Sample size Bayesian approaches

Monte Carlo data for y were generated according to with mean x, to simulate process sampling data. A window size of 25 was used here and to demonstrate the performance of the Bayesian approach. 

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... 

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 very unsatisfactory feature of conventional approaches to sample size calculation is that there is no mention of cost. This means that for any two quite different indications with the same effect size, that is to say the same ratio of clinically relevant difference to standard deviation, the sample size would be the same whatever the cost or difficulty of recruiting and treating patients. This is clearly illogical and trialists probably manage this issue informally by manipulating the clinically relevant difference in the way discussed in Section 13.2.3. Clearly, it would be better to include the cost explicitly, and this suggests decision-analytic approaches to sample size determination. There are various Bayesian suggestions and these will be discussed in the next section. 

Pezeshk H, Gittins J (2002) A fully Bayesian approach to calculating sample sizes for clinical trials with binary responses. Drug Information Journal 36 143 150. 

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 present paper will put forward the point that a Bayesian framework may be viewed as rather natural for tackling issues (a) and (b) altogether. Indeed, beyond the forceful epistemological and decision-theory feamres of a Bayesian approach, it includes by definition a double-level probabilistic model separating epistemic and aleatory components and offers a traceable process to mix the encoding of engineering expertise inside priors and the observations inside an updated epistemic layer that proves mathematically consistent even when dealing with very low-size samples. 

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). 

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. ... 

In this section we look at using the Gibbs sampler for a simple situation where we have a random sample of size n from a normal fj., a ) distribution with both parameters unknown. To do Bayesian inference we will need to find a joint prior density for the two parameters. Usually, we want to make inferences about the mean /i, and regard cr as a nuisance parameter. There are two approaches we can take to choosing the joint prior distributions for this problem. 

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