Herbert Robbins Bayesian Analysis

The EB approach is Herbert Robbins’ most influential contribution to statistical theory. Among his many fruitful ideas, EB approach, which he named as well as developed, had the biggest effect on statistical thinking. He published his first paper on this approach in 1951, and continued to develop these ideas in two later papers published in 1956 and 1961. Since then, it has become a powerful tool in statistical decision making and now a day, this approach to statistical design and analysis is emerging as an increasingly effective and practical alternative to the frequentist and Bayesian approaches.
When we have so much data, but so little data about each object, we can achieve great gains by sharing information across objects. We may not have

This uses one of the strengths of the Bayesian approach, the ability of derivation of estimate of the parameter. This estimator usually contains some unknown quantities that Bayesian specifies. Instead, EB constructs estimators from the available information on hand, and substitutes them into the Bayes quantity. This uses one of the strengths of conventional approach, the ability of construction of estimates of the unknown quantities. Thus, EB uses Bayesian approach for modelling a situation and conventional approach for constructing estimators of unknown quantities of the model. The EB estimation approach has facilitated the analysis of complex multi-faceted problems, which are often difficult to handle using the conventional or Bayesian approach. Broadly, EB approach has two essential components, likelihood function and prior distribution. Once the data has been observed, the sampling distribution can be considered as the function of unknown prior parameter. The likelihood function is constructed from the sampling distribution of the data. It contains the measurement model and gives the probability density of observing measurements given the parameter of

Bayesian analyst may have some hypotheses or beliefs about what he/she expect to find before the data is collected. These hypotheses are called the priors, which represent the information available before an actual data come available. However, EB analysts have pointed out that there is no clear-cut way from which Bayesian analyst can conclude that one prior is better than the other is. There is no single answer to the question, “What should be the right prior?” Selecting appropriate prior is perhaps the most important and most contentious aspect of Bayesian modelling. The standard Bayesian inference does not tell us how to select prior distributions. For much of the time the prior information is subjective and is based on the Bayesian’s own experience and judgment. Uncertainty or degree of belief with respect to the parameter can be quantified by a probability distribution, known as a prior distribution. The prior distribution is the probability density function measuring the likelihood of an event, chosen at random having a rate of occurrence of the parameter. According to Armitage & Colton (2005), the appearance of prior distributions is at once a strength and a weakness of the Bayesian approach: a strength because it allows information beyond the data at hand to be used in making inferences, and a weakness because inferences inevitably