Bayesian statistics sample size

SABRE Method. Acronym for Simulated Approach to Bayesian Reliability Evaluation. An advanced approach to designing a reliability test program developed at PicArsn, the objective of which was to design a test program of minimum sample size for artillery fired atomic projectiles. Called the SABRE method, the program uses mathematical modeling, Monte Carlo simulation techniques, and Bayesian statistics. It is a sophisticated system devised to test items that cannot be tested because of their atomic nature. The aim is to determine the risk factor and to predict what will happen when the projectile is fired... 

The cornerstone of Bayesian methods is Bayes Theorem, which was first published in 1763 (Box Tiao, 1973). Bayes Theorem provides a method for statistical inference in which a prior distribution, based upon subjective judgement, can be updated with empirical data, to create a posterior distribution that combines both judgement and data. As the sample size of the data becomes large, the posterior distribution will tend to converge to the same result that would be obtained with frequentist methods. In situations in which there are no relevant sample data, the analysis can be conducted based upon the prior distribution, without any updating. 

In Sections 2 to 4, we review the technology of synthetic oligonucleotide microarrays and describe some of the popular statistical methods that are used to discover genes with differential expression in simple comparative experiments. A novel Bayesian procedure is introduced in Section 5 to analyze differential expression that addresses some of the limitations of current procedures. We proceed, in Section 6, by discussing the issue of sample size and describe two approaches to sample size determination in screening experiments with microarrays. The first approach is based on the concept of reproducibility, and the second approach uses a Bayesian decision-theoretic criterion to trade off information gain and experimental costs. We conclude, in Section 7, with a discussion of some of the open problems in the design and analysis of microarray experiments that need further research. 

A sample space is generally defined and all probabilities are calculated with respect to that sample space. In many cases, however, we ate in a position to update the sample space based on new information. For example, like the fourth example of Example 2.3, if we just consider the case that two outcomes from roUing a die twice are the same, the size of the sample space is reduced from 36 to 6. General definitions of conditional probability and independence are introduced. The Bayes theorem is also introduced, which is the basis of a statistical methodology called Bayesian statistics. 

De Santis, E (2007) Using historical data for Bayesian sample size determination. Journal of the Royal Statistical Society, Series A 170,95-113. 

Wang, F. and Gelfand, A. E. (2002) A simulation-based approach to Bayesian sample size determination for performance under a given model and for separating models. Statistical Sciences 17,193-208. 

The overall goal of Bayesian inference is knowing the posterior. The fundamental idea behind nearly all statistical methods is that as the sample size increases, the distribution of a random sample from a population approaches the distribution of the population. Thus, the distribution of the random sample from the posterior will approach the true posterior distribution. Other inferences such as point and interval estimates of the parameters can be constructed from the posterior sample. For example, if we had a random sample from the posterior, any parameter could be estimated by the corresponding statistic calculated from that random sample. We could achieve any required level of accuracy for our estimates by making sure our random sample from the posterior is large enough. Existing exploratory data analysis (EDA) techniques can be used on the sample from the posterior to explore the relationships between parameters in the posterior. 

The computational approach to Bayesian statistics allows the posterior to be approached from a completely different direction. Instead of using the computer to calculate the posterior numerically, we use the computer to draw a Monte Carlo sample from the posterior. Fortunately, all we need to know is the shape of the posterior density, which is given by the prior times the likelihood. We do not need to know the scale factor necessary to make it the exact posterior density. These methods replace the very difficult numerical integration with the much easier process of drawing random samples. A Monte Carlo random sample from the posterior will approximate the true posterior when the sample size is large enough. We will base our inferences on the Monte Carlo random sample from the posterior, not from the numerically calculated posterior. Sometimes this approach to Bayesian inference is the only feasible method, particularly when the parameter space is high dimensional. 

The derivation of functional relationships between independent variables, i.e., a concentration or amount proportional quantity and dependent variables - the response - belongs to the daily work of an analytical chemist. The functional relation has to be established in the calibration step and the concentration of an unknown sample can be estimated by its inverse application. Really both the dependent and independent variables are superimposed by error. Statistical methods accounting for errors in both responses (y) and concentrations (x) can hardly be applied if only a small sample size is available because the estimates become poor. Furthermore, in comparison to the Bayesian approach the incorporation of prior knowledge or subjective aspects with respect to the uncertainty of the data is carried out more easily by fuzzy methods. Results relying on the Bayesian approach can be doubtful if standard model assumptions do not hold. ... 

Uncertainties owning to the limited size of the sample or the equivalent sufficient statistic k,T can be quantified following the Bayesian approach, proposed by Lindley (1965) and Bedford Cook (2001). 

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