Will Monroe August 9, with materials by Mehran Sahami and Chris Piech. image: Arito. Parameter learning

Transcription

1 Will Monroe August 9, 07 with aterials by Mehran Sahai and Chris Piech iage: Arito Paraeter learning

2 Announceent: Proble Set #6 Goes out tonight. Due the last day of class, Wednesday, August 6 (before class). Soe serious coding! Congressional votng No late days! Heart disease diagnosis

3 Review: Paraeter estaton Soetes we don t know things like the expectaton and variance of a distributon; we have to estate the fro incoplete inforaton. n n = X i S = ( X X X ) i n i= n i= ^ =arg ax LL()

4 Review: Central liit theore Sus and averages of IID rando variables are norally distributed. n σ X = X i N (μ, ) n i= n n Y =n X = X i N (n μ, n σ ) i=

5 Easily-confused principles Constant ultple Su of identcal of a noral norals X N (μ, σ ) CLT X i??? X i N (μ, σ ) (independent & identcal) n n X N (nμ, n σ ) (independent & identcal) n X i N (nμ,n σ ) X i N (nμ,n σ ) (exactly) (approxiately, for large n) i= i=

6 Central liit theore deo

7 Review: Approxiatng a Poisson with a noral X Poi (λ) Y N (λ, λ) (for large λ)

8 Paraeters X Ber(p) =p Poi(λ) =λ Uni(a, b) = [a, b] N(μ, σ²) = [μ, σ²]

9 Maxiu likelihood estaton Choose paraeters that axiize the likelihood (joint probability given paraeters) of the exaple data. ^ =arg ax LL() * x x x x * 0 x 3 4

10 How to: MLE. Copute the likelihood. L()=P ( X,, X ). Take its log. LL()=log L() 3. Maxiize this as a functon of the paraeters. * d LL()=0 d x4 x3 x x x * 0

11 Maxiu likelihood for Bernoulli The axiu likelihood p for Bernoulli rando variables is the saple ean. p^ = X i i=

12 Derivaton: MLE for Bernoulli. Copute the likelihood. = p L()=P( X,, X ) = P( X i ) i= = i= don t forget: IID eans independent! p if X i= ( p) if X i=0 {

13 Derivaton: MLE for Bernoulli. Copute the likelihood. = p L()=P( X,, X ) = P( X i ) i= = i= don t forget: IID eans independent! p if X i= ( p) if X i=0 { Xi X i = p ( p) i=

14 Derivaton: MLE for Bernoulli. Take its log. = p Xi X i L()= ( ) i= Xi X i LL()=log ( ) i= Xi X i = log [ ( ) ] i= = [ X i log +( X i )log( ) ] i=

15 Derivaton: MLE for Bernoulli 3. Maxiize this as a functon of the paraeters. = p LL()= [ X i log +( X i ) log ( ) ] i= ^ p^ =arg ax LL() = X i X i d LL()= d i= = X i ( X i ) i= i= ( = [ ] + ( )( i= + =0 ) )( ) Xi ( ) i= X i = i= X i = = ( ) i= Xi

16 Maxiu likelihood for noral The axiu likelihood μ for noral rando variables is the saple ean, and the axiu likelihood σ² is the uncorrected ean square deviaton. ^ μ= Xi i= ^ ^ σ = ( X i μ) i=

17 Derivaton: MLE for Noral. Take its log =[μ, σ ] L()= i= x μ σ ( e σ π [ ) x μ σ ( LL()= log e σ π i= ) ] x μ = log σ log π σ i= ( )

18 Derivaton: MLE for noral 3. Maxiize this as a functon of the paraeters. =[μ, σ ] X i μ LL()= log σ log π σ i= ^ ^ [ μ^, σ ]==arg ax LL() ( LL()= X i μ σ σ μ i= X i μ = σ i= μ = X i =0 σ i= σ μ= ( ( ) )( ) ( ) X i = X i= )

19 Derivaton: MLE for noral 3. Maxiize this as a functon of the paraeters. =[μ, σ ] X i μ LL()= log σ log π σ i= ^ ^ [ μ^, σ ]==arg ax LL() ( ) X i μ X i μ LL()= σ σ σ σ i= ( X i μ) = σ 3 σ i= [ [ ( )] ] = )( ( X μ) i 3 σ =0 σ i= ) σ = ( X i μ) = ( X i X i= i=

20 Break te!

21 Maxiu likelihood for unifor The axiu likelihood a and b for unifor rando variables are the iniu and axiu of the data. ^ b=ax Xi a^ =in X i i i b a a b

22 Derivaton: MLE for unifor. Copute the likelihood. =[a, b] L()= i= { b a 0 if a X i b otherwise. Take its log. LL()= i= log (b a) if a X i b otherwise { 3. Maxiize this as a functon of the paraeters. ^ =arg ^ [ a^, b]= ax LL()

23 Derivaton: MLE for unifor =[a, b] if a X i b L()= b a i= 0 otherwise ^ =arg ^ [ a^, b]= ax L() { Likelihood: L(a,) L(a, ) a^ =in X i i Likelihood: L(0, b) L(0, b) a b ^ b=ax Xi i

24 Maxiu a posteriori estaton Choose the ost likely paraeters given the exaple data. You ll need a prior probability over the paraeters. ^ =arg ax P ( X,, X n ) =arg ax [ LL()+log P () ]

25 Review: Multnoial rando variable An ultnoial rando variable records the nuber of tes each outcoe occurs, when an experient with ultple outcoes (e.g. die roll) is run ultple tes. X,, X MN (n, p, p,, p ) vector! P ( X =c, X =c,, X =c ) c c c n = p p p c, c,, c ( )

26 Roll all of the dice! A 6-sided die is rolled 7 tes. What is the probability we get: one two 0 threes fours 0 fves 3 sixes? X,, X 6 MN (7,,,,,, ) P ( X =, X =, X 3 =0, X 4 =, X 5 =0, X 6 =3) = =40 6,,0,,0, ( )( ) ( ) ( ) ( ) ( ) ( ) 7 ()

27 Maxiu likelihood with ultnoial A 6-sided die is rolled 7 tes. We get: one two 0 threes fours 0 fves 3 sixes What is the MLE for p₁,, p₆? 3 X,, X 6 MN (7,,, 0,, 0, ) you ll never roll a 3! not in a illion years!

28 Are we doing this backwards? ^ =arg ax P ( X,, X n ) ^ =arg ax P ( X,, X n )

29 Bayes to the rescue ^ =arg ax P ( X,, X n ) P( X,, X n ) P() =arg ax P ( X,, X n ) =arg ax P( X,, X n ) P () =arg ax [ log P( X,, X n )+log P () ]

30 Review: Beta rando variable An beta rando variable odels the probability of a trial s success, given previous trials. The PDF/CDF let you copute probabilites of probabilites! X Beta (a, b) f X ( x)= Cx { a b ( x) 0 if 0< x < otherwise

31 Review: Dirichlet distributon Beta is the distributon ( conjugate prior ) for the p in the Bernoulli and binoial. Dirichlet is the distributon for the p₁, p₂, in the ultnoial. X, X, Dir (a, a, ) fx, X, ( x, x, )= C x a x b if 0<{x, x, }<, x + x + = (0 otherwise)

32 Laplace soothing Also known as add-one soothing: assue you ve seen one iaginary occurrence of each possible outcoe. # ( X =i)+k pi = n+ k # ( X =i)+ pi = n+

33 Maxiu likelihood with ultnoial A 6-sided die is rolled 7 tes. We get: one two 0 threes fours 0 fves 3 sixes What is the MLE for p₁,, p₆? 3 X,, X 6 MN (7,,, 0,, 0, ) you ll never roll a 3! not in a illion years!

34 Laplace with ultnoial A 6-sided die is rolled 7 tes. We get: one two 0 threes fours 0 fves 3 sixes What is the Laplace estate for p₁,, p₆? 3 4 X,, X 6 MN (7,,,,,, ) stll a chance!

35 Paraeter priors X Ber(p) p ~ Beta(a, b) Bin(n, p) p ~ Beta(a, b) MN(p) p ~ Dir(a) Poi(λ) λ ~ Gaa(k, ) Exp(λ) λ ~ Gaa(k, ) N(μ, σ²) μ ~ N(μ, σ ²) σ²~ InvGaa(α, β)