Together Chew PRML-1.2 probability theory

Together Chew PRML-1.2 probability theory

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A key concept in the field of pattern recognition are that of uncertainty.

It can be seen that probability theory in pattern recognition is obviously a very important chunk.

Reading other books in the probability of this is also very tangled.

We also use an example to understand some important concepts in probability theory.

Imagine we have both boxes, one red and one blue, and the red box we have 2 apples and 6 oranges, and in the blue box we has 3 apples and 1 orange.

The green one is an apple, the yellow one is an orange.

Let's assume that there is a 60% probability that the blue box is selected, with a 40% probability that the red box is selected.

Before we discuss it, we have some variables to agree on:

In this example, the identity of the box that would be chosen are a random variable, which we shall denote by B. This random variable can take one of the possible values, namely r (corresponding to the red box) or B (corresponding to T He blue box). Similarly, the identity of the fruit is also a random variable and would be denoted by F. It can take either of the values a (for Apple) or O (for orange).

We have: p (b = r) = 4/10, p (b = b) = 6/10

Of course, the probability must be the number between [0,1].

If we had to make a choice between red and blue, then obviously all P added up to 1, which was like eating a piece of pie.

Then we might ask a lot of questions about probabilities, and, of course, before that we have to learn two tools, the well-known sum rule and the product rule.

Let's look at one more example:

This example tells us how to multiply the principle of additive principle

Of course, the proof process is nothing more than deduction and collation

So we finally got the following:

Here p (x, y) is the joint probability, which can be expressed as "X and Y probability". Similarly, p (Y | x) is the conditional probability, which can be expressed as "the probability of y under the condition given X", and P (X) is the edge probability, which can be simply expressed as "x probability". These two simple rules form the basis of all probability deduction we use in the book.

Based on the product rules, and the symmetry p (x, y) = P (y, X), we immediately get the relationship between the following two conditional probabilities:

is to use the multiplication principle of the two joint probabilities.

Finally, if the union distribution of the two variables can be decomposed into the product of two edge distributions, that is, p (x, y) = P (x) p (y), then we say that X and Y are independent of each other (independent). According to the product rules, we can get P (Y | x) = P (y), so that the conditional distribution of Y under the condition of the given x is actually independent of the value of X. For example, in our fruit box example, if each box contains the same proportions of apples and oranges, then P (F | B) = P (F), so the probability of choosing apples is irrelevant to which box is selected.

Together Chew PRML-1.2 probability theory