Statistics/Probability/Bayesian

Bayesian analysis is the branch of statistics based on the idea that we have some knowledge in advance about the probabilities that we are interested in, so called a priori probabilities. This might be your degree of belief in a particular event, the results from previous studies, or a general agreed-upon starting value for a probability. The terminology "Bayesian" comes from the Bayesian rule or law, a law about conditional probabilities. The opposite of "Bayesian" is sometimes referred to as "Frequentist Statistics."

Example

Consider a box with 3 coins, with probabilities of showing heads respectively 1/4, 1/2 and 3/4. We choose arbitrarily one of the coins. Hence we take 1/3 as the a priori probability $P(C_1)$ of having chosen coin number 1. After 5 throws, in which X=4 times heads came up, it seems less likely that the coin is coin number 1. We calculate the a posteriori probability that the coin is coin number 1, as:

$\begin{array}{cc}P(C_1|X=4)& =\frac{P(X=4|C_1)P(C_1)}{P(X=4)}\\ & =\frac{P(X=4|C_1)P(C_1)}{P(X=4|C_1)P(C_1)+P(X=4|C_2)P(C_2)+P(X=4|C_3)P(C_3)}\\ & =\frac{(\genfrac{}{}{0ex}{}{5}{4})(\frac{1}{4}{)}^4\frac{3}{4}\frac{1}{3}}{(\genfrac{}{}{0ex}{}{5}{4})(\frac{1}{4}{)}^4\frac{3}{4}\frac{1}{3}+(\genfrac{}{}{0ex}{}{5}{4})(\frac{1}{2}{)}^4\frac{1}{2}\frac{1}{3}+(\genfrac{}{}{0ex}{}{5}{4})(\frac{3}{4}{)}^4\frac{1}{4}\frac{1}{3}}\end{array}$

In words:

The probability that the Coin is the first Coin, given that we know heads came up 4 times... Is equal to the probability that heads came up 4 times given we know it's the first coin, times the probability that the coin is the first coin. All divided by the probability that heads comes up 4 times (ignoring which of the three Coins is chosen). The binomial coefficients cancel out as well as all denominators when expanding 1/2 to 2/4. This results in

$\frac{3}{3+32+81}=\frac{3}{116}$

In the same way we find:

$P(C_2|X=4)=\frac{32}{3+32+81}=\frac{32}{116}$

and

$P(C_3|X=4)=\frac{81}{3+32+81}=\frac{81}{116}$.

This shows us that after examining the outcome of the five throws, it is most likely we did choose coin number 3.

Actually for a given result the denominator does not matter, only the relative Probabilities $p(C_i)=P(C_i|X=4)/P(X=4)$ When the result is 3 times heads the Probabilities change in favor of Coin 2 and further as the following table shows:

Heads	$p(C_1)$	$p(C_2)$	$p(C_3)$
5	1	32	243
4	3	32	81
3	9	32	27
2	27	32	9
1	81	32	3
0	243	32	1