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# chapter3new - Sampling Importance Resampling(SIR Another...

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Sampling Importance Resampling (SIR) Another method to simulate (almost) from f based on g Assumption: f & g known up to a constant, i.e. f = c 1 ˜ f and g = c 2 ˜ g c 1 , c 2 > 0 unknown. 1. Generate y 1 , y 2 , . . . , y m iid g 2. Compute w j = ˜ f ( y j ) ˜ g ( y j ) for j = 1 , . . . , m 3. Make x = y j with prob w j , i.e., 1/14

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Notes: I The almost means: x | y 1 , . . . , y , d f , i.e., P [ x A | y 1 , . . . , y m ] m →∞ Z A f ( x 0 ) dx 0 I The sample y 1 , . . . , y m and the weights can be reused to generate x 1 , . . . , x n 00 g 2/14
Application in Bayesian Stat:Sensibility analysis Leave-one-out: How does the inference change when removing a datum. I Let y n = ( y 1 , . . . , y n ) be the vector observations and I y - i = ( y 1 , . . . , y i - 1 , y i + 1 , . . . , y n ) the observations after removing the i -th one I Assume p ( y n | θ ) = Q i p ( y i | θ ) I Assume we have a posterior sample θ 1 , . . . , θ m p ( θ | y n ) We can generate θ - i p ( θ | y - i ) by setting θ - i = θ j w.p. w j p ( θ | y - i ) p ( θ | y n ) == Q l 6 = i p ( y | θ j ) Q n l = 1 p ( y | θ j ) = 1 p ( y i | θ j ) 3/14

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Selecting importance function g I In principle, any g works, but I
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chapter3new - Sampling Importance Resampling(SIR Another...

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