Review of Probability Theory - Sorry

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The probability of the event X=x is denoted as P X (x) or P X (X=x). P X (x) is often called the probability mass (or density function) or the probability distribution for X. It follows that and . Example: Outcome Random variable X P X (X=x) head 0 P X (0) = 1/2 tail 1 P X (1) = 1/2 Expectation 1. The expectation (alias mean value, average) of a random variable X is defined as Example: 2. The expectation of a function F=F(X) is defined as Example: if we get
4. Bayes‘ Decision theory Bayes‘ decision theory has been developed in the book [Duda and Hart, 1973], from which we summarize the following fundamentals. Bayes‘ decision theory is a fundamental statistical approach to the problem of pattern classification. This approach is based on the assumption that the decision problem is posed in probabilistic terms, and that all of the relevant probability values are known. The aim of the decision is to assign an object to a class. This classification is only carried out by means of measurements taken over the objects. Example Let us reconsider the problem, posed in the previous pages, of designing a classifier to separate two kinds of marble pieces, Carrara marble and Thassos marble. Suppose that an observer, watching marble emerge from the mill, finds it so hard to predict what type will emerge next, that the sequence of types of marble appears to be random. Using decision-theoretic terminology, we say that as each piece of marble emerges, nature is in one or the other of the two possible states: either the marble is Carrara marble, or Thassos marble. We let • = 1 for Carrara, • = 2 for Thassos. denote the state of nature, with: Because the state of nature is so unpredictable, we consider to be a random variable. If the mill produced as much Carrara marble than Thassos marble, we would say that the next piece of marble is equally likely to be Carrara marble or Thassos marble. More generally, we assume that there is some a priori probability P( 1 ) that the next piece is Carrara marble, and some a priori probability P( 2 ) that the next piece is Thassos marble. These a priori probabilities reflect our prior knowledge of how likely we are to see Carrara marble or Thassos marble before the marble actually appears. P( 1 ) and P( 2 ) are non negative and sum to one. Now, if we have to make a decision about the type of marble that will appear next, without being allowed to see it, the only information we are allowed to use is the value of the a priori probabilities. It seems reasonable to use the following decision rule: In this situation, we always make the same decision, even though we know that both types of marble will appear. How well it works depends upon the values of the a priori probabilities:
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Page 5 and 6: where (4) Bayes‘rule shows how th
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Review of Probability Theory - Sorry

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