Probability

Observed variables (evidence): Agent knows certain things about the state of the world (e.g., sensor readings or symptoms)

Unobserved variables: Agent needs to reason about other aspects (e.g. where an object is or what disease is present)

 

Random Variables: Like variables in a CSP, random variables have domains

R in {true, false}   (often write as {+r, -r})

L in possible locations, maybe {(0,0), (0,1), …}

Discrete-boolean/binary(but not boolean)

Continuous

 

Probability Distributions

You almost should not put zeros in your probability unless it never ever happens.

 

Probability has distributions, P(W) is the whole table while P(W=rain) is a probability.

*In the property of probabilties, 0 is allowed!

 

Joint Distributions

Size of distribution if n variables with domain sizes d? d^n

Writing all the joint distributions is very expensive.

 

 

Events

Joint Distribution is just a distribution.

Events are the actual things that happen and are calculated.

 

 

Marginal Distributions

Marginal distributions are sub-tables which eliminate variables.

Marginalization (summing out): Combine collapsed rows by adding

Joint Distribution -> Marginal Distribution

**Be aware of the calculation

 

Conditional Probabilities

Of b, which part is also a? The middle part

Of b, which is the proportion that is P(a,b)?

 

Conditional Distributions

Given a joint distribution, the conditional distribution can be calculated.

 

Normalization Trick

Probablistic Inference

compute a desired probability from other known probabilities

1) Inference by Enumeration

2) Product Rule

3) Chain Rule

4) Bayes' Rule

 

Inference by Enumeration

all joint distributions(which is a lot)

Evidence Variables: choose some of the X variables

Query variable: the variable you want to know

ex) Given the information, do I need my unbrella?

Thus, P(Q|c1, c2, ..., ck)

Hidden variables: variables you don't know but also don't need

ex) Ghostbusters: sensor value of the variable you haven't sensed

You don't want it as part of your answer but you don't want it either

 

Evidence: Things you know

Query: Things you don't know

Hidden: Things you don't know but also don't want to know

Step 1: eliminate the rows that are not consistant with the evidence

(as done before)

Step 2: sum out! H

Step 3: normalize

 

* What is hard: Having the actual joint distribution tablem

 

Watch the change in probability

If it is winter, the probability of it being sunny decreases.

If it is winter and hot, the probability of it being sunny decreases.

 

In P(W), the hidden variables are T and S so they need to be removed.

 

*** Work on calcuations on conditional problems.

 

Obvious problems:

Worst-case time complexity O(d^n)

Space complexity O(d^n) to store the joint distribution

-> Thus you need and alternative way

 

The Product Rule

Sometimes have conditional distributions but want the joint

 

marginal*conditional = joint

This is just the definition of conditional probability.

 

The Chain Rule

 

Bayes Rule

How to use Baye's Rule: Change between sound and text(since you know one but don't know the other)

 

**Solve the last quiz.

 

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