21 Which of the following statements is always true 1 P A B P B A 2 P A B P A B

# 21 which of the following statements is always true 1

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2.1 Which of the following statements is always true? 1. P ( A | B ) = P ( B | A ) 2. P ( A, B ) = P ( A | B ) P ( B ) 3. P ( A, B ) = P ( A ) P ( B ) 4. P ( A | B ) = P ( A ) 5. P ( A, B, C ) = P ( A ) P ( C ) 6. P ( A, B, C ) = P ( A ) P ( B ) P ( C ) 1
7. P ( A, B, C ) = P ( A ) P ( B | A ) P ( C | A, B ) 8. P ( A ) = b domain ( B ) P ( A, B = b ) 9. P ( A ) = b domain ( B ) P ( A | B = b ) P ( B = b ) 2.2 Now assume that A , B , and C are all independent of each other. Which of these statements is true? 3 Logarithms 3.1 Log-probs Let p be a probability, so it is bounded to [0 , 1] (between 0 and 1, inclusive). What is the range of possible values for log( p ) ? Please be specific about open versus closed intervals. 3.2 Prob ratios Let p and q both be probabilities. What is the range of possible values for p/q ? 3.3 Log prob ratios What is the range of possible values for log( p/q ) ? 4 Deriving Bayes Rule The definition of conditional probability can be written as P ( A, B ) = P ( A | B ) P ( B ) or alternatively as P ( A | B ) = P ( A, B ) /P ( B ) . Starting from this, derive Bayes Rule, in this form: P ( A | B ) = P ( B | A ) P ( A ) P ( B ) It should take only a few lines to prove/derive this. Note that you can apply the definition of conditional probability not just for P ( A, B ) but also for P ( B, A ) . Here we are using standard notation where comma indicates conjunction/intersection so order of variables doesn’t matter when defining a joint event. 5 Survey 5.1 What natural languages do you know? Which is your favorite? 5.2 What programming languages do you know? Which is your favorite? 2
5.3 What do you hope to get out of this course? 3

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