Recursive Updating Fixed Parameter

Transcription

1 Recursive Udatig Fixed Parameter So far we ve cosidered the sceario where we collect a buch of data ad the use that (ad the rior PDF) to comute the coditioal PDF from which we ca the get the MAP, or ay other Bayesia estimator we wish. Give batch (vector) of data xx NN = (xx, xx 2, xx 3, xx NN ) ad the rior (θθ) The Bayes rule gives the coditioal PDF we eed: ( x ) θ = N ( θ) x N ( x ) ( θ) N or ( xn) ( ) θ θ ( x θ) Proortioality costat easy to fid: makes this itegrate to So o eed to fuss about keeig track of the roortioality costat! I fact, for MAP it is ever eeded! ( ) ( ) θ θ ( θ) xn xn Posterior Prior Likelihood N New Name!

2 But suose our data arrives sequetially as xx, xx 2, xx 3, xx NN, Each time we get a ew iece of data we wat to udate our coditioal PDF (the extract the MAP, etc. as desired) But we do NOT wat to comute with ALL the PAST data! Just use curret coditioal PDF ad the oe ew data oit!! A useful result for dealig with this sequetial settig is: (,,, θ) = ( θ) (, θ) (,, θ) ( x, θ) x x x x x x x x x x Note that this a geeralizatio of what we saw earlier ad called Decomosig the Joit PDF ( x, y) ( y x) ( x) XY = Y X X 2

3 Now we ca use this decomositio to see how to sequetially udate the osterior PDF o which Bayes estimatio is based: Cosider the osterior PDF after the first data oit xx ad the as we collect more ad more data: ( ) ( ) θ x θ x ( θ) Acts like a ew Prior! Acts like a ew Prior! Acts like a ew Likelihood! ( ) ( ) θ x, x θ ( x, x θ) 2 2 = ( θ) x ( θ) x ( 2 x, θ) ( θ ) x x ( x, θ) 2 ( ) ( ) ( ) θ x, x, x θ x ( θ) x ( x, θ) x ( x, x, θ) θ x, x x ( x, x, θ) This leads to the Bayesia Recursive Udate Viewoit: ( ) ( ) θ x θ x x ( x, θ) N N N N New Posterior Old Posterior New Likelihood Geeralizatio of Posterior Prior Likelihood 3

4 Recursive Udatig Evolvig State Based o of Moo & Stirlig, Mathematical Methods & Algorithms for Sigal Processig, Pretice-Hall, 2000 I the revious we cosider fixed state does ot chage w/ time Now the state chages w/ time: ss (we ow abado θθ otatio for arameter ) Goal: Observe xx 0, xx, xx 2, ad estimate ss 0, ss, ss 2, Note: State is vector! Measuremets here are scalars (but could be geeralized to vectors) Need some Notatio! X = [ x x x x ] T (All measuremets u to time ) 0 2 ss kk = estimate of state ss based o XX kk (data u to time k) 4

5 We ca oly roceed if we have a aroriate models! State Dyamics Model: The state is a Markov Radom Process: ( s s, s,, s ) = ( s s ) ( ( + ) 0) = f ( + ) f ( ) s s, s,, s s s Next state is coditioally ide of all ast states excet most recet oe! This catures the dyamics of the state How does the state trasitio from time to time + Holds also for fuctio of state ss + Measuremet Model: ( s,,,, ) = ( s ) x x x x x or equivaletly ( s, X ) = ( x s ) x e.g., Meas. xx + deeds oly o state ss + lus oise that is ide samle-to-samle Combiig these two gives a useful result: ( s, s, s,, s, X ) = ( x s ) x

6 Basic First Stes of Recursio Give Prior PDF o iitial state we ca get a estimate Thik like this: Udate Posterior ad State Estimate usig measuremet xx 0 : ( s x ) 0 0 = ( s ) s ˆ 0 0 ( s ) ( 0 s0) ( s0) ( x ) x 0 x 0 ŝ 0 0 Proagate (somehow!) this osterior to ss xx 0 (via.) We do t have = so view this as rior to data (via.) ( s x ) Pro. ( s x0) 0 0 ŝ 0 (via.) Iterate: Treat ss xx 0 as a ew rior ad go to Udate That results i ss XX, etc. We ow eed to flesh this idea out! 6

7 Bayesia Recursio for Evolvig State From the above Basic view we ca see that we really wat some fuctio F{ } to go from ss XX to ss + XX + ; that is { } ( s X ) = F ( s X ), x We ll see that this is doe i two stes: Udate & Proagate From Bayes Theorem we ca write ( s+ X+ ) = ( s+ X, + ) ( x+ s+, X) = ( x X ) x + ( s X ) From measuremet model, this = xx + ss + so we ca write: ( ) ( ) ( ) x Normalizer s X s s X Udate Ste Posterior Meas. Model Prior Measuremet Model gives the Likelihood Fuctio! 7

8 But the udate ste eeds that rior (ss + XX ) Fidig it yields the Proagatio ste! We fid it usig the cocet of Margial from Joit : ( ) (, ) s X = s s X ds + + The decomosig the PDF iside the itegral gives ( s, s X ) = ( s s, X ) ( s X ) + + ( ) ( ) = s + s s X deeds o ss ad oise that is ide samle-to-samle Usig this iside the itegral gives the roagatio ste: ( ) ( ) ( ) s X = s s s X ds + + Proagatio Ste Prior for Udate State Dyamic Model Posterior 8

9 Results for Bayesia Recursio for Evolvig State ( ) ( ) ( ) x s X s s X Udate Ste Posterior Measuremet Model Prior ( ) ( ) ( ) s X = s s s X ds + + Proagatio Ste Prior for Udate State Dyamic Model Posterior We eed: (i) Iitial Prior, (ii) State Model, ad (iii) Measuremet Model x Udate Proagate ( s X ) ( s X ) ( s X ) + s ˆ (State Predictio) sˆ (State Estimate) sˆ+ (State Predictio) Ca also extract covariaces from these geerated PDFs 9

10 x Udate Proagate ( s X ) ( s X ) ( s X ) + s ˆ (State Predictio) sˆ (State Estimate) sˆ+ (State Predictio) What haes whe all the PDFs are guarateed to be Gaussia? Oly eed to udate & roagate cod. meas ad covariaces! x s ˆ Udate sˆ Proagate s ˆ+ C C C + Cod. Mea = MMSE Cod. Mea = MMSE 0