9,070 research outputs found
Consistency for Parametric Interval Markov Chains
Interval Markov Chains (IMCs) are the base of a classic probabilistic specification theory by Larsen and Jonsson in 1991. They are also a popular abstraction for probabilistic systems. In this paper we introduce and study an extension of Interval Markov Chains with parametric intervals. In particular, we investigate the consistency problem for such models and propose an efficient solution for the subclass of parametric IMCs with local parameters only. We also show that this problem is still decidable for parametric IMCs with global parameters, although more complex in this case
Reachability in Parametric Interval Markov Chains using Constraints
Parametric Interval Markov Chains (pIMCs) are a specification formalism that
extend Markov Chains (MCs) and Interval Markov Chains (IMCs) by taking into
account imprecision in the transition probability values: transitions in pIMCs
are labeled with parametric intervals of probabilities. In this work, we study
the difference between pIMCs and other Markov Chain abstractions models and
investigate the two usual semantics for IMCs: once-and-for-all and
at-every-step. In particular, we prove that both semantics agree on the
maximal/minimal reachability probabilities of a given IMC. We then investigate
solutions to several parameter synthesis problems in the context of pIMCs --
consistency, qualitative reachability and quantitative reachability -- that
rely on constraint encodings. Finally, we propose a prototype implementation of
our constraint encodings with promising results
Parameter Synthesis for Parametric Interval Markov Chains
AELOS_HCERES2020, STR_HCERES2020Interval Markov Chains (IMCs) are the base of a classic probabilistic specification theory introduced by Larsen and Jonsson in 1991. They are also a popular abstraction for probabilistic systems. In this paper we study parameter synthesis for a parametric extension of Interval Markov Chains in which the endpoints of intervals may be replaced with parameters. In particular, we propose constructions for the synthesis of all parameter values ensuring several properties such as consistency and consistent reachability in both the existential and universal settings with respect to implementations. We also discuss how our constructions can be modified in order to synthesise all parameter values ensuring other typical properties
Regenerative block empirical likelihood for Markov chains
Empirical likelihood is a powerful semi-parametric method increasingly
investigated in the literature. However, most authors essentially focus on an
i.i.d. setting. In the case of dependent data, the classical empirical
likelihood method cannot be directly applied on the data but rather on blocks
of consecutive data catching the dependence structure. Generalization of
empirical likelihood based on the construction of blocks of increasing
nonrandom length have been proposed for time series satisfying mixing
conditions. Following some recent developments in the bootstrap literature, we
propose a generalization for a large class of Markov chains, based on small
blocks of various lengths. Our approach makes use of the regenerative structure
of Markov chains, which allows us to construct blocks which are almost
independent (independent in the atomic case). We obtain the asymptotic validity
of the method for positive recurrent Markov chains and present some simulation
results
Interest rate models with Markov chains
Imperial Users onl
Bivariate modelling of precipitation and temperature using a non-homogeneous hidden Markov model
Aiming to generate realistic synthetic times series of the bivariate process
of daily mean temperature and precipitations, we introduce a non-homogeneous
hidden Markov model. The non-homogeneity lies in periodic transition
probabilities between the hidden states, and time-dependent emission
distributions. This enables the model to account for the non-stationary
behaviour of weather variables. By carefully choosing the emission
distributions, it is also possible to model the dependance structure between
the two variables. The model is applied to several weather stations in Europe
with various climates, and we show that it is able to simulate realistic
bivariate time series
Selection of proposal distributions for generalized importance sampling estimators
The standard importance sampling (IS) estimator, generally does not work well
in examples involving simultaneous inference on several targets as the
importance weights can take arbitrarily large values making the estimator
highly unstable. In such situations, alternative generalized IS estimators
involving samples from multiple proposal distributions are preferred. Just like
the standard IS, the success of these multiple IS estimators crucially depends
on the choice of the proposal distributions. The selection of these proposal
distributions is the focus of this article. We propose three methods based on
(i) a geometric space filling coverage criterion, (ii) a minimax variance
approach, and (iii) a maximum entropy approach. The first two methods are
applicable to any multi-proposal IS estimator, whereas the third approach is
described in the context of Doss's (2010) two-stage IS estimator. For the first
method we propose a suitable measure of coverage based on the symmetric
Kullback-Leibler divergence, while the second and third approaches use
estimates of asymptotic variances of Doss's (2010) IS estimator and Geyer's
(1994) reverse logistic estimator, respectively. Thus, we provide consistent
spectral variance estimators for these asymptotic variances. The proposed
methods for selecting proposal densities are illustrated using various detailed
examples
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