Artificial Intelligence

Probability

Uncertainty: Agents may need to handle uncertainty, whether due to partial observability, nondeterminism, or a combination of the two. An agent may never know for certain what state it's in or where it will end up after a sequence of actions.

The frequentist position is that the numbers can come only from experiments: if we test 100 people and find that 10 of them have a cavity, then we can say that the probability of a cavity is approximately 0.1.

The objectivist view is that probabilities are real aspects of the universe rather than being just descriptions of an observer's degree of belief.

The subjectivist view describes probabilities as a way of characterizing an agent's beliefs, rather than as having any external physical significance. The subjective Bayesian view allows any self-consistent ascription of prior probabilities to propositions, but then insists on proper Bayesian updating as evidence arrives.

\begin{aligned} P(A) = p \Rightarrow P(-A) = 1-p \end{aligned}

Independence:

\begin{aligned} X \perp Y : P(X)P(Y) = P(X,Y) \end{aligned}

Dependence:

Conditional Probability

\begin{aligned} P( Y | X ) \end{aligned}

Total Probability

\begin{aligned} P(Y) = \sum_{i} P(Y | X=i) * P(X=i) \end{aligned}

Example

\begin{aligned} P(D_1 = sunny) =&\; 0.9 \\ P(D_2 = sunny | D_1 = sunny) =&\; 0.8 \\ P(D_2 = rainy | D_1 = sunny) =&\; 0.2 \\ P(D_2 = sunny | D_1 = rainy) =&\; 0.6 \\ P(D_2 = rainy | D_1 = rainy) =&\; 0.4 \end{aligned}