  When testing a statistical hypothesis, is it legitimate to deliberate on the
basis of initial data about whether and how to collect further data?
Game-theoretic probability's fundamental principle for testing by betting says
yes, provided that you are testing by betting and do not risk more capital than
initially committed. Standard statistical theory uses Cournot's principle,
which does not allow such optional continuation. Cournot's principle can be
extended to allow optional continuation when testing is carried out by
multiplying likelihood ratios, but the extension lacks the simplicity and
generality of testing by betting.
  Game-theoretic probability can also help us with descriptive data analysis.
To obtain a purely and honestly descriptive analysis using competing
probability distributions, we have them bet against each other using the Kelly
principle. The place of confidence intervals is then taken by a sets of
distributions that do relatively well in the competition. In the simplest
implementation, these sets coincide with R. A. Fisher's likelihood intervals.
