Possible results are mutually exclusive and exhaustive. Explaining this distinction is the purpose of this first column. The distinction between probability and likelihood is fundamentally important: Probability attaches to possible results likelihood attaches to hypotheses. Distinguishing Likelihood From Probability In searching for something to write about that would be of general interest, I settled on presenting the basics of Bayesian data analysis in what I hope is an accessible form. In these consultations, I am struck by how much misunderstanding there is about basics.
There is a sense these days that Bayesian data analysis is a coming thing, so colleagues often consult me about it.
(“You cannot prove the null.”) The realization of these absurdities made me a Bayesian. Moreover, we understand a priori that the null hypothesis can never be accepted the best it can do is not be rejected. If it fails, we conclude that our hypothesis is correct - without testing it against the data and without formulating it with the same exactitude with which we formulated the hypothesis we did test (i.e., the null). When I look back on the formulation of the statistical inference problem I was taught and used for many years, I am astonished that I saw no problem with it: To test our own hypothesis, we test a different hypothesis - the null hypothesis. I ended up teaching a Bayesian-oriented graduate course in statistics and now use Bayesian methods in analyzing my own data. I knew that Bayesian methods could provide support for null hypotheses, so I began to look into them. They said that all he had were insignificant results that could not be used to support his null hypothesis. He got strong pushback from the reviewers. Some years ago, a postdoctoral fellow in my lab tried to publish a series of experiments with results that - to his surprise - supported a theoretically important but extremely counterintuitive null hypothesis.
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