TIMEDEPENDENT PARAMETERESTIMATRON

Want to findperameterostnutefor

th perimeter OEIRD

You mighthave a prior0 plo Prooron permits

The problenistindependat

Xo potxolO Initiellondihu

Xan pfxu.xh.to Model

ywplyalxa.cl observations

Guldanpute

1 3 ECO sfoplxo.is yio dxo d0

where

plxo.slyiiD.JPYiilxoiqOplxosloploassure independentintine

plxo.si o sphceldI.TPGnlxu.i

ply Xo c O plya Xu Ok s

we could amputee

lo plotYiDOdo

bymerginabong i e

p foly o fpfxo.io yiddxobutthis is impractical

The generic strategy is touseRECURSION

MEHODSI

MAP MaximumAposteriori Estinate

NCHC SaphyEM ExpectatuMinimischief

A E Laplace Approximation

This is a way to approximate a pdfthat is model by a Gaussian

let p te s 1 E Iflade

want to approximatepH near mode 2 o

Find modedfDF 0

Now if ft were a Gaussiancentered

at Zo

In fth buffed Ate Zo 2

where D II hit t.ioSo

fed a flatexpEEG.z.gg

if g t is Gaussian the

gods µ KexpEEGany

g lThe generalization

biz but o El N

w be A 88bn ft hMxM matrix

tht t.dexp f.tkoTAGzoBa an approximation to pad Ifbeer Is fo

is g lLdetAftp.expf.EEolTtEoM

a

def ENERGYFUNCTIONAL

Write play see 9 19

9,10 log polyD logplo

Thesiplest strategy is MCMC

MCMC

Drew Oo go the proposaldistributor

for is1 NSayle O'tngComo

di min f eOE glo 101Fois

where OE 9,10 t 4,101

Draw u 2110,1

IEcar

Rule it Ong10 symmetric

Hen f 10 101

Gi1

ROBUST ADAPTIVE MC RAM

Draw On Polo andinitialize

So tobe thelowertriangleCholeskyfactor ofthe the

initial covariance

E soso

for ist NSayle Od O t Siri

rinkllon

Li shin 1 eOE

U U10,1

if u Ois0

tense oisoin

Find Si w positive diagonal

elements of

Sisi S Miki il Isiturn

Quit Criteria

RECURSIONFOR MARGINALLIKELIHOOD

Rwh thekey to sequential parameter estimatesis the marginalization andfor this therecurrent rule is th wg toperform thisefficiently

Recall that

ploty i s Ip XoSOlya dXo tis required to compute

Elo s fp lo ly 0 do

uihere p lo ly x ply lo p o

The challenge is hudeyplyi.to

plyi.sk ll plyulyik.i owhere

plyulyi.h.no

plyulxndplxulyHe.iO dxh

bbsenahus Predictive recursive

party b.no s plxulxu.yo plxalyiuo dxk

and

plxulyi.noPlYhlxkiOx.o1pCynlyi

h i 0y

ALGORITHM

statwith Polo loglplo

for 6 1 T

9h10 Cla lo loglplyhlyii.hn ODwhee

p lyn ly wO plyuhhO pknyimiddleIt

i i

AMAP MaximumAPosteriori Eshnote

We will approximate

ploty 8 0 8mm

thus ignoring the distributionspread apletely

The MAP is such that p Omm

We find Om argqaxfploly.ir

by finding ago.mu Glo

0mm

To do this we differentiate the distributionand find th extremizer

Fisher's Identity expresses gradientof 9,10 as the expectation of the derivative

of th anplete observant log likelihood

12 No needfor rewram in thisapproach

Once MAP is found we approximate

plot yr a IdfGM HCOmaryY

whereH istheHessian

i ln pkg In lolOmm In Ip1ohm1 ME In 2 it the detCH

thesetermspenalise complexity

in degreesoffreedom

H s 88 In p101ohm p

Omm

s Wh p f8mn40BK Bey

esinfocriterion

lu p lo a hip lol 0mm fdust ofobservatoriespermeter

Rude Simple non recursive

EM EXPECTATION MINIMIZATION

Iteratively findsthe MLestimator Thebasisis that though we can't anputeteenagedlikelihood we might beable toestimate

a lower band

Let g Go be an arbitraryPdf

L tog ply lol Ffglxo.at owhere

Flgfxoi OD fqfxo.dhogpkoi.IOGao

dxo

Thekey idea isthatwe maximize theLHS of

byiterativelymaximizingthe towerboundf I glxo.it o

Algorithm Schematic

Start go 00

for n s1,2Estep

gnn argues51910

stepon argnex FIFTH

In practiceThe Estep

gn Xo pHoi ly it OHnby F Ipluggingthis in

Ffg HoDOBJpHo ily 076gphlox y iii OldXo

fplxoi.ly th logplxoiqyi.siOldXo

dpdmOQ 10,0fplxoi.ly th logplxoiqyi.si A dXo

Manning F is equivalent tofinding 0

EMALGORITTMI

Start 00for n s1,2Cstep computeQ10,07IMstep anpute Ont'sargnfexQ1407

To find orgfaxQQOn

require JoQ10,07 0

Wecould use Fisher'sIdentity if we

evaluate th gradient of Q at 070we get exactly

gologplot ToQ1907loon

Notchugsuseful but a funnelway toapproximatethegradientof the energy

ofI yGoel is to find the0mL

In p N lo she pHXlot

step corresponds to a expecthwon pCHYO

t

Mstep maximise expectatin

Oh argmgxpcxly.org

Algorithmgivenplx.NO over observed

variablesYadshete X ad parameter0

Thegoelistonexinize pollo wrt O

1 Choose

2 eucleete paly Odd

3 Onewsargfax 0,00d

www.oold qplx.YOddllnplYxl04 Oney O

5 test for gait criteria