Bayes' Probability

10 Oct 2018

 

Bayes’ Theory

 

In probabilistic study, there are two main stream approaches:

1. Frequentist Approach( the classic probability )
   • Define probability as an event’s relative frequency in a large number of trials when performed infinite times
   • \[P(x) = \lim_{n_t -> \infty} \frac{n_x}{n_t}\]
   • \(n_t\) is total number of trials and \(n_x\) is number of that event \(x\) occurred and \(P(x)\) is the probability
   • However, application of frequentist approach is nearly impossible in the real word, because you cannot simply try everything infinite amount of times. It is difficult to apply real world problems…

   

1. Bayesian Approach
   • Define probability as a “degree of belief”. It is subjective, but not random
   • Experience and data are used to update the probability

   • Practical to apply in real world problems!!

   • Bayesian Theorem: \(P(H|E) = {P(H \bigcap E) \over P(E)} = \frac{P(E|H)*P(H)}{P(E)} = \frac{P(E|H)*P(H)}{P(H)*P(E|H) + P(H^c)*P(E|H^c)}\)

     • \(P(H)\) is called the prior probability, the probability that you know before the “update”

     • \(P(H \vert E)\) is called the posterior probability, the updated probability with new information

     • The Bayesian Theorem uses conditional probability to update prior probability to posterior probability

       

Conditional Probability

\(P(A \vert B) = {P(A \cap B) \over P(B)}\)

• Given that event B occurred, the chance that \(P(A)\) also occurred. (probability of A given B)
• if the two events A and B are completely independent, \(P(A \vert B) = P(A)\)
• This implies that information of B is irrelevant or useless when you want to know A

   

Example

Let’s say that a mother is in her 40s. She got a positive reaction from X-ray examination of breast cancer. What is her probability of really having a breast cancer?

• Probability that women in her 40s will have a breast cancer is 1%
• Probability that cancer patient of women in 40s will be diagnosed positive from X-ray examination is 90%
• Probability that healthy women in 40s will be diagnosed positive from X-ray examination is 5%

Solution

The probability that we want to know is \(P( c \vert p)\)

Given informations are:

• \(P(c)\) = 0.01
• \(P(p \vert c)\) = 0.9
• \(P (p \vert c^c)\) = 0.05

Using the Bayes theorem: \(P(c \vert p) = {P(c)*P(p \vert c) \over P(p)}\)

So all we need to know is the value of \(P(p)\) which is:

\(P(p) = P(p|c)*P(c) + P(p|c^c)*P(c^c) = 0.9*0.01 + 0.05*0.99 = 0.0585\)

The final calculation of what we want to know would be: 0.01 * 0.9 / 0.0585 = 0.15384…

Thus, the probability that a mother will have breast cancer, given that her X-ray examination was positive is about 15%