"The Subjective Statistician"

From Inference Review:

Paul Weirich is a Curators’ Professor in the Philosophy Department of the University of Missouri, Columbia. Recipient of grants from the NSF and the ACLS, he is an associate editor for the British Journal for the Philosophy of Science. Among his books are Decision Space (2001), Realistic Decision Theory (2004), Collective Rationality (2010), and Models of Decision-Making (2015).

Consider a coin. Assume that it is fair. Call this hypothesis H. Toss the coin. Use H to calculate the ensuing probabilities. Designate the result a significance test of H.
Assign due credit to Ronald Fisher.

Bayesian statistics is quite different. The axis of inferences goes the other way. It is the probability of H that is up for grabs.
Objective Bayesians ground probability assignments in physical symmetries:
Both sides are the same.

Empirical Bayesians use relative frequencies:
Fifty-fifty is fifty-fifty.

Subjective Bayesians appeal to personal judgments:
I am so into mauve.

Each method is unavoidably subjective, but, like unhappy families, each method is subjective in its own way.

Suppose that E expresses evidence that is new, or newly known. The conditional probability of H given E is P(H|E). The probability of H given E is P´(H). Whereupon P´(H) = P(H|E).

The old probability of H has become the new probability of H.
Thomas Bayes’s theorem offers a means of calculating P(H|E)

P(H|E) = P(H)P(E|H)/P(E).

Notice that

P(H|E)= P(H)P(E|H)/[P(H)P(E|H) + P(~H)P(E|~H)].

Consider the hypothesis H that a coin is fair. Remember that the coin has been tossed twenty times. Remember that on four occasions, it came up heads. Call that heads-up E.
Suppose that P(H) is 0.5.

P(E) is thus [(0.5 × 0.0046) + (0.5 × 0.0911)]

Now look at this:

P(H) = [0.5 × 0.0046]/[(0.5 × 0.0046) + (0.5 × 0.0911)] = 0.0481.

Up the coin has gone, but down has come the probability that the coin is fair.
In Fisher’s method of significance testing there is no appeal to the prior probabilities of H and E. There is just the probability of E given H.

Significance testing is meant to lead to a decision. It has an unavoidably active aspect. Either H should be accepted or not, but, in either case, something must be done.
The imperative of action is to reject H, if, and only if, the test results are improbable given the hypothesis.

In coin tossing, toss-ups are covered by a probability distribution curve. The coin has been tossed twenty times. At one end of the curve, there are four or fewer heads. At the other, four or fewer tails. Such are the critical regions of improbable results. The probability of a test result matching the real result is 0.012. This is less than a significance level of 0.05.

Give it up. The coin is not fair.
Well, neither is life.

But notice this: Fisher’s test shows only that either H is false or that the test results are improbable....MORE