I think I'd be wasting my time continuing with this. I'm not a statistician or a user of statistical methods, so I'm not really in a position to judge the book -- it's just not for me. A couple of thoughts anyway:

- The style is strange; it gave me the feeling of being dropped into the author's train of thought, or perhaps thrown headlong into its path. I don't think that was due to the complexity of the ideas, or even to the semantic density of the writing, but rather to some combination of stream-of-consciousness composition and assumed shared background between writer and reader. In a popular book I would simply call this bad writing/editing, but since Mayo is a specialist writing for specialists, I'm not in a position to lay down that kind of judgment. It didn't feel necessary, though, at least in the early sections that I read.

- (What follows may simply reveal my own ignorance and misunderstanding)

p. 53: "Analogous situations to the optional stopping example occur even without optional stopping, as with selecting a data-dependent, maximally likely, alternative. Here's an example from Cox and Hinkley [citation] attributed to Allan Birnbaum [citation]. A single observation is made on X, which can take values 1, 2, ..., 100. [...] If X is observed to be r, [...] then the most likely hypothesis is θ = r. In fact, Pr(X = r; θ = r) = 1. By contrast, Pr(X = r; θ = 0) = 0.01. Whatever value r that is observed, hypothesis θ = r is 100 times as likely as is θ = 0. [...] So "even if in fact θ = 0, we are certain to find evidence apparently pointing strongly against θ = 0, if we allow comparisons of likelihoods chosen in the light of the data" [citation]. This does not happen if the test is restricted to two preselected values. [...] Allan Birnbaum gets the prize for inventing chestnuts that deeply challenge both those who do, and those who do not, hold the Likelihood Principle."

(As far as I can remember, or find using the index, the Likelihood Principle has not yet been formally defined, but on page 30 it is said to be "related" to the "Law of Likelihood", i.e. "Data x are better evidence for hypothesis H1 than for H0 if x is more probable under H1 than under H0: Pr(x; H1) > Pr(x; H0), that is, the likelihood ratio (LR) of H1 over H0 exceeds 1.")

Earlier (p. 38) a similar case is used to illustrate "our most serious problem: The Law of Likelihood permits finding evidence in favor of a hypothesis deliberately arrived at using the data". A deck of cards is shuffled, and the top card turned over: it is the Ace of Diamonds. The LL tells us that "the hypothesis that the deck consists of 52 aces of diamonds [...] is [52 times] better supported than the hypothesis that the deck is normal". This is supposed to present a problem for 'Likelihoodists', one which they can only evade by insisting on the distinction between evidence and belief.

I don't understand what is supposed to be strange or threatening about the 'trick deck' case. The hypothesis that the deck consists of 52 aces of diamonds *has* just received some support, whether we formulated it in advance or not -- but this doesn't imply that we need to reduce our confidence that the deck is normal. All that has happened is the other 51 'the deck consists of 52 copies of one card' hypotheses have just been ruled out. If we gave (or would have given) each of them a 0.01% chance before looking at the data, we should now consider the AD hypothesis 0.52% likely to be true.

Similarly, I don't understand what is supposed to be puzzling about the Allan Birnbaum case, which seems to be presented almost as a paradox. If we had no other reason to favour the hypothesis that θ = r, then the observed value provides no evidence for or against θ = 0, only evidence in favour of θ = r. The ratio Pr(θ = r)/Pr(θ = 0) increases, but only because θ = [anything other than 0 or r] has been completely ruled out, and its probability mass transferred to θ = r.

Surely nobody really advocates the Bayesianism-without-prior-probabilities approach that these examples seem to target? Mayo must have a reason for presenting them, but I would have loved a clear explanation of what it is.