CHAPTER 8 - Decision Theoretic and Bayesian Approach in Testing and Estimation

At the start of new operation, a production process is commonly unstable. Then, a process condition is into being stable gradually over time. For the purpose of shifting to mass production, the stability in a process should be evaluated. This paper proposes a method of evaluating the process stability based on Bayesian statistics and information theory. In Bayesian statistics, the knowledge for a process is renewed by deriving posterior distribution based on current prior distribution and observations. Here, equivalency between prior and posterior distributions could be considered to be a criterion for the stability in a process. It is needed to define difference between prior and posterior distributions in order to evaluate the equivalency between prior and posterior distributions. We first formulate the relation between the prior distribution and the posterior distribution for process mean using Bayesian statistics. Secondly, we evaluate the difference between the both distributions based on Kullback‐Leibler (K‐L) divergence in information theory. Finally, some numerical examples in the method of evaluating the stability in process mean are illustrated. Also, we discuss about a decision rule for the stability in process mean through the numerical investigation.

A crucial decision in designing a spatial sample for soil survey is the number of sampling locations required to answer, with sufficient accuracy and precision, the questions posed by decision makers at different levels of geographic aggregation. In the Indian Soil Health Card (SHC) scheme, many thousands of locations are sampled per district. In this paper the SHC data are used to estimate the mean of a soil property within a defined study area, e.g., a district, or the areal fraction of the study area where some condition is satisfied, e.g., exceedence of a critical level. The central question is whether this large sample size is needed for this aim. The sample size required for a given maximum length of a confidence interval can be computed with formulas from classical sampling theory, using a prior estimate of the variance of the property of interest within the study area. Similarly, for the areal fraction a prior estimate of this fraction is required. In practice we are uncertain about these prior estimates, and our uncertainty is not accounted for in classical sample size determination (SSD). This deficiency can be overcome with a Bayesian approach, in which the prior estimate of the variance or areal fraction is replaced by a prior distribution. Once new data from the sample are available, this prior distribution is updated to a posterior distribution using Bayes’ rule. The apparent problem with a Bayesian approach prior to a sampling campaign is that the data are not yet available. This dilemma can be solved by computing, for a given sample size, the predictive distribution of the data, given a prior distribution on the population and design parameter. Thus we do not have a single vector with data values, but a finite or infinite set of possible data vectors. As a consequence, we have as many posterior distribution functions as we have data vectors. This leads to a probability distribution of lengths or coverages of Bayesian credible intervals, from which various criteria for SSD can be derived. Besides the fully Bayesian approach, a mixed Bayesian-likelihood approach for SSD is available. This is of interest when, after the data have been collected, we prefer to estimate the mean from these data only, using the frequentist approach, ignoring the prior distribution. The fully Bayesian and mixed Bayesian-likelihood approach are illustrated for estimating the mean of log-transformed Zn and the areal fraction with Zn-deficiency, defined as Zn concentration <0.9 mg kg −1, in the thirteen districts of Andhra Pradesh state. The SHC data from 2015–2017 are used to derive prior distributions. For all districts the Bayesian and mixed Bayesian-likelihood sample sizes are much smaller than the current sample sizes. The hyperparameters of the prior distributions have a strong effect on the sample sizes. We discuss methods to deal with this. Even at the mandal (sub-district) level the sample size can almost always be reduced substantially. Clearly SHC over-sampled, and here we show how to reduce the effort while still providing information required for decision-making. R scripts for SSD are provided as supplementary material.

Journal Article Market Structure and Innovation Get access Market Structure and Innovation. By Morton I. KAMIEN and NANCY L. SCHWARTZ. (Cambridge: Cambridge University Press, 1982. Pp. xi + 241. £17.50 hardback, £6.95 paperback.) P. Lesley Cook P. Lesley Cook University of Sussex Search for other works by this author on: Oxford Academic Google Scholar The Economic Journal, Volume 92, Issue 368, 1 December 1982, Pages 985–987, https://doi.org/10.2307/2232689 Published: 01 December 1982

Sample size estimation is a fundamental element of a clinical trial, and a binomial experiment is the most common situation faced in clinical trial design. A Bayesian method to determine sample size is an alternative solution to a frequentist design, especially for studies conducted on small sample sizes. The Bayesian approach uses the available knowledge, which is translated into a prior distribution, instead of a point estimate, to perform the final inference. This procedure takes the uncertainty in data prediction entirely into account. When objective data, historical information, and literature data are not available, it may be indispensable to use expert opinion to derive the prior distribution by performing an elicitation process. Expert elicitation is the process of translating expert opinion into a prior probability distribution. We investigated the estimation of a binomial sample size providing a generalized version of the average length, coverage criteria, and worst outcome criterion. The original method was proposed by Joseph and is defined in a parametric framework based on a Beta-Binomial model. We propose a more flexible approach for binary data sample size estimation in this theoretical setting by considering parametric approaches (Beta priors) and semiparametric priors based on B-splines.

The classical theory to determine sampling plans yields a large sample-size which is necessary to demonstrate the product reliability. Above all, the sample-size increases tremendously, if failures have to be taken into account. To use all information about the product given through the development process the application of Bayes procedure is recommended. In the paper, it was suggested how to generate a prior distribution from fatigue damage calculations, preceding tests or the analysis of warranty data of a former product. We considered a case where the information about the reliability may not be given for the desired product lifetime but for any other time. Additionally, it was taken into account that prior information may be available from an accelerated test. When prior information is used there is always the uncertainty as to what extent the information about the reliability is valid for the actual product conditions. It is obvious that the information of a former product or preceding tests may not be totally transferable to the actual product concerning reliability. In this paper the so-called /spl delta/ was introduced which artificially reduces the quality level of prior reliability information. A procedure for the estimation of the decrease-factor was suggested for a case where information about the failure behavior of a former product is utilized. In this context, the FMECA offers a useful foundation. In a synthetic example the reliability demonstration test was planned by using calculation results, results of a preceding test and the knowledge about the failure behavior of a former product. It was shown that the sample-size necessary to demonstrate the requirements increases with lower decrease-factors. Compared with the classical method it was possible to reduce the sample-size if prior knowledge is considered. The sample-size is reduced by 4 test items at least. For the best case where the prior information was totally used no subsequent test would be necessary. However, due to differences in test conditions or environment and function the total use of prior information is not recommended. To minimize the risk of failing to meet the product requirements the decrease-factor should be estimated with lower values than 1.

The aim of this study was to explore whether Bayesian reasoning can be applied to therapeutic questions in a way that is similar to its application in diagnostics. A clinically relevant, therapeutic question was formulated in accordance with Bayesian reasoning for the clinical management of patients with newly diagnosed rheumatoid arthritis (RA). Prior probability estimates of response to drug treatment (methotrexate, MTX) were obtained from the literature. As a marker of treatment response, changes in the Health Assessment Questionnaire (HAQ) scores were assessed after three months of treatment. Likelihood ratios for this marker were calculated on the basis of data from a clinical registry, using changes in the Disease Activity Score (DAS) as gold standard. Using Bayes' theorem, prior probability and likelihood ratios were combined to estimate posterior probabilities of treatment response in individual patients. On the basis of the literature, the prior probability of response of RA patients to MTX was estimated 45 percent. At 3 months follow-up, this probability increased to 80 percent or decreased to 23 percent, depending on the changes that were observed in Health Assessment Questionnaire scores. Bayesian reasoning can be applied to therapeutic issues in a way that is conceptually fully compatible with its use in diagnostics. As such, it can be used to bridge the gap between aggregate data and individual patient management.

Bayesian methods are increasingly used in proof-of-concept studies. An important benefit of these methods is the potential to use informative priors, thereby reducing sample size. This is particularly relevant for treatment arms where there is a substantial amount of historical information such as placebo and active comparators. One issue with using an informative prior is the possibility of a mismatch between the informative prior and the observed data, referred to as prior-data conflict. We focus on two methods for dealing with this: a testing approach and a mixture prior approach. The testing approach assesses prior-data conflict by comparing the observed data to the prior predictive distribution and resorting to a non-informative prior if prior-data conflict is declared. The mixture prior approach uses a prior with a precise and diffuse component. We assess these approaches for the normal case via simulation and show they have some attractive features as compared with the standard one-component informative prior. For example, when the discrepancy between the prior and the data is sufficiently marked, and intuitively, one feels less certain about the results, both the testing and mixture approaches typically yield wider posterior-credible intervals than when there is no discrepancy. In contrast, when there is no discrepancy, the results of these approaches are typically similar to the standard approach. Whilst for any specific study, the operating characteristics of any selected approach should be assessed and agreed at the design stage; we believe these two approaches are each worthy of consideration.

Bayesian approaches have been extensively used in animal breeding sciences, but similar approaches in the context of evolutionary quantitative genetics have been rare. We compared the performance of Bayesian and frequentist approaches in estimation of quantitative genetic parameters (viz. matrices of additive and dominance variances) in datasets typical of evolutionary studies and traits differing in their genetic architecture. Our results illustrate that it is difficult to disentangle the relative roles of different genetic components from small datasets, and that ignoring, e.g. dominance is likely to lead to biased estimates of additive variance. We suggest that a natural summary statistic for G-matrix comparisons can be obtained by examining how different the underlying multinormal probability distributions are, and illustrate our approach with data on the common frog (Rana temporaria). Furthermore, we derive a simple Monte Carlo method for computation of fraternity coefficients needed for the estimation of dominance variance, and use the pedigree of a natural Siberian jay (Perisoreus infaustus) population to illustrate that the commonly used approximate values can be substantially biased.

The aim of the study was evaluate and compare the efficiency of Bayesian and frequentist approach to describe the rumen degradation of NDF. Simulated data was composed by four scenarios: regular restriction in the number of incubation times, random loss of incubation times, loss of specific parts of degradation curves, variation in the precision of the incubations procedures. Two real datasets was used, these real data encompassed the evaluation of NDF degradation of a tropical grass (Brachiaria decumbes). The model was fitted according their characteristics approach and compared by plots and assessors. The Bayesian and frequentist approach presented reliable estimates of degradation parameters for the majority of the data tested. Therefore, in specific cases with short random records number, the Bayesian approach showed greater bias of the estimates of incubation residue and estimates of degradation rate without a biological coherence of the parameters, compared to frequentist inference. In another words, the Bayesian approach fitted with prior diffuse, presented less flexible. Nevertheless, it is emphasized the importance of the background information before the modeling, mainly for the Bayesian approach, in order to define proper prior distributions. Future thorough studies about the influence of non-informative prior for the parameters are necessary.

For more than three decades, Deborah Mayo has defended severe testing, a framework for evaluating the strength of scientific evidence that provides philosophical foundations for (variants of) classical statistical methods. In her newest book, Statistical Inference as Severe Testing, Mayo (1) extends and refines her views on severe testing, (2) takes aim at researchers who claim that the reproducibility crisis is a result of the use of classical statistics, and (3) exposes some contemporary Bayesian methods as lacking philosophical foundations. The book is engaging, sometimes funny, and often insightful. After reviewing the basics of severe testing, I summarize three lessons from Mayo's book that are valuable for philosophers and scientists alike. I then discuss one way in which I think Mayo's arguments could be tightened.Mayo argues that one has evidence for a hypothesis to the extent the hypothesis has passed a severe test. More precisely, she argues that “for data to warrant a hypothesis H requires not just that (S-1) H agrees with the data (H passes the test), but also (S-2) with high probability, H would not have passed the test so well, were H false” (92). Mayo's two criteria are inspired by the Neyman-Pearsonian maxim that good statistical tests minimize Type 1 and Type 2 errors, but they are novel in at least two ways. First, Mayo's criteria are applicable outside of statistical contexts. For example, let H be the hypothesis that the Cleveland Browns played an excellent team last weekend, and suppose our data is that the Cleveland Browns lost their game. If the Cleveland Browns lose to all opponents regardless of ability, then our data fails to satisfy S-2 and so does not severely test H. Since S-1 and S-2 are applicable in both statistical and nonquantitative contexts, Mayo's criteria unify the logic of statistical testing with qualitative logics of evidence.Second, severity is a function of data, unlike size and power, the two core quantities of orthodox Neyman-Pearsonian methodology. For this reason, Mayo's two criteria closely resemble Nozick's adherence and sensitivity conditions when those conditions are relativized to “methods” (Nozick 1981).1Mayo draws several important lessons from this seemingly simple, two-pronged framework. Here, I discuss three that recur throughout the newest book. First, Mayo carefully distinguishes between statistical hypotheses (e.g., about the frequency of a trait in a population) and “substantive” scientific hypotheses about, for example, causation or the existence of a theoretical entity (84, 93). Conflating the two can lead one to mistakenly infer that a substantive scientific hypothesis has been severely tested, when in fact the evidence against the hypothesis is stronger than that for it.As a vivid example, Mayo discusses a series of psychological experiments used to defend the hypothesis H that heterosexual “men's implicit self-esteem is lower when a [female] partner succeeds than when a [female] partner fails” (101). Mayo carefully distinguishes four statistical hypotheses that are plausibly consequences of ¬ H and then argues that the data were sufficient to reject only one of the four; call it ¬ H 1. Mayo concludes that the experimental data failed to fit H, as researchers could not reject three statistical hypotheses that were compatible with ¬ H. Further, Mayo argues that the statistical evidence against ¬ H 1 (i.e., the purported evidence for ¬¬ H = H ) was perfectly compatible with the falsity of the substantive hypothesis H. So the data also failed to satisfy S-2.Second, Mayo argues that, whereas the severe tester has a unified account of testing both statistical hypotheses and modeling assumptions, likelihoodists and Bayesians lack a cogent story about when to reject a bad statistical model (298–320). Some likelihoodists, for example, argue that measures of evidential strength must be comparative and that the likelihood ratio P (E|H 0)/P (E|H 1) quantifies the degree to which evidence E fits a hypothesis H 0 better than H 1 (Edwards 1984; Royall 1997). It is easy to construct cases in which such a likelihood ratio is enormous, because the data fit H 0 much better than H 1, and yet the data fit neither statistical hypothesis well. If H 0 and H 1 are the only statistical hypotheses under investigation, one might erroneously conclude that one has good evidence for H 0. The right conclusion, according to the severe tester, is to recognize that the two hypotheses do not exhaust logical space and to reject that underlying modeling assumption instead.Naive Bayesian reasoning falls victim to the same criticism. According to some Bayesians, experimenters ought to assign prior probabilities to a fixed set of hypotheses H and then update their degrees of belief by conditionalization. Again, suppose the data fit all hypotheses in H poorly but fit one H ∈ H much better than its rivals. Then a naive Bayesian's posterior degree of belief in H might be nearly one. Again, the right conclusion, according to the severe tester, is to consider more hypotheses. But orthodox Bayesianism is silent about when to do that.Finally, Mayo argues that several novel Bayesian approaches to statistical inference lack foundations (400–415). These parts of the book will be of special interest to philosophers who are unfamiliar with the substantial differences between Bayesianism in theory and in practice. According to textbook philosophical presentations, prior and posterior probabilities represent an experimenter's beliefs. However, some contemporary Bayesians have given up on the myth that the infinitely precise densities that appear in journal articles are appropriate idealizations of any person's beliefs. Those Bayesians, however, disagree about what justifies the choice of prior (e.g., transparent communication vs. frequentist coverage properties) and the use of further Bayesian mathematical machinery.Mayo levels several different criticisms against these contemporary Bayesian approaches, but I think all share the same underlying philosophical problem. Rationality, so says standard decision theory, requires one to maximize expected utility relative to one own's degrees of belief. So some contemporary “Bayesian” methods require a new theory of rationality to justify the behavior of an experimenter whose decisions employ posterior probabilities that do not even approximately represent her (or perhaps anyone's) beliefs.There are other important lessons scattered through Mayo's engaging and fairly accessible book. That leads me to my main criticism: accessibility sometimes comes at the cost of precision. I focus on one example that is directly relevant to Mayo's analysis of the causes of the replicability crisis, a central topic of the book.Mayo repeatedly suggests that a hypothesis is not severely tested if it is the result of “data-dredging.” Data-dredging is one of several pejorative terms (including p-hacking) that some statistical reformers use, and it refers to a loosely defined set of purportedly objectionable practices for finding associations in data. Often, “data-dredging” describes the practice of subjecting a single data set to many (sometimes hundreds or thousands of) statistical tests in an attempt to find a statistically significant result that is publishable.Critics of data-dredging allege that, because there is always a chance of finding a spurious association between two variables, one should be wary of “dredged” hypotheses. For severe testers, that default skepticism is justifiable: if an experiment is not designed to test a hypothesis, it is unlikely to be a severe test by chance. However, intuitions about data-dredging depend upon how the practice is described. Searching through data for patterns sounds exactly like what scientists should do!Mayo illustrates the dangers of data-dredging via several thought experiments. For example, she imagines a pharmaceutical company that hires scientists to find the benefits of a recently developed drug (267–68). When the hired researchers find no evidence that the drug is salutary in any of ways the manufacturer had hoped, they mine their data to find some salutary effect. Sure enough, they find the drug has an “impressive benefit on factor B” and begin marketing the drug with that promise. Mayo concludes that the hired scientists fail to subject the hypothesis to a severe test because they would have found an effect no matter what.Unfortunately, Mayo never explicitly applies her two criteria for severity to this example. Which hypothesis exactly is being tested? Here are two candidates. Let H 1 be the hypothesis “The drug has some salutary effect” and let H 2 be the hypothesis “The drug has the specific salutary effect found on B.” Even if the drug manufacturer is guaranteed to find some effect (i.e., H 1 is not severely tested), it does not follow that the manufacturer lacks evidence that the drug has the specific effect found by mining the data (i.e., it does not follow that H 2 is not severely tested). When I grade student logic exams, I am guaranteed to find that some student scored highest, but I do not thereby lack evidence that Roshan scored best when I tally her score and see it is highest. Similarly, cigarette manufacturers cannot avoid liability for causing lung cancer by (1) collecting enormous amounts of data, (2) investigating the relationship between cigarette smoking and hundreds of health problems, and then (3) claiming they were destined to find some statistically significant effect or another.2Some accusations of data-dredging, I claim, are instances of a fallacy that Mayo typically distinguishes for her readers with great care: the conflation of the pre-experimental reliability of a test with the postexperimental evidence provided by a particular data set.3 The history of science is replete with examples of unlikely, happy discoveries, where the data unequivocally support a hypothesis that scientists did not intend to test. In accessible, vivid prose, Mayo illustrates why one should typically be wary of data-dredging, but readers will be disappointed if they wish to learn how Mayo's two criteria can be used to distinguish between (a) hypotheses that are illicitly “dredged” and (b) hypotheses that are supported by one's data, despite being formulated postexperiment.Quibbles aside, Mayo's newest book is a yet another deep and engaging contribution to the philosophy of statistics and philosophy of science. Most importantly, it shows how foundational philosophical debates matter to pressing issues in science policy.

We study variable sampling plans for exponential distributions based on type-I hybrid censored samples. For this problem, two sampling plans based on the non-failure sample proportion and the conditional maximum likelihood estimator are proposed by Chen et al. [J. Chen, W. Chou, H. Wu, and H. Zhou, Designing acceptance sampling schemes for life testing with mixed censoring, Naval Res. Logist. 51 (2004), pp. 597–612] and Lin et al. [C.-T. Lin, Y.-L. Huang, and N. Balakrishnan, Exact Bayesian variable sampling plans for the exponential distribution based on type-I and type-II censored samples, Commun. Statist. Simul. Comput. 37 (2008), pp. 1101–1116], respectively. From the theoretic decision point of view, the preceding two sampling plans are not optimal due to their decision functions not being the Bayes decision functions. In this article, we consider the decision theoretic approach, and the optimal Bayesian sampling plan based on sufficient statistics is derived under a general loss function. Furthermore, for the conjugate prior distribution, the closed-form formula of the Bayes decision rule can be obtained under either the linear or quadratic decision loss. The resulting Bayesian sampling plan has the minimum Bayes risk, and hence it is better than the sampling plans proposed by Chen et al. (2004) and Lin et al. (2008). Numerical comparisons are given and demonstrate that the performance of the proposed Bayesian sampling plan is superior to that of Chen et al. (2004) and Lin et al. (2008).

Comparison of Fourier and Bayesian analysis of nmr signals. I. Well-separated resonances (the single-frequency case)

A trending approach for analysis in critical care research: Bayesian statistics.

Bayesian approaches have been demonstrated as effective methods for reliability analysis of complex systems with small-amount data, which integrate prior information and sample data using Bayes’ theorem. However, there is an assumption that precise prior probability distributions are available for unknown parameters, yet these prior distributions are sometimes unavailable in practical engineering. A possible way to avoiding this assumption is to generalize Bayesian reliability analysis approach by using imprecise probability theory. In this paper, we adopt a set of imprecise Dirichlet distributions as priors to quantify uncertainty of unknown parameters and extend traditional Bayesian reliability analysis approach by introducing an imprecise Dirichlet model (IDM). When the prior information is rare, the result of imprecise Bayesian analysis method is too rough to support engineering decision-making, so we proposed an optimization model to reduce the imprecision of the new method. Spindles are crucial for machine tools and reliability data related to spindles of new-developed machine tools are often rare. We can then use the imprecise Bayesian reliability analysis method to assess its reliability. In this paper, we mainly investigate the reliability assessment of a motorized spindle to illustrate the effectiveness of the proposed method.

On the estimation of parameters of log-unsaturated conductivity covariance from solute transport data