Bayesian Methods for Nonlinear Classification and Regression

3 Probable and Improbable data (the Bayesian equivalent of a conŽ dence interval). 4 Description of Distributions: real x (one-dimensional and multidimensional Gaussian, chi-squared, exponential, t). 5 Description of Distributions: natural x (binomial, multinomial, Poisson). A typo in (5.5) is one of the few that I detected (good typesetting). 6 Form Invariance: real x. Delightful picture and discussion of snow akes illustrate rotational symmetry. 7 Examples of Invariant Measures. The uniform measure is invariant under translation. The exponential measure is invariant under dilations. The Gaussian is invariant under both translation and dilation. 8 Linear Representation of Form Invariance. Linear spaces are introduced, and the notion of linear spaces is used in Chapter 11 for a generalized form of form invariance. 9 Beyond Form Invariance: the Geometric Prior. Jeffreys’ prior is introduced as a way to avoid having to analyze the symmetry group and the “multiplication function” (the group operation). This is a well-known result, and I assume that the geometric interpretation is standard, but I appreciated inclusion of this topic. 10 Inferring the Mean or Standard Deviation. Interesting examples from nuclear physics are provided. Often, these parameters are fundamental “constants” so are intrinsically far more “interesting” than the mean of some temporary “population” that is often the subject of statistical analysis. Integration over “uninteresting” parameters is included. I prefer the usual term “nuisance” parameter in this context. 11 Form Invariance II: natural x. A generalized notion of form invariance. 12 Independence of Parameters. The notion is to have individual parameters such that inference about one of them is not impacted by later, more precise estimation of the other. This is possible under a strong form of independence, if the model p factors into two models allowing one to infer ́1 from one set of events and ́2 from another set. This is a new topic for me, and the “hint of quantum mechanics” via commuting operators reminded me that I never really understood my undergraduate course in quantum mechanics. 13 The Art of Fitting I: real x. This is a refreshingly different treatment of the common topic of Ž tting a family of functions to data. 14 Judging a Fit I: real x. The Bayesian notion of whether a parameter estimate is consistent with the data is considered. It is possible that no member of the function family Ž ts the data well. The chi-squared criterion is introduced, in the Bayesian framework. 15 The Art of Fitting II: natural x. Count rate data with “coherent alternatives” is the key example. 16 Judging a Fit II: real x. Model selection (for example: to choose a Poison or a multinomial model for the data) of a speciŽ c type is introduced. 17 Summary. Again, I caution prospective readers that this is not a text from which to learn Bayesian methods. The analogy with quantum mechanics is discussed again, so I hope to Ž nd time to reread Chapter 12.