E Frequentists, Bayesians, and the Philosophy of Hypothesis Testing
Despite the fact that frequentist and Bayesian statistics have their own specialized niches, frequentists and Bayesians do not always coexist peacefully. Frequentists, who hold the dominant position in most areas of experimental science, tend to ignore Bayesian thinking altogether in their textbooks.
Bayesians, on the other hand, often make much of the Bayesian-frequentist split and call attention to the deficiencies of frequentism. While Bayesian reasoning has not made a mark in many areas of biology, it is increasingly important in social sciences and certain kinds of Big Data analyses (Chapter 15).Box 6.2 Falsification Using Bayesian Model Selection Methods
Here's a trivial example to show how reasoning would work. As a neuroscientist, you are investigating brain mechanisms of drug-seeking behavior in cocaine-addicted mice. Your idea is that within a group of cells, the (make- believe) nucleus Sniffens (n. Snf.), there is a dedicated neuronal circuit that regulates this behavior. There are only two sorts of neurons in n. Snf., Locals and Fars, one or both of them must mediate the behavior. The Locals outnumber Fars by 50:1. If a mouse pushes the correct lever press, a minute pinch of powdered cocaine is automatically deposited onto a tiny mirror nearby. Your H1 says that most of the cells that are active during the behavior (B) are Locals; that is, that the posterior odds ratio, p(B/L)/p(B/F) is >1.0. H2 says that most of the cells are Fars; that is, p(B/L)/p(B/F) is 10 (or 10 and favors H1, you can conclude that most of the activated neurons are Locals. You reject H2.
NHST remains the major statistical hypothesis testing mode in basic science. If you think that probabilities are properties of the external world, then the fixed values of frequentist probabilities form natural cutoffs for all-or-none decisions involving falsification.
Of course scientific decisions, including those based on falsification, are always tentative. We use statistics to aid in making judgments, and judgments are by definition subjective. Nevertheless, frequentist methods obvious mesh nicely with the methods of Fisher and Popper.On the other hand, the Neyman-Pearson philosophy has much in common with the typical Bayesian perspective. Both focus on practical problems and strive for good, predictive models. The Neyman-Pearson program emphasizes the importance of taking errors into account and of playing their costs of various kinds of errors off against each another in deciding what to do. Their philosophy is close in spirit to that of the Bayesian program of weighing and incrementally improving models by selecting for the better fitting ones. Moreover, since a modified Bayesian approach is compatible with hypothesis testing and rejection, scientists could create a parameter that would serve the same purpose as the p-value does now. The affinities between the Neyman-Pearson methods and Bayesian approaches suggest several potential ways of improving our scientific decision-making.
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