Saturday, July 26, 2014

Wald: Sequential Analysis (1947)

I've always thought that Shannon's insights in the 1940s papers seemed like pure magic, but this book suggests that at parts of his way of thinking were already in the air at the time: From his own frequentist perspective, Wald comes oddly close to defining a version of information theory.

The central question of the book is:
How can we devise decision procedures that map observations into {Accept, Reject, Continue} in such a way that (1) the probability of wrongly choosing Accept or Reject is low, and (2) the expected number of Continue decisions is low?
Throughout the book, the answer that he proposes is to use likelihood ratio tests. This puts him strangely close to the Bayesian tradition, including for instance Chapter 5 of Jeffreys' Theory of Probability.

 A sequential binomial test ending in rejection (p. 94)

In particular, the coin flipping example that Jeffreys considers in his Chapter 5.1 is very close in spirit to the sequential binomial test that Wald considers in Chapter 5 of Sequential Analysis. However, some differences are:
• Wald compares two given parameter values p0 and p1, while Jeffreys compares a model with a free parameter to one with a fixed value for that parameter.
• Jeffreys assigns prior probabilities to everything; but Wald only uses the likelihoods given the two parameters. From his perspective, the statistical test will thus have different characteristics depending on what the underlying situation is, and he performs no averaging over these values.
This last point also means that Wald is barred from actually inventing the notion of mutual information, although he comes very close. Since he cannot take a single average over the log-likelihood ratio, he cannot compute any single statistic, but always has to bet on two horses simultanously.