perameterostnutefor - oregon state university
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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
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