General Introduction to Modern Bayesian Statistics

Abstract

With Bayesian statistics there is no traditional null (H0) and alternative hypothesis (H1) like there is with standard null hypothesis testing. Instead there is a standardized likelihood distribution to assess whether a new treatment is better or worse than control.

The term odds, otherwise called the ratio of [the chance of having a disease]/[chance of having no disease], plays a key role not only in logistic regressions and Cox regressions, but also in traditional Bayes statistical analyses as post-test odds and pre-test odds, where post-test odds = prior-test odds x likelihood ratio.

Modern Bayes does not work with normal distributions, but likelihood distributions, that are approximated differently. The traditional Bayes factor is not the area under the curve (AUC) of a likelihood distribution curve, but, rather, the ratio of the AUCs of two likelihood distributions. We should add, that the ratio of two odds values, has often been named Bayes factor with traditional Bayesian statistics, but with modern Bayesian statistics the Bayes factors are mostly based on the ratios of two likelihood distributions.

In the past the non exact intuitive definition of the prior was the Achilles heal of Bayes. Fortunately, the intuitive prior and the posterior odds have been replaced with more exact likelihood distributions and interpretations based on intervals of uncertainty.

Differences between traditional and Bayesian Statistics may include.

• 1. A better underlying structure of the alternative hypothesis H1 and the null hypothesis H0 may be provided.

• 2. Bayesian tests work with 95% credible intervals that are usually somewhat wider and this may reduce the chance of statistical significances with little clinical relevance.

• 3. Maximal likelihoods of likelihood distributions are not always identical to the mean effect of traditional tests, and this may be fine, because biological likelihoods may better fit biological questions than numerical means of non-representative subgroups do.

• 4. Bayes uses ratios of likelihood distributions rather than ratios of Gaussian distributions, which are notorious for ill data fit.

• 5. Bayesian integral computations are very advanced, and, therefore, give optimal precisions of complex functions, and better so than traditional multiple mean calculations of non representative subsamples do.

• 6. With Bayesian testing type I (alpha) and II (beta) errors need not being taken into account.