Poit Estimatio b MLE Lesso 5
Review Defied Likelihood Maximum Likelihood Estimatio Step : Costruct Likelihood Step : Maximize fuctio Take Log of likelihood fuctio Take derivative of fuctio Set derivative 0 Solve for parameter
Poit estimatio Goal: 'fit' a model to data Sigle best value of a parameter that fids most support i the data set Cotrasts with iterval estimatio (e.g. 95% cofidece iterval) Examples: N SS x x i N N SS xi x N
Maximum Likelihood Step : Write a likelihood fuctio describig the likelihood of the observatio Step : Fid the value of the model parameter that maximized the likelihood dl 0 d
a L l L l a a0 0. da a l L ML e
A secod data poit Suppose a secod plat dies at da 4 Step : Defie the likelihood L Pr (a, a ρ) Pr (a a, ρ) Pr (a ρ) Pr (a ρ) Pr (a ρ) Exp(a ρ) Exp(a ρ) Assume measuremets are idepedet
Step : Fid the maximum a L l L l a a a a 0 0.0833 da a a l L ML e e a
A whole data set Step : Defie Likelihood L Pr a, a,, a Pr ai Exp ai Assume measuremets are idepedet
Step : Fid the maximum L e ai l L l ai l ai l L ML ai 0 ai / a
Mortalit Alterate data First example based o kowig times idividuals died What if istead we ol kew the fial outcome N umber of idividual plats umber that survived Wat to fid the MLE for q probabilit of survivig to the ed of the expm't
Step : Write the likelihood L Biom N, N
Step : Write the likelihood L Biom N, Step : Fid the maximum ll l N l N
Step : Write the likelihood L Biom N, Step : Fid the maximum ll l N l ll N N
Step : Write the likelihood L Biom N, N Step : Fid the maximum ll l N l ll N N
Step : Write the likelihood L Biom N, N Step : Fid the maximum ll l N l ll N N N ML
Mortalit combiig age of death ad survival till ed Step Write the likelihood L Biom N, N
Mortalit combiig age of death ad survival till ed Step Write the likelihood L Biom N, N P survive to T P dead b T
Mortalit combiig age of death ad survival till ed Step Write the likelihood L Biom N, N P survive to T P dead b T T e e T Expoetial CDF
Mortalit combiig age of death ad survival till ed Step Write the likelihood L Biom N, N P survive to T P dead b T T e e T L Biom N, e T T N e
Step : Fid the maximum Le T e T N
Step : Fid the maximum Le ll T e T N T N l e T
Step : Fid the maximum Le ll ll T e T N T N l e T T Te T N T e
Step : Fid the maximum Le ll ll ML T e T N T N l e T T Te T N T e l / N T
Survival Aalsis L Biom N, Died N Survived
Survival Aalsis L Biom N, L i i N
Survival Aalsis L Biom N, L i i Mortalit at time a L e ai N
Survival Aalsis L Biom N, L i i Mortalit at time a L e Cesored at time c ai k e ci N
Survival Aalsis L Biom N, N L i i Mortalit at time a L e Survived to time T Cesored at time c a i k e c i e T k
L e a i k e c i e T k
L e a i k e c i e T k k ll l ai c i T k
L e a i k e c i e T k k ll l ai c i T k k ll ai c i T k 0
L e a i k e c i e T k k ll l ai c i T k k ll ai c i T k 0 ML k ai c i T k
MLE for the Normal L N i, [ i exp ]
L ll [ i exp ] l l i
L ll l L exp [ i ] l l i i 0
L ll l L exp [ i ] l l i i i 0
L exp [ i ] l l i ll l L i ML i 0
Normal Variace ll l l i
Normal Variace ll l L l l i 3 i 0
Normal Variace ll l L l l i 3 i 0 3 i
Normal Variace ll l L l l i 3 i 0 3 i i
Populatio Growth Rate t t t e t t e r r t l t l t r t x t x t r t t ~ N 0, x t ~ N x t r,
Step : Write Likelihood T L N x t x t r, t [ T T exp x t x t r t ]
Step : Write Likelihood T L N x t x t r, t [ T T exp x t x t r t Step : Maximize Likelihood Take logs T l L T l x t x t r t ]
Solve for mea T l L T l x t x t r t T l L x t x t r 0 r t
Solve for mea T l L T l x t x t r t T l L x t x t r 0 r t T T r x t x t t
Solve for mea T l L T l x t x t r t T l L x t x t r 0 r t T T r x t x t t x T x 0 r ML T
Solve for variace T l L T l x t x t r t
Solve for variace T l L T l x t x t r t T l L T 3 x t x t r 0 t
Solve for variace T l L T l x t x t r t T l L T 3 x t x t r 0 t ML T x t x t r T t
Homework You wat to kow the desit of fish i a set of experimetal pods You observe the followig couts i te pods: 5,6,7,3,6,5,8,4,4,3 What is our process model? What is our data model? Solve for the aaltical MLE What is the estimate for this populatio?