Common formulae of probability theory

Some probability formulas are often used for a period of time, but often forget, here is a note

1. Multiplication rule

\ (P\left (x, y \right) =p\left (x|y \right) p\left (y \right) =p\left (y|x \right) p\left (x \right) \)

is actually an equivalent form of the conditional probability formula

2. Independence

If \ (x\) and \ (y\) are independent of each other, then there are:

\ (P\left (x, y \right) = P\left (x\right) p\left (y\right) \)

3. Bayesian rules (Bayes ' rule)

Bayesian rule has become a Bayesian formula, which has been widely used in many fields, and its formula is as follows:

\ (P\left (y|x \right) =\frac{p\left (x|y \right) p\left (y \right)}{p\left (x \right)}\)

The denominator is a normalized constant, which ensures that the sum of all possible values of the left posterior probability is 1. Therefore, we can usually write:

\ (P\left (y|x \right) =\eta p\left (x|y \right) p\left (x \right) \)

Given the background knowledge \ (e\) Given, Bayes becomes:

\ (P\left (y|x,e \right) =\frac{p\left (x|y,e \right) p\left (y|e \right)}{p\left (x|e \right)}\)

4, The Edge of

The edge probability formula is as follows:

\ (P\left (x \right) = \int_{y}^{} p\left (x, y \right) dy\)

In discrete cases, the integral becomes summation:

\ (P\left (x \right) = \sum_{y}^{} p\left (x, y \right) \)

5. The law of full probability

The full probability is a variant of the edge probability, which can be deduced from the multiplication rule, namely:

\ (P\left (x \right) = \int_{y}^{} p\left (X|y \right) p\left (y \right) dy\)

And, for the discrete case, the corresponding probability of the sum, namely:

\ (P\left (x \right) = \sum_{y}^{} p\left (X|y \right) p\left (y \right) dy\)

6. Markov hypothesis

Markov hypothesis refers to the variable \ (x_{t}\), which is related only to its direct state of the previous moment (x_{t-1}\), and \ (x_{t^{'}-1}\), where \ (t^{'}<t-1\), then there is

\ (P\left (x_{t}|x_{1:t-1} \right) = P\left (x_{t}|x_{t-1} \right) \)

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Resources

[1]. Cyrill Stachniss, Chen Bai fan, Liu Lijing (translated). Robot map creation and Environment exploration, 2013.

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Common formulae of probability theory