A type or type class is basically an empirical distribution for a sequence of n independent events. For instance, suppose we flip coin n=2 times. We have 3 type classes, all heads, all tails, one head/one tail. Size of the type class is the number of sequences in that class. Here, two of our classes have size 1 and "one head/one tail" has size 2.

Since the number of types grows polynomially with n, while number of observation sequences grows exponentially, some type classes end up getting an exponential number of sequences. Since constants of exponential growth are different for different types, probability mass ends up getting concentrated in a small number of types and several important properties follow from this concentration behavior.

In particular, Thomas/Cover use method of types to formalize and prove the following statements in Chapter 12 of "Information Theory and Statistics" (1991).

- Probability of hypothesis E under distribution P is determined by the probability of closest member of E to P (Sanov's theorem)
- To test whether data x was generated by p or q, it's sufficient to only look at the ratio of probabilities of x under p and under q (Neyman-Pearson lemma)
- When data is generated by p or q and you are trying to tell which one it is from data, probability of making a mistake for an optimal classifier is bounded by exp(-n max(KL(p,q),KL(q,p)). A tighter bound is to use KL(x,p)=KL(x,q) where x is the "halfway point" between distributions (Chernoff's theorem)

Rodriguez says it outperforms BIC and AIC by orders of magnitude

- Thomas, Cover "Information Theory and Statistics"
- Csiszar, Shields Notes on Information Theory and Statistics

## 1 comment:

Excellent machine learning blog,thanks for sharing...

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