Bayes and health care research.

Bayes’ rule shows how one might rationally change one’s beliefs in the light of evidence. It is the foundation of a statistical method called Bayesianism. In health care research, Bayesianism has its advocates but the dominant statistical method is frequentism. There are at least two important philosophical differences between these methods. First, Bayesianism takes a subjectivist view of probability (i.e. that probability scores are statements of subjective belief, not objective fact) whilst frequentism takes an objectivist view. Second, Bayesianism is explicitly inductive (i.e. it shows how we may induce views about the world based on partial data from it) whereas frequentism is at least compatible with non-inductive views of scientific method, particularly the critical realism of Popper. Popper and others detail significant problems with induction. Frequentism’s apparent ability to avoid these, plus its ability to give a seemingly more scientific and objective take on probability, lies behind its philosophical appeal to health care researchers. However, there are also significant problems with frequentism, particularly its inability to assign probability scores to single events. Popper thus proposed an alternative objectivist view of probability, called propensity theory, which he allies to a theory of corroboration; but this too has significant problems, in particular, it may not successfully avoid induction. If this is so then Bayesianism might be philosophically the strongest of the statistical approaches. The article sets out a number of its philosophical and methodological attractions. Finally, it outlines a way in which critical realism and Bayesianism might work together. </p

Superstatistical distributions from a maximum entropy principle

We deal with a generalized statistical description of nonequilibrium complex systems based on least biased distributions given some prior information. A maximum entropy principle is introduced that allows for the determination of the distribution of the fluctuating intensive parameter $\beta$ of a superstatistical system, given certain constraints on the complex system under consideration. We apply the theory to three examples: The superstatistical quantum mechanical harmonic oscillator, the superstatistical classical ideal gas, and velocity time series as measured in a turbulent Taylor-Couette flow

Approximation algorithms for NEXTtime-hard periodically specified problems and domino problems

We study the efficient approximability of two general class of problems: (1) optimization versions of the domino problems studies in [Ha85, Ha86, vEB83, SB84] and (2) graph and satisfiability problems when specified using various kinds of periodic specifications. Both easiness and hardness results are obtained. Our efficient approximation algorithms and schemes are based on extensions of the ideas. Two of properties of our results obtained here are: (1) For the first time, efficient approximation algorithms and schemes have been developed for natural NEXPTIME-complete problems. (2) Our results are the first polynomial time approximation algorithms with good performance guarantees for `hard` problems specified using various kinds of periodic specifications considered in this paper. Our results significantly extend the results in [HW94, Wa93, MH+94]

The diagnostic accuracy of high b-value diffusion- and T2-weighted imaging for the detection of prostate cancer: a meta-analysis

Purpose: This study aims to investigate the role of diffusion-weighted imaging (DWI) and T2-weighted imaging (T2WI) in combination for the detection of prostate cancer, specifically assessing the role of high b-values (> 1000 s/mm2), with a systematic review and meta-analysis of the existing published data.  Methods: The electronic databases MEDLINE, EMBASE, and OpenSIGLE were searched between inception and September 1, 2017. Eligible studies were those that reported the sensitivity and specificity of DWI and T2WI for the diagnosis of prostate cancer by visual assessment using a histopathologic reference standard. The QUADAS-2 critical appraisal tool was used to assess the quality of included studies. A meta-analysis with pooling of sensitivity, specificity, likelihood, and diagnostic odds ratios was undertaken, and a summary receiver-operating characteristics (sROC) curve was constructed. Predetermined subgroup analysis was also performed.  Results: Thirty-three studies were included in the final analysis, evaluating 2949 patients. The pooled sensitivity and specificity were 0.69 (95% CI 0.68–0.69) and 0.84 (95% CI 0.83–0.85), respectively, and the sROC AUC was 0.84 (95% CI 0.81–0.87). Subgroup analysis showed significantly better sensitivity with high b-values (> 1000 s/mm2). There was high statistical heterogeneity between studies.  Conclusion: The diagnostic accuracy of combined DWI and T2WI is good with high b-values (> 1000 s/mm2) seeming to improve overall sensitivity while maintaining specificity. However, further large-scale studies specifically looking at b-value choice are required before a categorical recommendation can be made

Periodically specified satisfiability problems with applications: An alternative to domino problems

We characterize the complexities of several basic generalized CNF satisfiability problems SAT(S), when instances are specified using various kinds of 1- and 2-dimensional periodic specifications. We outline how this characterization can be used to prove a number of new hardness results for the complexity classes DSPACE(n), NSPACE(n), DEXPTIME, NEXPTIME, EXPSPACE etc. The hardness results presented significantly extend the known hardness results for periodically specified problems. Several advantages axe outlined of the use of periodically specified satisfiability problems over the use of domino problems in proving both hardness and easiness results. As one corollary, we show that a number of basic NP-hard problems become EXPSPACE hard when inputs axe represented using 1-dimensional infinite periodic wide specifications. This answers a long standing open question posed by Orlin

Parameter Estimation Error Dependency on the Acquisition Protocol in Diffusion Kurtosis Imaging

Mono-exponential kurtosis model is routinely fitted on diffusion weighted, magnetic resonance imaging data to describe non-Gaussian diffusion. Here, the purpose was to optimize acquisitions for this model to minimize the errors in estimating diffusion coefficient and kurtosis. Similar to a previous study, covariance matrix calculations were used, and coefficients of variation in estimating each parameter of this model were calculated. The acquisition parameter, b values, varied in discrete grids to find the optimum ones that minimize the coefficient of variation in estimating the two non-Gaussian parameters. Also, the effect of variation of the target values on the optimized values was investigated. Additionally, the results were benchmarked with Monte Carlo noise simulations. Simple correlations were found between the optimized b values and target values of diffusion and kurtosis. For small target values of the two parameters, there is higher chance of having significant errors; this is caused by maximum b value limits imposed by the scanner than the mathematical bounds. The results here, cover a wide range of parameters D and K so that they could be used in many directionally averaged diffusion weighted cases such as head and neck, prostate, etc

Generalized entropy optimized by an arbitrary distribution

We construct the generalized entropy optimized by a given arbitrary statistical distribution with a finite linear expectation value of a random quantity of interest. This offers, via the maximum entropy principle, a unified basis for a great variety of distributions observed in nature, which can hardly be described by the conventional methods. As a simple example, we explicitly derive the entropy associated with the stretched exponential distribution. To include the distributions with the divergent moments (e.g., the Levy stable distributions), it is necessary to modify the definition of the expectation value.Comment: 10 pages, no figure