Summary. We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure pD for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general pD approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the ‘hat’ matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding pD to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis.

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Bayesian measures of model complexity and fit

We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.

/pdf/general-methods-for-monitoring-convergence-of-iterative-45g63vyekv.pdf

General methods for monitoring convergence of iterative simulations

Approaches for Statistical Inference. The Bayes Approach. The Empirical Bayes Approach. Performance of Bayes Procedures. Bayesian Computation. Model Criticism and Selection. Special Methods and Models. Case Studies. Appendices.

/pdf/bayes-and-empirical-bayes-methods-for-data-analysis-3936z5a6so.pdf

Bayes and Empirical Bayes Methods for Data Analysis

The field of viscous liquid and glassy solid dynamics is reviewed by a process of posing the key questions that need to be answered, and then providing the best answers available to the authors and their advisors at this time. The subject is divided into four parts, three of them dealing with behavior in different domains of temperature with respect to the glass transition temperature, Tg , and a fourth dealing with ‘‘short time processes.’’ The first part tackles the high temperature regime T.Tg ,i n which the system is ergodic and the evolution of the viscous liquid toward the condition at Tg is in focus. The second part deals with the regime T;Tg , where the system is nonergodic except for very long annealing times, hence has time-dependent properties ~aging and annealing!. The third part discusses behavior when the system is completely frozen with respect to the primary relaxation process but in which secondary processes, particularly those responsible for ‘‘superionic’’ conductivity, and dopart mobility in amorphous silicon, remain active. In the fourth part we focus on the behavior of the system at the crossover between the low frequency vibrational components of the molecular motion and its high frequency relaxational components, paying particular attention to very recent developments in the short time dielectric response and the high Q mechanical response. © 2000 American Institute of Physics.@S0021-8979~00!02213-1#

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1065&context=mse_pubs

Relaxation in glassforming liquids and amorphous solids

Matrix Differential Calculus with Applications in Statistics and Econometrics

: This paper discusses some aspects of selecting and testing random and pseudorandom number generators. The outputs of such generators may he used in many cryptographic applications, such as the generation of key material. Generators suitable for use in cryptographic applications may need to meet stronger requirements than for other applications. In particular, their outputs must he unpredictable in the absence of knowledge of the inputs. Some criteria for characterizing and selecting appropriate generators are discussed in this document. The subject of statistical testing and its relation to cryptanalysis is also discussed, and some recommended statistical tests are provided. These tests may he useful as a first step in determining whether or not a generator is suitable for a particular cryptographic application. The design and cryptanalysis of generators is outside the scope of this paper.

/pdf/a-statistical-test-suite-for-random-and-pseudorandom-number-xf8s068cdk.pdf

A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications

(1997). Bayes and Empirical Bayes Methods for Data Analysis. Technometrics: Vol. 39, No. 3, pp. 337-337.

This paper discusses some aspects of selecting and testing random and pseudorandom number generators. The outputs of such generators may be used in many cryptographic applications, such as the generation of key material. Generators suitable for use in cryptographic applications may need to meet stronger requirements than for other applications. In particular, their outputs must be unpredictable in the absence of knowledge of the inputs. Some criteria for characterizing and selecting appropriate generators are discussed in this document. The subject of statistical testing and its relation to cryptanalysis is also discussed, and some recommended statistical tests are provided. These tests may be useful as a first step in determining whether or not a generator is suitable for a particular cryptographic application. However, no set of statistical tests can absolutely certify a generator as appropriate for usage in a particular application, i.e., statistical testing cannot serve as a substitute for cryptanalysis. The design and cryptanalysis of generators is outside the scope of this paper.

SP 800-22 Rev. 1a. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications

(2002). Analysis of Time Series Structure SSA and Related Techniques. Technometrics: Vol. 44, No. 3, pp. 290-290.

Analysis of Time Series Structure SSA and Related Techniques

Random Number Generators (RNGs) are an important building block for algorithms and protocols in cryptography. They are paramount in the construction of encryption keys and other cryptographic algorithm parameters. In practice, statistical testing is employed to gather evidence that a generator indeed produces numbers that appear to be random. Few resources are readily available to researchers in academia and industry who wish to analyze their newly developed RNG. To address this problem, NIST has developed new metrics that may be employed to investigate the randomness of cryptographic RNGs. In this paper, issues such as statistical test suites, evaluation frameworks, and the interpretation of results are addressed.

Andrew L. Rukhin

Papers

A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications

Bayes and Empirical Bayes Methods for Data Analysis

SP 800-22 Rev. 1a. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications

Analysis of Time Series Structure SSA and Related Techniques

Statistical Testing of Random Number Generators