Conjugate Priors-

Case of Normal,@-uI

Recall Normal density feat e 202

EK3=µ T×= o' x~N(ye, f)

If Xi , Xz, - - - , Xn are independentand Xin N(Mi , E)

then II. aini - NCE airi , Eairi)[ sum of independent normals is normal ]

Consider the problem

f-Calm = re- CETI T is known

µ is unknown

manner. " a.at:1?.;::re7hnfWhen we don't know somethingwe assign a prior to it .

Let f-fu) be our prior .

What can you say about µ if you are observing z?

ffrtnl-zf.gg#fa Difficulty Denominator !

From that point on , you can find

PIM >Nola) , E [Mla] , etc . . .

How to compute the integral ? [it could be hard]

Solution : Choose ffa) to make the math work !

Is this a limitation ?

Justification for solution above :

mnmv_choice of prior does not matter in practice ifwe have a large amount of observations .

- [typical] choose a prior that will produce a

posterior of the same form as the prior . [conjugate]

( update parameters without changing the model )-- =

prior& Flobserration

(original parameters)

Flute) = SCHAALSffafutffndm

say f-(pi ) = 1- Causbyit feltfu-PT)+0

Denominator : ftp.oe-EI?*#.,.jdv..- A

Observation : If Mr N (p , E) o then µ In nNYT)new parameters

since posterior will have same formas prior, we say the prior is conjugate

f-Calm) f- (a)

firm =

Tp&f none•

wForm 2-=- form 1

prior is conjugate to Normed

Nerd is conjugate to Normed

conjugate prior means : you figured out the integral !

false, =t.EE#.f....e-4ttI-

pDrop

constants .

n:::::i e.ee#.e.az#fHntxft4HfH)we say f-Ca ) x god iff f = c. fee)

a"

proportional to"

constant

theres a e- 4¥

.

e- HE22(n-a)'

+ Tcu-pj= e

-

Else'

tu'- 2am) +04µFp2- yup)

= e-#

Zora 2

= e

-

2. OZ 2

-

@7- E)M2- zu (En + Tp)x e s

= e

-

I.

TIZZ

- e

le - 3¥32-#

2 They

= e

-

0422

where m= Entry¥2

VIRZI92+22

mix - NCEEEI.EE. )

More data : Ni , Zz, - - - , Nn n i. i. d . Conditioned onM\- -

- - -- - - - - - - - - - - - -

"

f-feel a , - - - inn) & Ffa , - - - pen In) ffu) i

= II faints- -- - -

-

-

"

xi÷e- EEE. e- a¥

What if we look at E = &?ci/n (average)a- IM ~ N(µ , of) [sum of independent Normals)

ffntx) - Ftalntffu )g kN(pic)Nfa, %)

rhi - NC , )If we work on the problem f-(MIM,. . - -xn)

we will find f- (a) oh , -→ an ) = flu II) .

n-zoo : Hula ) n NCE, %)

Prior"doesn't matter

"

Thu

Observe N- e- no and let y be such that

ly - not → a

trainee -

- ftp.?;f?ffqY-yjAssumeHIFlyin = no) omyaxfucy)> °

> a :c:ti÷i÷=a:Ei÷:i::Yi÷- ÷Chebyshev PC tri -Ml > e) say → o