# AIMA: Quantifying Uncertainty

> Source: https://ollybritton.com/notes/textbooks/ai-a-modern-approach/uncertain-knowledge-and-reasoning/quantifying-uncertainty/ · Updated: 2021-04-04 · Tags: aima, notes

> In which we see how to tame uncertainty with numeric degrees of belief.

## Flashcards
##### Why do we need probability theory??
Because it provides a reasonable framework for dealing with uncertainty.

##### What is the decision theory "equation"??
$$
\text{decision theory} = \text{probability theory} + \text{utility theory}
$$

##### What is the principle of maximum expected utility (MEU)??
> An agent is rational if and only if it chooses the action that yields the highest expected utility, averaged over all possible outcomes of the action.

##### If logical assertions say whether a possible world exists or not, what do probabilistic assertions say??
How likely a given world is.

##### What is a prior or unconditional probability??
Degrees of belief in an event without any other information.

##### What is a posterior or conditional probability??
Degrees of belief in an event with additional information given.

##### Is $$P(\text{Total} = 11)$$ a prior or posterior probability??
Prior.

##### Is $$P(\text{Total} = 11 | \text{Die}_1 = 6)$$ a prior or posterior probability??
Posterior.

##### How do you pronounce $$P(A | B)$$??
"The probability of $A$ _given_ $B$"

##### What is the product rule for $$P(a \land b)$$??
$$
P(a | b)P(b)
$$

##### What is the difference between $$\pmb{P}(\pmb{x})$$ and $$P(x)$$??
The former is a probability distribution showing the different probabilities for different values of a vector $\pmb{x}$ and the latter is the probability of $x$ occuring.

##### What is a joint probability distribution??
A way of describing the probability for different combinations of variables.

##### What is the full joint probability distribution??
A way of describing the probability for every different combinations of random variables under consideration.

##### How can you reduce the size of a full joint probability distribution??
Use absolute and conditional independence to factor it down.

##### What is Bayes' rule??
$$
P(A | B) = \frac{P(B | A)P(A)}{P(B)}
$$

##### What is Bayes' rule, for determing the probability of a $\text{Cause}$ given its potential $\text{Effect}$??
$$
P(\text{Cause} | \text{Effect}) = \frac{P(\text{Effect}) | P(\text{Cause})P(\text{Cause})}{\text{Effect}}
$$

##### Why is Bayes' rule sometimes written as $$\pmb{P}(A | B) = \alpha \pmb{P}(B | A) \pmb{P}(B)$$??
Because $\alpha$ is a normalisation factor that makes the values of the probability distribution sum to one.

##### What is conditional independence of $A$ and $B$ given $C$??
Where $A$ and $B$ are only independent after observing $C$ as it is the direct cause of both of them.

##### How could you state the conditional independence of $A$ and $B$ given $C$ in probability notation??
$$
P(A \land B | C) = P(A | C)P(B | C)
$$

##### What is as naive Bayes model??
A technique for estimating probabilities in a situation where you have a single cause with conditional independence opf all effect variables.

##### What is an example cause and effect in a naive Bayes model??
* Cause: The category of a piece of text
* Effect: Certain keywords appearing in the text

##### What is the formula for $P(\text{Cause} | \pmb{e})$ in a naive Bayes model??
$$
\alpha P(\text{Cause}) \prod_{i} P(e_i | \text{Cause})
$$

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