Three-Toed Sloth

Books to Read While the Algae Grow in Your Fur, August 2023

Attention conservation notice: I have no taste. (For a change, this month I am qualified, sort of, to comment on my non-fiction reading.) Also, most of my reading this month was done at odd hours and/or while bottle-feeding a baby, so I'm less reliable and more cranky than usual.

(Left almost-finished in 2023, because I got interrupted, and posted in 2026, because I wanted to procrastinate about half-a-dozen research projects.)

Tore Schweder and Nils Lid Hjort, Confidence, Likelihood, Probability: Statistical Inference with Confidence Distributions [doi:10.1017/CBO9781139046671]

We observe a random variable \( X \), which came from a probability distribution \( p(x) \). We hope that we have a well-specified probability model, a family of distributions \( m(x;\theta) \), parameterized by some variable \( \theta \) (possibly very high or even infinite-dimensional), and we hope that \( p(x) = m(x;\theta_0) \) for some \( \theta_0 \). (That last is the "well-specified" part.) A confidence set for \( \theta \) is a random, data-dependent set \( C_{\alpha}(X) \) which contains or "covers" the true parameter with the specified probability, the "confidence level" \( \alpha \): \[ \Pr{(\theta_0 \in C_{\alpha}(X))} = \alpha \] Actually, I will (following convention) drop the explicit dependence on \( X \) at this point, but keep the capital \( C \) as a reminder that this is a random set (because it's a function of the data, which are random).

It's natural to ask "wait, what distribution is used to calculate this?"; the answer is that the expression above is a little bit of a lie simplification, the real requirement is \[ \min_{\theta}{m(\theta_0 \in C_{\alpha})} = \alpha \] This means one of two things must be true: either

1. The true parameter \( \theta_0 \) is in the confidence set; or
2. We got unlucky with data, and something which is relatively unlikely under any distribution (probability at most \( 1-\alpha \) ) happened.

If you have really not seen the idea before, I elaborate elsewhere.

Leaving aside some pathological cases, any sensible way of building confidence sets will nest lower-confidence sets inside higher-confidence ones: if \( \beta \leq \alpha \), then \( C_{\beta} \subseteq C_{\alpha} \). (The 90% confidence set is a subset of the 95% confidence set, which is a subset of the 99% confidence set, etc.) This is, of course, exactly what would happen if one had a probability distribution on the parameter space. The idea of a "confidence distribution" is to take this literally, and build a probability distribution for \( \theta \) from nested confidence sets. Let's call this distribution \( D \), again with the capital letter \( D \) as a reminder that this is a random object. If \( D(a) = \alpha \) for some set \( a \), then \( a \) is a level-\( \alpha \) confidence set. This is a frequentist alternative to Bayesian posterior distributions for parameters.

The idea of confidence distributions has a rich (i.e., complicated and disputed) history going back at least to Fisher, who introduced what he called "fiducial" inference, when one can identify a "pivot", i.e., a function of \( X \) and \( \theta \) whose distribution is the same for all \( \theta \); some algebraic trickery then lets one convert that into an \( X \)-dependent distribution for \( \theta \). (*) The idea of frequentist coverage distributions has been revived since, roughly, 2000, as people have developed more practical ways to construct them.

I must admit that I did not pay much attention to confidence distributions, even after hearing a couple of seminars on them, because those constructions still eluded me. But after the kth such talk, when I finally decided I needed to understand more, I saw there was a book by someone whose work, and writing, I admire...

The authors do their best to be clear, but this is very much a book where reading means grinding through the proofs and exercises. Having done so, I now have a much better grasp of what's going on. (For example: there are usually different ways of constructing confidence sets for the same parameter; if one confidence set is smaller than another, at the same confidence level, we generally prefer the smaller set, it's more "efficient". This book finally explains, in a way I can understand, what corresponds to such efficiency for confidence distributions. [This is part of why this book spends so much time on likelihood-based methods.]) I now get confidence distributions in a way I simply didn't before, to the point where I think I can see where to make a (minor) contribution to the subject of likelihood-free, simulation-based confidence distributions. (We'll see if referees agree.) At the very least, I see the point now!

I strongly recommended this book to anyone with a solid grasp of theoretical statistics (at the level of van der Vaart or Schervish), and an interest in confidence sets and/or statistical foundations. §

*: A standard example is to consider a single sample \( X \) from the family of Gaussian distributions with variance 1 but unknown mean, i.e., \( X \sim \mathcal{N}(\theta, 1) \). Then \( X - \theta \sim \mathcal{N}(0,1) \), regardless of \( \theta \). One can test any particular value of \( \theta \) by seeing whether \( X - \theta \) falls into a high-probability interval for a standard Gaussian; the set of all \( \theta \) which pass is a confidence interval. Fisher's proposal was basically: let's make our confidence distribution for \( \theta \) be \( \mathcal{N}(X, 1) \). This is all simple enough for the Gaussian-location family, but mathematical complications arise for other situations, even when one can find a pivot. ^

Lindsay Buroker, Star Kingdom, vols. 1, 2, 3, 4, 5, 6, 7, 8

Buroker is a reliable author of very fluffy mind candy, either science fiction or fantasy. A lot of her output is re-working material from identifiable prior art, without hewing so closely to the canon as to constitute fan fiction. (Thus, at the risk of seeming to damn them with faint praise, the Fallen Empire books are much better Star Wars sequels than the sequel movies.) The present books are notable to me because what's being reworked here is actually Lois McMaster Bujold's Vorkosigan books: a short, fast-talking, "frenetically scheming" protagonist prone to catastrophic success and seizures; a budding interstellar empire with backwards views on biological engineering; a military genius with clones; a catgirl supersoldier; and other props and properties that I neglected to note as I read...

Bujold is a much more powerful writer than Buroker --- there are always moments in LMB's comedy where the reader sees the abysses of moral horror her characters are dancing above, and there's nothing like that here. But sometimes one wants to stay on the surface, while enjoying some of the same pleasures... §

Walter Jon Williams, Imperium Restored

The conclusion of the space-opera trilogy that opened with The Accidental War and continued with Fleet Elements. As such, it is incomprehensible without those first two books. Rather than summarize the plot, I will just say that Williams continues to combine a sharp political intelligence, story-telling that actually needs the scale of space opera, and wrenching emotional drama told in good prose. It's not his very best, but that's a high bar to clear. If you find my taste in science fiction congenial at all, I urge you to read the whole saga. §

Chelsea Cain et al., Spy Island

Comic book mind candy, affectionately parodying several genres at once: glamorous spies, mysterious islands, etc. §

Books to Read While the Algae Grow in Your Fur; Scientifiction and Fantastica; Enigmas of Chance