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

Efficient computation of updated lower expectations for imprecise continuous-time hidden Markov chains

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

Imprecise Continuous-Time Markov Chains

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 (ICTMC's), 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 ICTMC's 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

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

Knowledge-based bias correction : a case study in veterinary decision support

In collaboration with experts from veterinary research institutes throughout Europe, we developed a decision-support system for the early detection of Classical Swine Fever in pigs. For evaluating our system's diagnostic performance, practitioners and researchers collected data from the real-world field and from laboratory experiments. Originating from different sources, these data could not be viewed as constituting an unbiased sample from a single probability distribution. In this paper, we present a knowledge-based method for correcting the biases in estimates from such divergent data. We demonstrate the use of our method for estimating the sensitivity and specificity characteristics of our veterinary decision-support system

Two-state imprecise Markov chains for statistical modelling of two-state non-Markovian processes.

This paper proposes a method for fitting a two-state imprecise Markov chain to time series data from a twostate non-Markovian process. Such non-Markovian processes are common in practical applications. We focus on how to fit modelling parameters based on data from a process where time to transition is not exponentially distributed, thereby violating the Markov assumption. We do so by first fitting a many-state (i.e. having more than two states) Markov chain to the data, through its associated phase-type distribution. Then, we lump the process to a two-state imprecise Markov chain. In practical applications, a two-state imprecise Markov chain might be more convenient than a many-state Markov chain, as we have closed analytic expressions for typical quantities of interest (including the lower and upper expectation of any function of the state at any point in time). A numerical example demonstrates how the entire inference process (fitting and prediction) can be done using Markov chain Monte Carlo, for a given set of prior distributions on the parameters. In particular, we numerically identify the set of posterior densities and posterior lower and upper expectations on all model parameters and predictive quantities. We compare our inferences under a range of sample sizes and model assumptions. Keywords: imprecise Markov chain, estimation, reliability, Markov assumption, MCM

FDG PET and PET/CT: EANM procedure guidelines for tumour PET imaging: version 1.0

The aim of this guideline is to provide a minimum standard for the acquisition and interpretation of PET and PET/CT scans with [18F]-fluorodeoxyglucose (FDG). This guideline will therefore address general information about [18F]-fluorodeoxyglucose (FDG) positron emission tomography-computed tomography (PET/CT) and is provided to help the physician and physicist to assist to carrying out, interpret, and document quantitative FDG PET/CT examinations, but will concentrate on the optimisation of diagnostic quality and quantitative information

Evolutionary dynamics of emblematic Araucaria species (Araucariaceae) in New Caledonia:Nuclear and chloroplast markers suggest recent diversification, introgression, and a tight link between genetics and geography within species

BACKGROUND: New Caledonia harbours a highly diverse and endemic flora, and 13 (out of the 19 worldwide) species of Araucaria are endemic to this territory. Their phylogenetic relationships remain largely unresolved. Using nuclear microsatellites and chloroplast DNA sequencing, we focused on five closely related Araucaria species to investigate among-species relationships and the distribution of within-species genetic diversity across New Caledonia. RESULTS: The species could be clearly distinguished here, except A. montana and A. laubenfelsii that were not differentiated and, at most, form a genetic cline. Given their apparent morphological and ecological similarity, we suggested that these two species may be considered as a single evolutionary unit. We observed cases of nuclear admixture and incongruence between nuclear and chloroplast data, probably explained by introgression and shared ancestral polymorphism. Ancient hybridization was evidenced between A. biramulata and A. laubenfelsii in Mt Do, and is strongly suspected between A. biramulata and A. rulei in Mt Tonta. In both cases, extensive asymmetrical backcrossing eliminated the influence of one parent in the nuclear DNA composition. Shared ancestral polymorphism was also observed for cpDNA, suggesting that species diverged recently, have large effective sizes and/or that cpDNA experienced slow rates of molecular evolution. Within-species genetic structure was pronounced, probably because of low gene flow and significant inbreeding, and appeared clearly influenced by geography. This may be due to survival in distinct refugia during Quaternary climatic oscillations. CONCLUSIONS: The study species probably diverged recently and/or are characterized by a slow rate of cpDNA sequence evolution, and introgression is strongly suspected. Within-species genetic structure is tightly linked with geography. We underline the conservation implications of our results, and highlight several perspectives. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/s12862-014-0171-6) contains supplementary material, which is available to authorized users