Compute P X using joint distribution P X Y Z Can do this with marginalization P

Compute p x using joint distribution p x y z can do

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Compute P ( X ) using joint distribution P ( X , Y , Z ) Can do this with marginalization P ( X ) = X y Ω y X z Ω z P ( X , Y = y , Z = z ) Proof. X y Ω y P ( X , Y = y ) = X y Ω y P ( X ) P ( Y = y | X ) = P ( X ) X y Ω y P ( Y = y | X ) = P ( X ) COS 424/SML 302 Probability and Statistics Review February 6, 2019 22 / 69 Subscribe to view the full document.

Bayes rule From the chain rule and marginalization, we obtain Bayes rule . P ( Y | X ) = P ( X , Y ) P ( X ) = P ( X | Y ) P ( Y ) P ( X ) = P ( X | Y ) P ( Y ) y P ( X | Y = y ) P ( Y = y ) Example: Bayes rule Let Y be a disease and X be a symptom. From P ( X | Y ), P ( X ) and P ( Y ), we can compute P ( Y | X ). Bayes rule is useful because we can flip a conditional probability. More on the interpretation of Bayes rule in the next lecture COS 424/SML 302 Probability and Statistics Review February 6, 2019 23 / 69 Independence Random variables are independent if knowing the outcome X does not change the probability of Y : P ( Y | X ) = P ( Y ) This means that their joint distribution factorizes: X ⊥⊥ Y ⇐⇒ P ( X , Y ) = P ( X ) P ( Y ) . Why? The chain rule P ( X , Y ) = P ( X ) P ( Y | X ) = P ( X ) P ( Y ) COS 424/SML 302 Probability and Statistics Review February 6, 2019 24 / 69 Subscribe to view the full document.

Independent RVs: Examples Examples of independent random variables Flipping a coin once / flipping the same coin a second time You use an electric toothbrush / blue is your favorite color Shoe size / voted for Trump Examples of not independent random variables Registered as a Republican / voted for Trump The color of the sky / time of day Age / shoe size COS 424/SML 302 Probability and Statistics Review February 6, 2019 25 / 69 Are these independent? Rolls from two twenty-sided dice Roll three dice to compute two random variables ( D 1 + D 2 , D 2 + D 3 ) # enrolled students and the temperature outside today # attending students and the temperature outside today COS 424/SML 302 Probability and Statistics Review February 6, 2019 26 / 69 Subscribe to view the full document.

Two coins Suppose we have two coins, one biased and one fair, P ( C 1 = H ) = 0 . 5 P ( C 2 = H ) = 0 . 7 . We choose one of the coins at random Z ∈ { 1 , 2 } , flip C Z twice, and record the outcome ( X , Y ). Are X and Y independent? COS 424/SML 302 Probability and Statistics Review February 6, 2019 27 / 69 Two coins Suppose we have two coins, one biased and one fair, P ( C 1 = H ) = 0 . 5 P ( C 2 = H ) = 0 . 7 . We choose one of the coins at random Z ∈ { 1 , 2 } , flip C Z twice, and record the outcome ( X , Y ). If the answer is not straightforward, consider P ( C 2 = H ) = 0 . 99 COS 424/SML 302 Probability and Statistics Review February 6, 2019 28 / 69 Subscribe to view the full document.

Two coins Suppose we have two coins, one biased and one fair, P ( C 1 = H ) = 0 . 5 P ( C 2 = H ) = 0 . 7 . We choose one of the coins at random Z ∈ { 1 , 2 } , flip C Z twice, and record the outcome ( X , Y ). What if we knew which coin was flipped Z ? What if the coins have equal probability of heads? COS 424/SML 302 Probability and Statistics Review February 6, 2019 29 / 69 Conditional independence X and Y are conditionally independent given Z : P ( Y | X , Z = z ) = P ( Y | Z = z ) for all possible values of z . Again, this implies a factorization X ⊥⊥ Y | Z ⇐⇒ P ( X , Y | Z = z ) = P ( X | Z = z ) P ( Y | Z = z ) , for all possible values of z . Subscribe to view the full document. • Spring '09
• Probability theory

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