Intro to AI - Uncertainty

Probability is the likelihood of a certain event \$a\$ to happen. Every possible outcome of the event can be thought of as a world \$omega\$ with its probability \$P(omega)\$.
The probability of every possible event when summed together is \$1\$.
\$sum_(omega in Omega) P(omega) = 1\$
for the probability of a single event we always have
\$ 0 le P(omega) le 1\$
where \$0\$ means the event is impossible and \$1\$ the event is certain.

Unconditional probability or marginal probability is the degree of believe in a proposition in absence of any other evidence.
The probability is unaffected by previous or future events. It can be determined by adding up the outcomes of the event \$a\$ and dividing by the total number of possible outcomes.

\$P(a) = (text{number of times 'a' occurs}) / (text{total number of possible outcomes})\$

Conditional probability is the degree of believe in a proposition given some evidence has already been revealed.
For example the probability of \$a\$ given that \$b\$ is true:
\$P(a|b) = (P(a wedge b)) / (P (b)) \$
We are interested in the events where both \$a\$ and \$b\$ are true (the numerator), but only from the worlds where we know \$b\$ to be true (the denominator).

Joint probability is the probability of event a and b.
The Formula is derived from conditional probability by multiplying with \$P(a)\$ or \$P(b)\$.
\$P(a wedge b) = P(a|b) \ P (b) \$
\$P(a wedge b) = P(b|a) \ P (a) \$
\$P(a wedge b)\$ can also be written as \$P(a, b)\$

A random variable in probability theory is a variable with a domain of possible (discrete) values it can take on, each possible value might have a different probability.

Probability distribution is a function that gives the probability for each possible value of a random variable, in the case of discrete values this can be just a vector of probabilities for each possible value of the variable.

Independence is the knowledge that the outcome of one event doesn’t affect the outcome of another event. For two independent event the following is true:
\$P(a|b) = P(a) \$
used in the formula for joint probability,t his gives us for two independent events \$a\$ and \$b\$: \$P(a wedge b) = P(a) \ P (b) \$

Using the joint probability formulas \$P(a wedge b) = P(a|b) \ P (b) \$
\$P(a wedge b) = P(b|a) \ P (a) \$

gives us:
\$P(a|b) \ P (b) = P(b|a) \ P (a) \$
and:
\$P(a|b) = ( P(b|a) \ P (a) ) / (P (b))\$

A Bayesian Network is a directed graph, each node represents a random variable. Each edge from node \$X\$ to \$Y\$ represent a parent child relationship. The probability distribution of \$Y\$ depends on the value of \$X\$. Each node \$Y\$ has probability distribution \$P(Y | Parents(Y))\$.

\$P(not a) = 1 - P(a)\$

\$P(a vee b) = P(a) + P(b) - P(a wedge b)\$
Both events \$a\$ and \$b\$ includes one case of \$a wedge b\$, actually one includes \$(a wedge b)\$ the other includes \$(b wedge a)\$, we have to subtract the probability once.

for a simple boolean value b:
\$P(a) = P(a wedge b) + P(a wedge not b)\$
or more general:
\$P(X = x_i) = sum_(j) P(X = x_i wedge Y = y_j)\$

for a single boolean value b:
\$P(a) = P(a|b) \ P(b) + P(a|not b) \ P(not b) \$
or more general:
\$P(X = x_i) = sum_(j) P(X = x_i | Y = y_j) \ P( Y = y_j)\$

To sample means randomly picking a value for a variable according to its probability distribution.

The Markov assumption is an assumption that the current state depends on only a finite fixed number of previous states.

A Markov chain is a sequence of random variables where the distribution of each variable follows the Markov assumption. That is, each event in the chain occurs based on the probability of the event before it.

the transition model of a markov chain creates the probability distributions of the next event based on the possible values of the current event.

A hidden Markov model is a type of Markov model for a system with hidden states that generate some observed event. This means that sometimes, the AI has some measurement of the world but no access to the precise state of the world.