Monte Carlo Methods: Lecture 3 : Importance Sampling

Transcription

1 Mote Carlo Methods: Lecture 3 : Importace Samplig Nick Whiteley Course material origially by Adam Johase ad Ludger Evers 2007

2 Overview of this lecture What we have see... Rejectio samplig. This lecture will cover... Importace samplig. Basic importace samplig Importace samplig usig self-ormalised weights Fiite variace estimates Optimal proposals Example

3 Recall rejectio samplig Algorithm 2.1: Rejectio samplig Give two desities f, g with f(x) < M g(x) for all x, we ca geerate a sample from f by 1. Draw X g. 2. Accept X as a sample from f with probability otherwise go back to step 1. Drawbacks: f(x) M g(x), We eed that f(x) < M g(x) O average we eed to repeat the first step M times before we ca accept a value proposed by g.

4

5 The fudametal idetities behid importace samplig (1) Assume that g(x) > 0 for (almost) all x with f(x) > 0. The for a measurable set A: P(X A) = A f(x) dx = A g(x) f(x) g(x) }{{} =:w(x) dx = A g(x)w(x) dx For some itegrable test fuctio h, assume that g(x) > 0 for (almost) all x with f(x) h(x) 0 = E f (h(x)) = f(x)h(x) dx = g(x)w(x)h(x) dx = E g (w(x) h(x)), g(x) f(x) h(x) dx g(x) }{{} =:w(x)

6 The fudametal idetities behid importace samplig (2) How ca we make use of E f (h(x)) = E g (w(x) h(x))? Cosider X 1,..., X g ad E g w(x) h(x) < +. The 1 a.s. w(x i )h(x i ) E g (w(x) h(x)) (law of large umbers), which implies 1 a.s. w(x i )h(x i ) E f (h(x)). Thus we ca estimate µ := E f (h(x)) by 1 Sample X 1,..., X g 2 µ := 1 w(x i)h(x i )

7 The importace samplig algorithm Algorithm 2.1a: Importace Samplig Choose g such that supp(g) supp(f h). 1. For i = 1,..., : i. Geerate X i g. ii. Set w(x i ) = f(xi) g(x. i) 2. Retur µ = w(w i)h(x i ) as a estimate of E f (h(x)). Cotrary to rejectio samplig, importace samplig does ot yield realisatios from f, but a weighted sample (X i, W i ). The weighted sample ca be used for estimatig expectatios E f (h(x)) (ad thus probabilities, etc.)

8 Basic properties of the importace samplig estimate We have already see that µ is cosistet if supp(g) supp(f h) ad E g w(x) h(x) < +, as µ := 1 a.s. w(x i )h(x i ) E f (h(x)) The expected value of the weights is E g (w(x)) = 1. µ is ubiased (see theorem below) Theorem 2.2: Bias ad Variace of Importace Samplig E g ( µ) = µ Var g ( µ) = Var g(w(x) h(x))

9 Is it eough to kow f up to a multiplicative costat? Assume f(x) = Cπ(x). The µ = w(x i)h(x i ) = 1 Cπ(X i ) g(x i ) h(x i) C does ot cacel out kowig π( ) is ot eough. Idea: Replace ormalisato by by ormalisatio by w(x i), i.e. cosider the self-ormalised estimator ˆµ = w(x i)h(x i ) w(x i) Now we have that ˆµ = w(x i)h(x i ) w(x = i) π(x i ) g(x i ) h(x i), π(x i ) g(x i ) ˆµ does ot deped o C eough to kow f up to a multiplicative costat

10 The importace samplig algorithm (2) Algorithm 2.1b: Importace Samplig usig self-ormalised weights Choose g such that supp(g) supp(f h). 1. For i = 1,..., : i. Geerate X i g. ii. Set w(x i ) = f(xi) g(x. i) 2. Retur ˆµ = w(x i)h(x i ) w(x i) as a estimate of E f (h(x)).

11 Basic properties of the self-ormalised estimate ˆµ is cosistet as ˆµ = w(x i)h(x i ) }{{ } = µ E f (h(x)) w(x i) } {{ } 1 a.s. E f (h(x)), (provided supp(g) supp(f h) ad E g w(x) h(x) < + ) ˆµ is biased, but asymptotically ubiased (see theorem below) Theorem 2.2: Bias ad Variace (ctd.) E g (ˆµ) = µ + µvar g(w(x)) Cov g (w(x), w(x) h(x)) Var g (ˆµ) = Var g(w(x) h(x)) 2µCov g (w(x), w(x) h(x)) + µ2 Var g (w(x)) + O( 2 ) + O( 2 )

12 Fiite variace estimators Importace samplig estimate cosistet for large choice of g. (oly eed that...) More importat i practice: fiite variace estimators, i.e. ( Var( µ) = Var w(x ) i)h(x i ) < + Necessary (albeit very restrictive) coditios for fiite variace of µ: f(x) < M g(x) ad Var f (h(x)) <, or E is compact, f is bouded above o E, ad g is bouded below o E. Note: If f has heavier tails the g, the the weights will have ifiite variace!

13 Optimal proposals Theorem 2.3: Optimal proposal The proposal distributio g that miimises the variace of µ is g (x) = h(x) f(x) h(t) f(t) dt. Theorem of little practical use: the optimal proposal ivolves h(t) f(t) dt, which is the itegral we wat to estimate! Practical relevace of theorem 2.3: Choose g such that it is close to h(x) f(x)

14 Super-efficiecy of importace samplig For the optimal g we have that Var f ( h(x1 ) h(x ) if h is ot almost surely costat. Superefficiecy of importace samplig ) > Var g ( µ), The variace of the importace samplig estimate ca be less tha the variace obtaied whe samplig directly from the target f. Ituitio: Importace samplig allows us to choose g such that we focus o areas which cotribute most to the itegral h(x)f(x) dx. Eve sub-optimal proposals ca be super-efficiet.

15 Example 2.5: Setup Compute E f X for X t 3 by... (a) samplig directly from t 3. (b) usig a t 1 distributio as istrumetal distributio. (c) usig a N(0, 1) distributio as istrumetal distributio.

16 Example 2.5: Desities x f(x) (Target) f(x) (direct samplig) gt 1 (x) (IS t 1 ) g N(0,1) (x) (IS N(0, 1)) x

17 Example 2.5: Estimates obtaied Samplig directly from t3 IS usig t1 as istrumetal dist IS usig N(0, 1) as istrumetal dist IS estimate over time Iteratio

18 Example 2.5: Weights Samplig directly from t3 IS usig t1 as istrumetal dist IS usig N(0, 1) as istrumetal dist Weights Wi Sample Xi from the istrumetal distributio