# Probability and Inference

maths

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 Theorem**

$$ \frac{p(A,B)}{p(B)}=p(A|B)=\frac{p(B|A)p(A)}{p(B)} $$

Two: **Partition Theorem**

$$ P(A) = \sum^{n}_{i=1} p(A|B_i)p(B_i)\qquad \text{ for some partion of the space: } S=\cup^{n}_{i=1}B_i $$

Three: **Chain Rule**

$$ p(A_1,A_2,\ldots,A_k) = \prod^{k}_{j=1}p(A_j|A_1,\ldots,A_{j-1}) $$

Four: **Naive Bayes**

$$ p(A_2|B)=\alpha p(B|A_2) p(A_2) \\ \ldots \\ p(A_m|B)=\alpha p(B|A_m) p(A_m) $$

For some partition of the space: $S=\cup^{m}_{i=1}A_i$. Note this will probably require conditional independence. i.e. for $A$ and $B$ to be CI given $C$ we can write: $p(A,B|C)=p(A|C)p(B|C) also: p(A|B,C)=p(A|C)$

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

Here were given:

$$ \begin{bmatrix} p(A)=0.75 & p(I)=0.2 & p(W)=0.05 \ p(wh|A)=0.15 & p(wh|I)=0.5 & p(wh|W)=0.1 \ p(b|A)=0.15 & p(b|I)=0.1 & p(b|W)=0.2 \end{bmatrix} $$

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 independent. thus

$$ \alpha p(wh,b|I) p(I) = \alpha p(wh|I) p(b|I) p(I) $$

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

$$ p(A|wh,b)=\alpha p(wh,b|A) p(A) = \alpha p(wh|A) p(b|A) p(A) \ p(W|wh,b)=\alpha p(wh,b|W) p(W) = \alpha p(wh|W) p(b|W) p(W) $$

or:

$$ p(A|wh,b) = \alpha 0.15 \times 0.15 \times 0.75 \ p(W|wh,b) = \alpha 0.1 \times 0.2 \times 0.05\ $$

Note that this is itself a partition so they must add up to 1: $$ \alpha (0.5 \times 0.1 \times 0.2) + (0.15 \times 0.15 \times 0.75) + (0.1 \times 0.2 \times 0.05) = 1 $$

thus, $$ \alpha = \frac{1}{(0.5 \times 0.1 \times 0.2) + (0.15 \times 0.15 \times 0.75) + (0.1 \times 0.2 \times 0.05)} =\frac{80}{223} $$

Now we have found alpha, its easy to calculate whichever value we need: $$ p(I|wh,b)= \frac{80}{223} p(wh|I) p(b|I) p(I) = \frac{80}{223} (0.5 \times 0.1 \times 0.2) \approx 0.3587 $$

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