Statistical modelling and latent variables (2)
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1 Statstcal modellg ad latet varables (2 Mxg latet varables ad parameters statstcal erece Trod Reta (Dvso o statstcs ad surace mathematcs, Departmet o Mathematcs, Uversty o Oslo
2 State spaces We typcally have a parametrc model or the latet varables, represetg the true state o a system. Also, the dstrbuto o the observatos may deped o parameters as well as latet varables. Observatos may ote be see as osy versos o the actual state o a system. Examples o states could be: Use gree arrows or oe-way parametrc depedecy (or whch you do t provde a probablty dstrbuto requetst statstcs. D 1. The physcal state o a rocet (posto, oretato, velocty, uel-state. 2. Real water temperature (as opposed to measured temperature. 3. Occupacy a area. 4. Carryg capacty a area.
3 Observatos, latet varables ad parameters - erece Sometmes we are terested the parameters, sometmes the state o the latet varables, sometmes both. Impossble to do erece o the latet varables wthout also dealg wth the parameters ad vce versa. Ote, other parameters aect the latet varables tha the observatos. D D D
4 Observatos, latet varables ad parameters M estmato A latet varable model wll specy the dstrbuto o the latet varables gve the parameters ad the dstrbuto o the observatos gve both the parameters ad the latet varables. Ths wll gve the dstrbuto o data *ad* latet varables: (D, =( (D, But a M aalyss, we wat the lelhood, (D! D Theory (law o total probablty aga: Pr( D Pr( D, Pr( D, Pr( or ( D ( D, d ( D, ( d
5 Observatos, latet varables ad parameters M estmato elhood: Pr( D or ( D Pr( D,, ( D, d ( D The tegral ca ote ot be obtaed aalytcally. Pr( D,, Pr( ( d I occupacy, the sum s easy (oly two possble states Kalma lter: For latet varables as lear ormal Marov chas wth ormal observatos depedg learly o them, ths ca be doe aalytcally. Alteratve whe aalytcal methods al: umercal tegrato, partcle lters, Bayesa statstcs usg MCMC.
6 Occupacy as a state-space model the model words Assume a set areas, (1,,A. Each area has a set o trasects. Each trasect has a depedet detecto probablty, p, gve the occupacy. Occupacy s a latet varable or each area,. Assume depedecy betwee the occupacy state deret areas. The probablty o occupacy s labelled. So, the parameters are =(p,. Pr( =1 =. Start wth dstrbuto o observatos gve the latet varable: Pr(x,j =1 =1, =p. Pr(x,j =0 =1, =1-p, Pr(x,j =1 =0, =0. Pr(x,j =0 =0, =1. So, or 5 trasects wth outcome 00101, we wll get Pr(00101 =1, =(1-p(1-pp(1-pp=p 2 (1-p 3. Pr(00101 =0, = =0
7 Occupacy as a state-space model graphc model Parameters ( : p =occupacy rate p=detecto rate gve occupacy. Oe latet varable per area (area occupacy A Pr( =1 = Pr( =0 =1- The area occupaces are depedet. x 1,1 x 1,2 x 1,3 x 1,1 Data: Detectos sgle trasects. Pr(x,j =1 =1, =p. Pr(x,j =0 =1, =1-p, Pr(x,j =1 =0, =0. Pr(x,j =0 =0, =1 The detectos are depedet *codtoed* o the occupacy. Importat to eep such thgs md whe modellg! PS: What we ve doe so ar s eough to start aalyzg usg WBUGS.
8 Occupacy as a state-space model probablty dstrbuto or a set o trasects Probablty or a set o trasects to gve >0 detectos a gve order s Pr( 1, p (1 p, Pr( 0, 0 whle wth o detectos Pr( We ca represet ths more compactly we troduce the detcato ucto. I(A=1 A s true. I(A=0 A s alse. The 0 Pr( 1, 0, (1 I( p 0, Pr( 0 0, 1 Wth o gve order o the detecto, we pc up the bomal coecet: Pr( 1, p (1 p, Pr( 0, I( 0 (Not relevat at all or erece. The or a gve dataset, the costat s just sttg there.
9 Occupacy as a state-space model area-specc margal detecto probablty (lelhood For a gve area wth a uow occupacy state, the detecto probablty wll the be (law o tot. prob.: Pr( Pr( 1, Pr( 1 Pr( 0, Pr( p (1 p I( 0(1 0 Bomal (p=0.6 Occupacy (p=0.6, =0.6 Occupacy s a zero-lated bomal model
10 Occupacy as a state-space model ull lelhood Each area s depedet, so the ull lelhood s: Pr( A 1 p (1 p I( 0(1 We ca ow do erece o the parameters, =(p,, usg M estmato (or usg Bayesa statstcs.
11 Occupacy as a state-space model occupacy erece Ierece o, gve the parameters, Pr( Pr( 1 1, Pr( Pr( 0, (1 (1 p 1 Pr( 1 Pr( p 1 1 (1 (Bayes theorem: 100% or 0 PS: We preted that s ow here. However, s estmated rom the data ad s ot certa at all. We are usg data twce! Oce to estmate ad oce to do erece o the latet varables. Avoded a Bayesa settg.
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