalised type of
Markov chains that allows one to take into account partially specified
probabilities and other types of model uncertainty. The class of inferences
that we consider contains, as special cases, tight lower and upper bounds on
expected hitting times, on hitting probabilities and on expectations of
functions that are a sum or product of simpler ones. Our algorithm exploits the
specific structure that is inherent in all these inferences: they admit a
general recursive decomposition. This allows us to achieve a computational
complexity that scales linearly in the number of time points on which the
inference depends, instead of the exponential scaling that is typical for a
naive approach
Imprecise Markov chains and their limit behaviour
When the initial and transition probabilities of a finite Markov chain in
discrete time are not well known, we should perform a sensitivity analysis.
This can be done by considering as basic uncertainty models the so-called
credal sets that these probabilities are known or believed to belong to, and by
allowing the probabilities to vary over such sets. This leads to the definition
of an imprecise Markov chain. We show that the time evolution of such a system
can be studied very efficiently using so-called lower and upper expectations,
which are equivalent mathematical representations of credal sets. We also study
how the inferred credal set about the state at time n evolves as n goes to
infinity: under quite unrestrictive conditions, it converges to a uniquely
invariant credal set, regardless of the credal set given for the initial state.
This leads to a non-trivial generalisation of the classical Perron-Frobenius
Theorem to imprecise Markov chains.Comment: v1: 28 pages, 8 figures; v2: 31 pages, 9 figures, major revision
after review: added, modified, and removed material (no results dropped,
results added), moved proofs to an appendi
Perturbation bounds and degree of imprecision for uniquely convergent imprecise Markov chains
The effect of perturbations of parameters for uniquely convergent imprecise
Markov chains is studied. We provide the maximal distance between the
distributions of original and perturbed chain and maximal degree of
imprecision, given the imprecision of the initial distribution. The bounds on
the errors and degrees of imprecision are found for the distributions at finite
time steps, and for the stationary distributions as well.Comment: 20 pages, 2 figure
A classification of invariant distributions and convergence of imprecise Markov chains
We analyse the structure of imprecise Markov chains and study their
convergence by means of accessibility relations. We first identify the sets of
states, so-called minimal permanent classes, that are the minimal sets capable
of containing and preserving the whole probability mass of the chain. These
classes generalise the essential classes known from the classical theory. We
then define a class of extremal imprecise invariant distributions and show that
they are uniquely determined by the values of the upper probability on minimal
permanent classes. Moreover, we give conditions for unique convergence to these
extremal invariant distributions
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