Probability and Inference

For the AI module in the Computer Science department you have to have a basic understanding of Probability and Inference. Below is an introduction to the probability details covered.

First off there are a few things we have to cover:

  • \(p(A)=0.5\) means that event \(A\) has a \(0.5\) or \(50\%\) chance of occurring
  • \(p(A,B)=0.5\) means that the events \(A\) and \(B\) have a \(0.5\) or \(50\%\) chance of both occurring at the same time
  • \(p(A,B C)=0.5\) means that the events \(A\) and \(B\) have a \(0.5\) or \(50\%\) chance of both occurring at the same time given that event \(C\) has occurred. This can be written as \(\frac{(p(A,B,C)}{p(C)}\)

These are pretty basic concepts, and we only really need a few identities to solve all the problems in the exam:

ONE: Bayes Theorm

Two: Partition Theorm

Three: Chain Rule

Four: Naive Bayes for some partion of the space: \( S=\cup^{m}_{i=1}A_i\). Note this will probably require conditional indipendance. ie for \(A\) and \(B\) to be CI given \(C\) we can write:

These 3 are used whenever a question comes up. There are lots of possible Questions but there will always be the requred combinations given. Splitting up what is asked for you buy the question in a certain way will provide an evaluatable line. Some examples are below:

Naive-Bayes-question

Here were given:

Where: \(A=\)android, \(I=\)iOS, \(W=\)windows, \(wh=\)white & \(b=\)british. We are asked to find:

Here, notice it says being white and a british sim are indipendant. thus

Now, we just need to find \(\alpha\) by computing this for all of the partion sections: \(A=\)android, \(I=\)iOS, \(W=\)windows:

or:

Note that this is itself a partition so they must add up to 1:

thus,

Now we have found alpha, its easy to calculate whichever value we need:

And were done for this question. Boom, 10 marks!