Flashcards in Uncertainty Deck (8)

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1

## What is the difference between a random variable and a probability distribution? Provide an example

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A random variable is only 1 variable of a certain world,

A probability distribution is a table of outcomes and their probabilities of the given RVs.

Example

RV: Weather ( sunny cloudy rainy)

PD using the above RV is the table of probability the world is in any of the states in weather (sunny, cloudy, rainy)

2

## In probability theory, what is a possible world? Provide an example.

### A possible world can be any tuple in a JPD. Using the same example as question 1, all the provided examples in the probability distribution would be considered possible worlds

3

## Explain, both in words and using math, the difference between P(H | E) and P (H.E)

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P (h|e) is the conditional probability of h given evidence e, while P (H E) is the joint probability of h and e

P(H|E) = P(H ^ E) / P(E)

P(H,E) = P(H ^ E)

4

## Consider the above joint probability distribution over the three random variables X, Y, Z. compute the marginal probability distributions for P(X), P(Y) and P(Z).

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P(X) = [1, 0], P(Y) = [0,1], P(Z) = [1, 0]

Process: Marginalize each variable independently (e.g. we sum all columns where X is true, then all col where X is false; so we get that X is 1 when it’s true, 0 otherwise).

5

## Name an operation that could be performed on a factor f that could decrease f's dimensionality and explain how much smaller the new factor would be

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Summing out a variable, the new factor would be half the size, because you are removing a variable from the distribution, which means you are decreasing it by a multiple of 2.

(size of the table is 2^k, where k is the number of the variable assuming each variable with a domain of 2.)

Assigning a variable is another option, similar explanation

6

## Name an operation that could be performed on a factor f that could increase f's dimensionality and explain how much bigger the new factor could be.

### Multiplying factors (doing a join on common variables), The increased size would depend on how many common and uncommon variables there between the two factors. Increases by a factor of 2 for every uncommon variable from one factor to the other.

7

## 7. (Uncertainty) [8 points] Consider a node N in a Bnet. N's domain has size d. N has k parents, each with also domain size d. What is the size of the CPT for P(N|parentsOfN)? Is it true that the sum of all the entries in this CPT is equal to d? why or why not?

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The size is d^k * (d-1)??

No because it is possible that there are more or less entries.

8