Advanced Statistical Computing

Transcription

1 Advanced Statistical Computing Fall 206 Steve Qin

2 Outline Collapsing, predictive updating Sequential Monte Carlo 2

3 Collapsing and grouping Want to sample from = Regular Gibbs sampler: Sample t+ from π π Sample 2 t+ from Sample t+ d from Alternatively: π t + t + 2, 2,..., d t 2 t Grouping: d ' = d, d. Collapsing, i.e., integrate out d :,,, t 3 t 3,...,,...,,..., t d t d t+ t t t d 2 3 d =,,,, 2,..., d 3

4 The three-schemes standard grouping collapsing 4

5 Some theory Hilbert space L 2 π of functions h. Define h, g = E { h g }, thus h = varπ h. Define forward operator F as Fh = π K, y h y dy = { t+ t h = }. 2 F = suph Fh for all fucntions with E h = The convergence of Markov chains is tied to the norms of the corresponding forward operators. E π. 5

6 Three-scheme theorem Standard F s : Grouping F g : Collapsing F c :! 2 d! { d };, 2 d! Theorem The norms of the three forward operators are ordered as. 2 d ; F c F g F s 6

7 Murray s data Eamples Bivariate Gaussian with mean 0 and unknown covariance matri Σ standard collapsing Σ y mis y obs, ymis, ymis, i yobs, ymis,[ i]. y, Σ. obs 7

8 Remarks Avoid introducing unnecessary parameters into a Gibbs sampler, Do as much analytical work as possible, However, introducing some clever auiliary variables can greatly improve computation efficiency. 8

9 Sequential Monte Carlo We wish to evaluate an integral assume h 0. θ = h π d = Eπ [ h X ]. ℵ Riemann sum on grid points as approimation. Alternatively, use Monte Carlo. Select random samples uniformly on its support. 9

10 An eample Both grid-point method and vanilla Monte Carlo methods wasted resources on boring desert area. 0

11 The basic idea Marshall 956 suggested that one should focus on the regions of importance so as to save computational resources importance sampling. Essential in high-dimensional models.

12 To evaluate The algorithm m Draw,..., from a trial distribution g. Calculate the importance weight w j Approimate µ by Remark: estimator µˆ = π µ π j = E [ h X ] = h π d. / g j is better than the unbiased why? 2 ℵ, for j =,..., m. w h w +! + w +! + w m ˆ = m µ µ = { w h +! + w m ~ m m h }. h m.

13 An eample cont. Use proposal function with,y ϵ [,] [,], a truncated miture of bivariate Gaussian Vanilla Monte Carlo Importance Sampling ˆ µ = ˆ µ = std ˆ µ = std ˆ µ =

14 Rao-Blackwellization Basic principle in Monte Carlo: carry out analytical computation as much as possible. 4

15 Rao-Blackwellization 5

16 Rao-Blackwellization 6

17 Rao-Blackwellization Conditioning an inferior estimator on the vale of sufficient statistics leads to the optimal estimator. 7

18 Sequential importance sampling For high dimensional problem, how to design trial distribution is challenging. Suppose the target density of can be decomposed as π π π 2! π d then constructed trial density as =, 2,..., d =,.., d g = g g2 2! gd d,.., d 8

19 Sequential importance sampling Suggest a recursive way of computing and monitoring importance weight. Denote then we have 9,..,,.., = d d d d d g g g w!!π π π,...,, 2 t = t t- t- t- t t t t t t g w w π =

20 Sequential importance sampling Advantages of the recursion scheme Can stop generating further components of if the partial weight is too small. Can take advantage of π in designing g t t t - t t - However, the scheme is impractical since requires the knowledge of marginal distribution π t. 20

21 Sequential importance sampling Add another layer of compleity: Introduce a sequence of auiliary distributions π π π such that π t t 2 2 d is a reasonable approimation of the marginal distribution π t, for t =,,d - and π d = π. Note the π d are only required to be known up to a normalizing constant. 2

22 For t = 2,,d, Draw X t = t from t = t-, t Compute The SIS procedure and let w t = w t - u t u t : incremental weight. u t t, and let The key idea is to breaks a difficult task into manageable pieces. If w t is getting too small, reject. t g t π t t = π g t t- t t t- 22

23 References Hammersley and Morton 954. Rosenbluth and Rosenbluth 955. Liu JS 200 Doucet et al

24 Eamples of SIS Growing a polymer Self avoid walk Sequential imputation for statistical missing data problem. More and details of these eamples, see Liu

25 Future topics Multigrid Monte Carlo MGMC, densityscaling Monte Carlo, hybrid Monte Carlo HMC, evolutionary Monte Carlo, echange Monte Carlo. Cluster method, data augmentation. Parameter epansion, multicanonical sampling, umbrella sampling, simulated tempering, multi-try Metropolis, particle filtering, 25