A Simple Gibbs Sampler. Biostatistics 615/815 Lecture 20

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1 A Smple Gbbs Sampler Bostatstcs 615/815 Lecture 20

2 Schedulng No Class November 30 Revew sesson Thursday, December 7 Fnal Assessment Tuesday, December 12

3 Optmzaton Strateges Sngle Varable Golden Search Quadratc Approxmatons Multple Varables Smplex Method E-M Algorthm Smulated Annealng

4 Smulated Annealng Stochastc Method Sometmes takes up-hll steps Avods local mnma Soluton s gradually frozen Values of parameters wth largest mpact on functon values are fxed earler

5 Gbbs Sampler Another MCMC Method Update a sngle parameter at a tme Sample from condtonal dstrbuton when other parameters are fxed

6 Gbbs Sampler Algorthm Consder a partcular choce of parameter values ( t Defne the next set of parameter values by : a.selectng component to update, say b.sample value for ( t+ 1 from p( x, 1, 2,... 1, + 1,... k Increment t and repeat prevous steps.

7 Alternatve Algorthm Consder a partcular choce of parameter values ( t Defne the next set of parameter values by : a. Update each component,1.. k, n turn b.sample value for c.sample value for... z.sample value for ( t+ 1 1 ( t+ 1 2 ( t+ 1 k from from from p( p( p( 1 2 k x, 2 x, 1 x, 1,,, 3 3 3,...,...,... k k k 1 Increment t and repeat prevous steps.

8 Key Property: Statonary Dstrbuton from ther posteror dstrbuton we expect the Gbbs sampler to sample parameter values Eventually, ( ~ ( ~ In fact...,...,, (,..., (,,..., ( s dstrbuted as,...,, Then (,...,, ( ~,...,, Suppose that ( 1 ( ( ( ( 2 1 ( ( ( 2 ( 1 x p x p x p x p x p x p t t k k k t k t t k t k t t + + =

9 Gbbs Samplng for Mxture Dstrbutons Sample each of the mxture parameters from condtonal dstrbuton Drchlet, Normal and Gamma dstrbutons are typcal Smple alternatve s to sample the source of each observaton Assgn observaton to specfc component

10 Samplng A Component Calculate the probablty that the observaton orgnated from a specfc component can you recall how random numbers can be used to sample from one of a few dscrete categores? = = l l j l j j j x f x f, x Z, (, (,, Pr( η φ π η φ π π φ η

11 C Code: Samplng A Component nt sample_group(double x, nt k, double * probs, double * mean, double * sgma { nt group; double p = Random(; double lk = dmx(x, k, probs, mean, sgma; for (group = 0; group < k - 1; group++ { double pgroup = probs[group] * dnorm(x, mean[group], sgma[group]/lk; f (p < pgroup return group; p -= pgroup; } return k - 1; }

12 Calculatng Mxture Parameters Z j j Z j j Z j n x n x s n x x n n p n j j j = = = = = = = : 2 2 : : 1 Before samplng a new orgn for an observaton update mxture parameters gven current assgnments Could be expensve!

13 C Code: Updatng Parameters vod update_estmates(nt k, nt n, double * prob, double * mean, double * sgma, double * counts, double * sum, double * sumsq { nt ; for ( = 0; < k; ++ { prob[] = counts[]/ n; mean[] = sum[] / counts[]; sgma[] = sqrt((sumsq[] - mean[]*mean[]*counts[] / counts[] + 1e-7; } }

14 C Code: Updatng Mxture Parameters II vod remove_observaton(double x, nt group, double * counts, double * sum, double * sumsq { counts[group] --; sum[group] -= x; sumsq[group] -= x * x; } vod add_observaton(double x, nt group, double * counts, double * sum, double * sumsq { counts[group] ++; sum[group] += x; sumsq[group] += x * x; }

15 Selectng a Startng State Must start wth an assgnment of observatons to groupngs Many alternatves are possble, I chose to perform random assgnments wth equal probabltes

16 C Code: Startng State vod ntal_state(nt k, nt * group, double * counts, double * sum, double * sumsq { nt ; for ( = 0; < k; ++ counts[] = sum[] = sumsq[] = 0.0; for ( = 0; < n; ++ { group[] = Random( * k; } counts[group[]] ++; sum[group[]] += data[]; sumsq[group[]] += data[] * data[]; }

17 The Gbbs Sampler Select ntal state Repeat a large number of tmes: Select an element Update condtonal on other elements If approprate, output summary for each run

18 C Code: Core of The Gbbs Sampler ntal_state(k, probs, mean, sgma, group, counts, sum, sumsq; for ( = 0; < ; ++ { nt d = Random( * n; f (counts[group[d]] < MIN_GROUP contnue; remove_observaton(data[d], group[d], counts, sum, sumsq; update_estmates(k, n - 1, probs, mean, sgma, counts, sum, sumsq; group[d] = sample_group(data[d], k, probs, mean, sgma; add_observaton(data[d], group[d], counts, sum, sumsq; f (( > BURN_IN && ( % THIN_INTERVAL == 0 /* Collect statstcs */ }

19 Gbbs Sampler: Memory Allocaton and Freeng vod gbbs(nt k, double * probs, double * mean, double * sgma { nt, * group = (nt * malloc(szeof(nt * n; double * sum = alloc_vector(k; double * sumsq = alloc_vector(k; double * counts = alloc_vector(k; /* Core of the Gbbs Sampler goes here */ free_vector(sum, k; free_vector(sumsq, k; free_vector(counts, k; free(group; }

20 Example Applcaton Old Fathful Eruptons (n = 272 Old Fathful Eruptons Frequency Duraton (mns

21 Notes My frst few runs found excellent solutons by fttng components accountng for very few observatons but wth varance near 0 Why? Repeated values due to roundng To avod ths, I set MIN_GROUP to 13 (5%

22 Gbbs Sampler Burn-In -250 LogLkelhood Log-Lkelhood Iteraton

23 Gbbs Sampler Burn-In 5 Mxture Means 4 Mean Iteraton

24 Gbbs Sampler After Burn-In Lkelhood Lkelhood Iteraton (Mllons

25 Gbbs Sampler After Burn-In Mean for Frst Component Component Mean Iteraton (Mllons

26 Gbbs Sampler After Burn-In Component Mean Iteraton (Mllons

27 Notes on Gbbs Sampler Prevous optmzers settled on a mnmum eventually The Gbbs sampler contnues wanderng through the statonary dstrbuton Forever!

28 Drawng Inferences To draw nferences, summarze parameter values from statonary dstrbuton For example, mght calculate the mean, medan, etc.

29 Component Means Densty Densty Mean Mean Mean Densty

30 Component Probabltes Densty Densty Densty Probablty Probablty Probablty

31 Overall Parameter Estmates The means of the posteror dstrbutons for the three components were: Frequences of 0.073, and Means of 1.85, 2.08 and 4.28 Varances of 0.001, and Our prevous estmates were: Components contrbutng.160, and Component means are 1.856, and Varances are , and 0.172

32 Jont Dstrbutons Gbbs Sampler provdes other nterestng nformaton and nsghts For example, we can evaluate jont dstrbuton of two parameters

33 Component Probabltes Group 3 Probablty Group 1 Probablty

34 So far today Introducton to Gbbs samplng Generatng posteror dstrbutons of parameters condtonal on data Provdng nsght nto jont dstrbutons

35 A lttle bt of theory Hghlght connecton between Smulated Annealng and the Gbbs sampler Fll n some detals of the Metropols algorthm

36 Both Methods Are Markov Chans The probablty of any state beng chosen depends only on the prevous state Pr( Sn = n Sn 1 = n 1,..., S0 = 0 = Pr( Sn = n Sn 1 = n 1 States are updated accordng to transton matrx wth elements p j. Ths matrx defnes mportant propertes, ncludng perodcty and rreducblty.

37 Metropols-Hastngs Acceptance Probablty Let qj = q(propose Sn+ 1 Let π and π j = j S n = be the relatve probabltes of each state The Metropols - Hastngs acceptance probablty s : a j = mn 1, π π j q q j j or a j = mn 1, π π j f q j = q j Only the rato π π j must be known, not the actual values of π

38 Metropols-Hastngs Equlbrum If we use the Metropols - Hastngs algorthm to update a Markov Chan, t wll reach an equlbrum dstrbuton where Pr(S = = π For ths to happen, the proposal states to communcate. densty must allow all

39 Gbbs Sampler The Gbbs sampler ensures that π q j = π q j j As a consequence, a j = mn π jq 1, π q j j = 1

40 Smulated Annealng Gven a temperature parameter τ, replace π ( wth π τ = π j 1 τ π 1 τ j At hgh temperatures, the probablty dstrbuton s flattened At low temperatures, larger weghts are gven to hgh probablty states

41 Addtonal Readng If you need a refresher on Gbbs samplng Bayesan Methods for Mxture Dstrbutons M. Stephens ( Numercal Analyss for Statstcans Kenneth Lange (1999 Chapter 24

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