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Probability and Statistics: Foundations of CA

Probability and statistics are essential to CA, in which targets are represented and, in given images, are described by their statistical models. A framework of probability theory is employed for constructing the statistical models and for describing the targets in given images. In this subsection, some basics of the probability theory are described.

Sum Rule and Product Rule of Probability

First, discrete probability is described. In this case, the random variable X takes discrete values such as x1, x2,..., xn. If the frequency in the case that X takes Xi is ci, the probability that X takes xi is represented as

where N = J2”=i ci. When there is another random variable Y that represents another aspect of the event mentioned above, Y takes values of y1; y2,...,ym. The frequency in the case that Y takes yj is dj, and the frequency in the case that X takes xi and Y takes yj simultaneously is rij. The joint probability is described as


Then, the following equation is derived from Eqs. (2.76), (2.77), and (2.78):

This equation is called the sum rule of probability. Because the left side of Eq. (2.79) means marginalization in terms of the random variable Y, it is called a marginalprobability. Assuming that X is fixed on x;, the ratio of frequency of the case of Y = yj to the cases of all Y is described as p(Y = yjX = xi). This is called a conditional probability, because this is the probability of Y = yj under the condition of X = xi. This conditional probability is calculated as

By substituting Eqs. (2.76) and (2.80) into Eq. (2.77), the equation is transformed as

This equation is called the product rule of probability. The probabilistic distribution of the random variable X is denoted as p(X), and the probability in the case that X takes a specific value is denoted as p(x,). By using these notations, the sum rule and the product rule of probability are written thus: The sum rule of probability is represented as

and the product rule of probability is represented as

Because p(X, Y) is symmetrical with respect to the random variables X and Y, p(X, Y) = p(Y,X) or p(YX)p(X) = p(XY)p(Y). By transforming this equation, the relationship between two conditional probabilities is derived:

This relationship is referred to as Bayes’ theorem. By substituting Eq. (2.83) into Eq. (2.82) after swapping X with Y on the right side of Eq. (2.83), the denominator of the right side of Eq. (2.84) is written as

Assuming the necessity of finding the probabilities of the random variable Y in Eq. (2.84), p(Y) and p(Y|X) are referred to as a prior probability distribution (or simply, a “prior”) and a posterior probability. These probabilities are so named because the former is a known probability before the actual value of the random variable X is known, and the latter is known after the actual value of the random variable X is known. When the joint probability p(X, Y) is equal to a product of p(X) and p(Y), i.e., p(X, Y) = p(X)p(Y) holds, the random variables X and Y are independent. In this case Eq. (2.84) (the product rule of probability) is transformed thus:

and hence p(Y) = p(Y|X). This means that the probability of Y is unaffected by X, and vice versa. When the random variables have continuous value, Bayes’ theorem is described thus:

where x and в are random variables, p is a posterior probability density function, q is a prior probability density function of в, and f is a likelihood function. The denominator of the right-hand side of Eq. (2.87) is a marginal probability density function.

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