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

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

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 $p(A_1|B) =\alpha p(B|A_1) p(A_1) \\\\ p(A_2|B)=\alpha p(B|A_2) p(A_2) \\\\ \ldots \\\\ p(A_m|B)=\alpha p(B|A_m) p(A_m)$ for some 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: $p(A,B|C)=p(A|C)p(B|C) \text{ also: } p(A|B,C) = p(A|C)$

These 3 are used whenever a question comes up. There are lots of possible Questions but there will always be the 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:

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

or:

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!