Probabilistic abstract interpretation: From trace semantics to DTMC’s and linear regression

In order to perform probabilistic program analysis we need to consider probabilistic languages or languages with a probabilistic semantics, as well as a corresponding framework for the analysis which is able to accommodate probabilistic properties and properties of probabilistic computations. To this purpose we investigate the relationship between three different types of probabilistic semantics for a core imperative language, namely Kozen’s Fixpoint Semantics, our Linear Operator Semantics and probabilistic versions of Maximal Trace Semantics. We also discuss the relationship between Probabilistic Abstract Interpretation (PAI) and statistical or linear regression analysis. While classical Abstract Interpretation, based on Galois connection, allows only for worst-case analyses, the use of the Moore-Penrose pseudo inverse in PAI opens the possibility of exploiting statistical and noisy observations in order to analyse and identify various system properties

Where do statistical models come from? Revisiting the problem of specification

R. A. Fisher founded modern statistical inference in 1922 and identified its fundamental problems to be: specification, estimation and distribution. Since then the problem of statistical model specification has received scant attention in the statistics literature. The paper traces the history of statistical model specification, focusing primarily on pioneers like Fisher, Neyman, and more recently Lehmann and Cox, and attempts a synthesis of their views in the context of the Probabilistic Reduction (PR) approach. As argued by Lehmann [11], a major stumbling block for a general approach to statistical model specification has been the delineation of the appropriate role for substantive subject matter information. The PR approach demarcates the interrelated but complemenatry roles of substantive and statistical information summarized ab initio in the form of a structural and a statistical model, respectively. In an attempt to preserve the integrity of both sources of information, as well as to ensure the reliability of their fusing, a purely probabilistic construal of statistical models is advocated. This probabilistic construal is then used to shed light on a number of issues relating to specification, including the role of preliminary data analysis, structural vs. statistical models, model specification vs. model selection, statistical vs. substantive adequacy and model validation.Comment: Published at http://dx.doi.org/10.1214/074921706000000419 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Valuing information from mesoscale forecasts

The development of meso-gamma scale numerical weather prediction (NWP) models requires a substantial investment in research, development and computational resources. Traditional objective verification of deterministic model output fails to demonstrate the added value of high-resolution forecasts made by such models. It is generally accepted from subjective verification that these models nevertheless have a predictive potential for small-scale weather phenomena and extreme weather events. This has prompted an extensive body of research into new verification techniques and scores aimed at developing mesoscale performance measures that objectively demonstrate the return on investment in meso-gamma NWP. In this article it is argued that the evaluation of the information in mesoscale forecasts should be essentially connected to the method that is used to extract this information from the direct model output (DMO). This could be an evaluation by a forecaster, but, given the probabilistic nature of small-scale weather, is more likely a form of statistical post-processing. Using model output statistics (MOS) and traditional verification scores, the potential of this approach is demonstrated both on an educational abstraction and a real world example. The MOS approach for this article incorporates concepts from fuzzy verification. This MOS approach objectively weighs different forecast quality measures and as such it is an essential extension of fuzzy methods

A note on p-values interpreted as plausibilities

P-values are a mainstay in statistics but are often misinterpreted. We propose a new interpretation of p-value as a meaningful plausibility, where this is to be interpreted formally within the inferential model framework. We show that, for most practical hypothesis testing problems, there exists an inferential model such that the corresponding plausibility function, evaluated at the null hypothesis, is exactly the p-value. The advantages of this representation are that the notion of plausibility is consistent with the way practitioners use and interpret p-values, and the plausibility calculation avoids the troublesome conditioning on the truthfulness of the null. This connection with plausibilities also reveals a shortcoming of standard p-values in problems with non-trivial parameter constraints.Comment: 13 pages, 1 figur

EPR Paradox,Locality and Completeness of Quantum Theory

The quantum theory (QT) and new stochastic approaches have no deterministic prediction for a single measurement or for a single time -series of events observed for a trapped ion, electron or any other individual physical system. The predictions of QT being of probabilistic character apply to the statistical distribution of the results obtained in various experiments. The probability distribution is not an attribute of a dice but it is a characteristic of a whole random experiment : '' rolling a dice''. and statistical long range correlations between two random variables X and Y are not a proof of any causal relation between these variable. Moreover any probabilistic model used to describe a random experiment is consistent only with a specific protocol telling how the random experiment has to be performed.In this sense the quantum theory is a statistical and contextual theory of phenomena. In this paper we discuss these important topics in some detail. Besides we discuss in historical perspective various prerequisites used in the proofs of Bell and CHSH inequalities concluding that the violation of these inequalities in spin polarization correlation experiments is neither a proof of the completeness of QT nor of its nonlocality. The question whether QT is predictably complete is still open and it should be answered by a careful and unconventional analysis of the experimental data. It is sufficient to analyze more in detail the existing experimental data by using various non-parametric purity tests and other specific statistical tools invented to study the fine structure of the time-series. The correct understanding of statistical and contextual character of QT has far reaching consequences for the quantum information and quantum computing.Comment: 16 pages, 59 references,the contribution to the conference QTRF-4 held in Vaxjo, Sweden, 11-16 june 2007. To be published in the Proceeding

Lab Retriever: a software tool for calculating likelihood ratios incorporating a probability of drop-out for forensic DNA profiles.

BackgroundTechnological advances have enabled the analysis of very small amounts of DNA in forensic cases. However, the DNA profiles from such evidence are frequently incomplete and can contain contributions from multiple individuals. The complexity of such samples confounds the assessment of the statistical weight of such evidence. One approach to account for this uncertainty is to use a likelihood ratio framework to compare the probability of the evidence profile under different scenarios. While researchers favor the likelihood ratio framework, few open-source software solutions with a graphical user interface implementing these calculations are available for practicing forensic scientists.ResultsTo address this need, we developed Lab Retriever, an open-source, freely available program that forensic scientists can use to calculate likelihood ratios for complex DNA profiles. Lab Retriever adds a graphical user interface, written primarily in JavaScript, on top of a C++ implementation of the previously published R code of Balding. We redesigned parts of the original Balding algorithm to improve computational speed. In addition to incorporating a probability of allelic drop-out and other critical parameters, Lab Retriever computes likelihood ratios for hypotheses that can include up to four unknown contributors to a mixed sample. These computations are completed nearly instantaneously on a modern PC or Mac computer.ConclusionsLab Retriever provides a practical software solution to forensic scientists who wish to assess the statistical weight of evidence for complex DNA profiles. Executable versions of the program are freely available for Mac OSX and Windows operating systems