Probabilistic Reachability Analysis for Large Scale Stochastic Hybrid Systems

This paper studies probabilistic reachability analysis for large scale stochastic hybrid systems (SHS) as a problem of rare event estimation. In literature, advanced rare event estimation theory has recently been embedded within a stochastic analysis framework, and this has led to significant novel results in rare event estimation for a diffusion process using sequential MC simulation. This paper presents this rare event estimation theory directly in terms of probabilistic reachability analysis of an SHS, and develops novel theory which allows to extend the novel results for application to a large scale SHS where a very huge number of rare discrete modes may contribute significantly to the reach probability. Essentially, the approach taken is to introduce an aggregation of the discrete modes, and to develop importance sampling relative to the rare switching between the aggregation modes. The practical working of this approach is demonstrated for the safety verification of an advanced air traffic control example

Optimal probabilistic estimation of quantum states

We extend the concept of probabilistic unambiguous discrimination of quantum states to quantum state estimation. We consider a scenario where the measurement device can output either an estimate of the unknown input state or an inconclusive result. We present a general method how to evaluate the maximum fidelity achievable by the probabilistic estimation strategy. We illustrate our method on two explicit examples: estimation of a qudit from a pair of conjugate qudits and phase covariant estimation of a qubit from N copies. We show that by allowing for inconclusive results it is possible to reach estimation fidelity higher than that achievable by the best deterministic estimation strategy.Comment: 7 pages, 2 figures, ReVTeX

Collective Animal Behavior from Bayesian Estimation and Probability Matching

Animals living in groups make movement decisions that depend, among other factors, on social interactions with other group members. Our present understanding of social rules in animal collectives is based on empirical fits to observations and we lack first-principles approaches that allow their derivation. Here we show that patterns of collective decisions can be derived from the basic ability of animals to make probabilistic estimations in the presence of uncertainty. We build a decision-making model with two stages: Bayesian estimation and probabilistic matching.
In the first stage, each animal makes a Bayesian estimation of which behavior is best to perform taking into account personal information about the environment and social information collected by observing the behaviors of other animals. In the probability matching stage, each animal chooses a behavior with a probability given by the Bayesian estimation that this behavior is the most appropriate one. This model derives very simple rules of interaction in animal collectives that depend only on two types of reliability parameters, one that each animal assigns to the other animals and another given by the quality of the non-social information. We test our model by obtaining theoretically a rich set of observed collective patterns of decisions in three-spined sticklebacks, Gasterosteus aculeatus, a shoaling fish species. The quantitative link shown between probabilistic estimation and collective rules of behavior allows a better contact with other fields such as foraging, mate selection, neurobiology and psychology, and gives predictions for experiments directly testing the relationship between estimation and collective behavior

Forecast verification for extreme value distributions with an application to probabilistic peak wind prediction

Predictions of the uncertainty associated with extreme events are a vital component of any prediction system for such events. Consequently, the prediction system ought to be probabilistic in nature, with the predictions taking the form of probability distributions. This paper concerns probabilistic prediction systems where the data is assumed to follow either a generalized extreme value distribution (GEV) or a generalized Pareto distribution (GPD). In this setting, the properties of proper scoring rules which facilitate the assessment of the prediction uncertainty are investigated and closed-from expressions for the continuous ranked probability score (CRPS) are provided. In an application to peak wind prediction, the predictive performance of a GEV model under maximum likelihood estimation, optimum score estimation with the CRPS, and a Bayesian framework are compared. The Bayesian inference yields the highest overall prediction skill and is shown to be a valuable tool for covariate selection, while the predictions obtained under optimum CRPS estimation are the sharpest and give the best performance for high thresholds and quantiles

Loopy belief propagation and probabilistic image processing

Estimation of hyperparameters by maximization of the marginal likelihood in probabilistic image processing is investigated by using the cluster variation method. The algorithms are substantially equivalent to generalized loopy belief propagation