BASIC INFERENCE

statisticalSignificancethe challenges are

propose areasonablehullhypothesis

collectingdata tosupportordismiss darn

To assess tle statisticalvalidity of the

null hypothesis Rudybestquantityto

assess

ex Nollhypothesis theaverageflight time

between SFOand JFK is 5 hours or less

to betested we will accept orrejectThe air

Let 2 s XIGstandard deviation

low2 means unlikely bychance

Pick an 2 value predefinedthreshold

Pr Xs x 1H right tail

Pr Xs x 1H left tail

a value is the significance level

The p value ismeasured forestimated

The lo wer th p vchetle moremeaningful the

result will be but really whatmattersis

whether p valueis smaller than a valve

or otherwise

The p wakereflects how surprisingtheresult

way be

CollectData foundthetitle near flighttime islonger that stirs with p wake of 0.01

in theworld whereflight ties are 5hrs or less there's

a 1 chancewe'd see a flight tire of 5.3hrs due

to variability randomness

Zcollectdata frm n s36flights

Orn I Cdata new 5.2 hrs

hypothesismean 55 hrs

data standard deniehu 0.5hrs

Is the evidence to supportnull hypothesis atan alphavalueof a 0.05

Clair Ho s shrDaba t s s 2h r

2s 2s.oo.sk GO 4 24

Ix 0.9918

p G I G 9918 0.01

Since p 22 we REJECT theHo hypothesis

Rene's enouphevidencetosupport downthat

flight ties between SFO JFK one

greater teak stars

Rule if we'dchosen a 0.01 or lowertheHo

would have been accepted I

BBYESLAWvs FREQUENTIST APPROACHESlikelihood

priorto making inferences y

BAYES RUE PfHD PANTHPCDp

posterior normalization

antihwel evidence

H hypothesis D data

In the physicalsciences It is oftenth statevariable Knightcontain dynamicqualities as

well as parameters etc

Frequentist Approach eshnoteth posterior

using fh likelihood only

ex Search for an animal in a perk We want

tolocate it it has a beaconThebeaconsends

out a signal that decaysalldistances

Thefrequntist gather thesignal and anporesdate

with his her familiarity ofthe pal

P Dl HL porkgeograph

Signal observatory

Obtainssamples frm P DlH and maximizePCDHH

The Bayesian in addition to familianyaithtlepark

wewill use priors PCH of places wherethe

animal hasbeen observed

a

Technical Background

Argmax Imax

The argmox of fat defined on aset xeD

argmax ft sfx fix fly HyeDXED

i e thesetof fun D thatachieve thehighest

functionvalue

ex argmax SMX I IXe IR

ex argmax fxD 0XE IR

The wax of thefunohufix defined on thesetXED is

wax ftp.fcx for anya cargmaxfatX cD X ED

The wax functiongives largest possiblevalue of

fix for any H D

The argmaxset is not unique butthe maxi's

BASIC ESTIMATIONTHEORYKary FundofStatsignalProcessy

PrenticeHall93

Basic estimationproblem i given yn nof observations where yn c

112k let

ply O beth pdf of tle N pointdataFind 8 an estimate of 0

E gly Ois tle estimator function

Challenges

Describe how good is as an estimateofO

Quantify howHe estinator ampereslametricand agrees with our expectationsofresult

Common optimal criterion isTle RMSEroot near square error

RMSECO co opHowdo wewritethis in terms ofestimators of

the data on4 Notice

MSE O is E CO Eld E Edo OTVarColt O op

However Eloi OJ is a functionofO andvor loisonly afunction of teedata

We assume or showthat ElIO O O

andthen MSE E vcr 8

MVV MinimumVarianceUnbiased Estimator

Bmw I argmin var OT

argmonECO ECON

estimator is unbiased 0107 0

for OED whereDistle rangeofinterest

EX Assume Wn ore white noise samples

E Wn 0 does notdepend on n

El wawn Elwnwnth

only depends on n buthot k andhes

nonzero valveofhWnWn 628mi

finite

Take Yn O tWn n O I N I

O is a parameterN I

Propose I f E gnn so

isthis an MVV estimatorN

Eto Eftp ign stnEElynM u

INON O

Unbiased

Min variance estimatorN 1

var co var ft yn f EvalynL a of

So is it minium variance

var co 3 In for all otherCRAMERRAO LOWERBOUND CLRBThe variance ofany

unbiasedestimator 0 must

be lowerbounded by the CLRB withthevariance

of th NVUestimatoraltainytle ARB That is

var co1

EffoplyDJ

and var Drew 1E 4.04

Furthermore if forsomefinches g I

ZolnplyD Ilo gg o

then we find the MVVestimator as

mw glythe minium variance is 0

For ble Kodinensive vector OE IRKthe guided arditin is

co I Yo

i e CE I Yo is positivesemidefinite

where

co co Elo KO ElloTis the covariancematrix

THE FISHER MATRIX 1510 is givenby

Ilo ijs.EETg.Ho5fICO is a KHL matrix

Note that ln ply a s Ilo gag Q

then we find the MVV estimator as

0mW Ghg and the

minimum variance is the I l O µEX Consider yn Ot Wn n 0,1 n i

Wu 7 0 and Vaiace Constant GZ

Plyn o expffolyn off

exp lyn or

Ipo tn lyn OD Ng lo o

22For EThe MVV estimator

varlomw ftp.ybutwe know frm previousexayle that

or E s of o Hesample near is

the MW estimator

Alsowhether

ftp.N fffyn o spI9o lgcyl oF

w f synTle MW is indeed 8mW InEynwith minimum variance

g ofex x HOTW

XE 112N HEIR P OeIRPW E IRNH

w n te 0,05in

Rubytee CLRB the FsgCxwillbe MVV eshwetov if

3hfpcx.gs Ilo gatowith Co I O

Q s CHTH H'xand arco eco o4HTHI

BLUE Best hear Unbiased estimator

unbiased minium Variance

Take Os A XL data

Elol A Elkwhich canbesatisfied if

E x HO AH I

The BLUEeshwter is found by seekingAwhich minutes tle Variance

co ACAT

Cs EtKx EG Cx EIGHT

subject toanstraint AH

Carryingout the muniz ahu

Os Dx CHE HI HE IxCo s IH C H

the formulafortheBLUE isthesarees HeMVUfer wear problems Thecrucial difference isthat HeMUU assured Gaussianstatistics

Ex Gauss Markov Eshmetor

Teke x Hot n

n n Lh s 0

flu theBLUE

G CHC HJ H'C

Co s H'C HI isthe urinecovariance µ