Lecture 9: September 19

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1 36700: Probability ad Mathematical Statistics I Fall 206 Lecturer: Siva Balakrisha Lecture 9: September 9 9. Review ad Outlie Last class we discussed: Statistical estimatio broadly Pot estimatio BiasVariace tradeoff Cofidece sets I this lecture we will go ito more detail about poit estimatio. Particularly, we will discuss methods for costructig estimators. This is i Chapter 7 of Casella ad Berger. I the ext lecture we will discuss methods for evaluatig estimators. 9.2 Fidig estimators Broadly poit estimatio is aliged with the geeral statistical pursuit of tryig to uderstad a uderlyig populatio from a sample. Particularly, uder the hypothesis that the sample was geerated from some parametric statistical model, a atural way to uderstad the uderlyig populatio is by estimatig the parameters of the statistical model. Oe ca of course woder what happes if the model is wrog? Or where do models really come from? I some rare cases, we actually kow eough about our data to hypothesise a reasoable model. Most ofte however, whe we specify a model, we do so hopig that it ca provide a useful approximatio to the data geeratio mechaism. The George Box quote is worth rememberig i this cotext: all models are wrog, but some are useful. 9
2 92 Lecture 9: September Method of momets We saw this idea before we will see a couple of other examples today. Broadly, the idea is that we ca estimate the sample momets i a straightforward way: m = m 2 =. m k = X i, Xi 2, Xi k. O the other had the populatio momets are some fuctios of the ukow parameters. We ca deote the populatio momets as: µ (θ,..., θ k ),..., µ k (θ,..., θ k ). The methodofmomets prescribes estimatig the parameters: θ,..., θ k by solvig the system of equatios: m = µ (θ,..., θ k ). m k = µ k (θ,..., θ k ). We already saw a applicatio of this idea to estimatig the mea ad variace of a Gaussia: Example : to obtai the estimators: If X,..., X N(θ, σ 2 ), we would solve: X i = θ, ˆθ = ˆσ 2 = = ad X i Xi 2 ( ( X i Xi 2 = θ 2 + σ 2, ) 2 X i ) 2 X j, j=
3 Lecture 9: September which of course are atural estimators of these quatities. You ca see that the methodofmomets estimators ca be biased sice i this example the estimator for the variace has ozero bias. Example 2: Suppose X,..., X Bi(k, p), i.e. each X i is the sum of k Ber(p) RVs, ad that both k ad p are ukow. Casella ad Berger poit out that this model has bee used to model crimes, where usually either the total umber of crimes k or the reportig rate p are kow. Let us first compute the first two momets of the biomial: µ = kp µ 2 = kp( p) + k 2 p 2. Deotig, X = X i, we solve the system of equatios: This yields the estimators: X = kp Xi 2 = kp( p) + k 2 p 2. ˆp = X (X i X) 2, X X ˆk 2 = X (X i X). 2 It is worth poitig out that i this case there are o apriori obvious estimators for the parameters of iterest. Further, the methodofmomets estimators i this case ca be egative (while the true parameters caot be). However, whe the methodofmomets estimator is egative it is because the sample mea is domiated by the sample variace  which is a sig of excessive variability. Aother importat use of the methodofmomets i statistics is i approximatig complex distributios by simpler oes. This is ofte called momet matchig. Example 3: The simplest example of momet matchig comes from the cetral limit theorem. Suppose that RVs X,..., X are i.i.d with kow distributio (it suffices if just
4 94 Lecture 9: September 9 their meas ad variaces are kow). Typically the exact distributio of: S = X i, ca be quite uwieldy. Oe possibility would be to try to approximate this distributio by that of a Gaussia distributio. We do this via the method of momets: ˆθ = E(S ) = µ i, ˆσ 2 = E(S E(S )) 2 = σi 2. Thus we could approximate the distributio of the sum by the Gaussia N(ˆθ, ˆσ 2 ). More geerally, we ca always try to approximate a complex distributio by a much simpler oe i this way Maximum Likelihood Estimatio The most popular techique to derive estimators is via the priciple of maximum likelihood. Suppose that X,..., X f θ where f θ deotes either the pmf or pdf of the populatio. The likelihood fuctio is defied by: L(θ X,..., X ) = f θ (X i ). A maximum likelihood estimator of θ based o the sample {X,..., X } is ay value of the parameter at which the fuctio L(θ X,..., X ) is maximized (as a fuctio of θ with X,..., X held fixed). Ulike the methodofmomets estimator we ca see that the MLE will always obey atural parameter restrictios, i.e. the rage of the MLE estimator is the same as the rage of the parameter. Ituitively, the MLE is just the value of the parameter which is most likely to have geerated the sample ad thus is a atural choice as a poit estimate. It also has may optimality properties i certai settigs: ideed, much of the early theoretical developmet i statistics revolved aroud showig the optimality of MLE i various seses. There are some commo difficulties with usig the MLE: the first is computatioal, how do we fid the global maximum of the likelihood fuctio ad how do we verify/certify that it is i fact the global maximum? The ext two are related to umerical sesitivity: () we ca ofte oly approximately maximize the likelihood (usig a umerical procedure like gradiet descet) so we eed that the likelihood is a wellbehaved fuctio of the parameters. (2)
5 Lecture 9: September Similarly, it is also atural to hope that the estimator is a wellbehaved fuctio of the data, i.e., we would hope that a slightly differet sample would ot result i a vastly differet MLE. The typical way to compute the MLE (suppose that we have k ukow parameters) is to either aalytically or umerically solve the system of equatios: Example : θ i L(θ X,..., X ) = 0 i =,..., k. Suppose X,..., X N(θ, ), the the likelihood fuctio is give as: L(θ X,..., X ) = exp( (X i θ) 2 /2). 2π A frequetly useful simplificatio is to observe that if θ maximizes L(θ X,..., X ) the θ also maximizes log L(θ X,..., X ). We ca also of course igore costats. So we have that: So that we obtai: ˆθ = arg max log L(θ X,..., X ) = arg mi θ θ ˆθ = X i. (X i θ) 2. We eed to further verify that the secod derivative (of the egative loglikelihood) is egative: i this case the secod derivative is 2. Example 2: which is maximized at Suppose that X,..., X Ber(p), the the loglikelihood is give by log L(p X,..., X ) X i log p + ( X i ) log( p) = X log p + ( X) log( p), ˆp = X Ivariace of the MLE Roughly, the ivariace property of the MLE states: suppose that the MLE for a parameter θ is give by ˆθ the the MLE for a parameter τ(θ) is τ(ˆθ). Let us deote the likelihood as a fuctio of the trasformed parameter by L (η X,..., X ). For ivariace we eed to cosider two cases:
6 96 Lecture 9: September 9. τ is a ivertible trasformatio: I this case there is a mappig from the likelihood of a poit η to a poit θ = τ (η), i.e. ad we have that the MLE for η: ˆη = arg sup η L (η X,..., X ) = L(τ (η) X,..., X ), L (η X,..., X ) = arg sup η L(τ (η) X,..., X ) = τ(arg sup L(θ X,..., X )) = τ(ˆθ). θ 2. Whe τ is ot ivertible: I this case, it is possible that several parameters θ, θ 2,... map to the same value η, so we eed some care i defiig the iduced likelihood. Particularly, if we defie: L (η X,..., X ) = the oe ca verify that the MLE is ivariat. sup L(θ X,..., X ), θ:τ(θ)=η We will discuss other properties of the MLE i future lectures. 9.3 Bayes Estimators The third geeral method to derive estimators is a bit differet philosophically from the first two. At a highlevel we eed to uderstad the Bayesia approach (we will of course stay clear of ay philosophical questios): i our approaches so far (socalled frequetist approaches) we assumed that there was a fixed but ukow true parameter, ad that we observed samples draw i.i.d from the populatio (whose desity/mass fuctio/distributio was idexed by the ukow parameter). I the Bayesia approach we cosider the parameter θ to be radom. We have a prior belief of the distributio of the parameter, which we update after seeig the samples X,..., X. We update our belief usig Bayes rule ad the updated distributio is kow as the posterior distributio. We deote the prior distributio by π(θ), ad the posterior distributio as π(θ X,..., X ). Usig Bayes rule we have that: π(θ X,..., X ) = π(θ)l(θ X,..., X ) θ π(θ)l(θ X,..., X ). The posterior distributio is a distributio over the possible parameter values. I this lecture we are focusig o poit estimatio so oe commo cadidate is the posterior mea (i.e. the expected value of the posterior distributio).
7 Lecture 9: September Igorig ay philosophical questios, oe ca view this methodology as a way to geerate cadidate poit estimators, by specifyig reasoable prior distributios. I practice however this calculatio is hard to do aalytically, so we ofte ed up specifyig priors out of coveiece rather tha ay real prior belief. Particularly, a coveiet choice of prior distributio is oe for which the posterior distributio belogs to the same family as the prior distributio: such priors are called cojugate priors. Example : Biomial Bayes estimator: Suppose X,..., X Ber(p). We will first eed to defie the Beta distributio: a RV has a Beta distributio with parameters α ad β if its desity o [0, ] is f(p) = Γ(α + β) Γ(α)Γ(β) pα ( p) β. For us it will be sufficiet to igore the ormalizig part ad just remember that the Beta desity: f(p) p α ( p) β. The mea of the Beta distributio is: α/(α + β). Let us deote X = There are two cadidate priors we will cosider: X i.. The flat/uiformative prior: π(p) = for 0 p. I this case, the posterior desity is: f(p X,..., X ) p X( ( X) p) = p ( X+) ( p) (( X)+). This is just a Beta( X +, ( X) + ) distributio. So our estimate (the posterior mea) would be: ˆp = X = + 2 X = w X + ( w) 2, which ca be viewed as a covex combiatio of the MLE ad the prior mea /2. 2
8 98 Lecture 9: September 9 2. The other commo prior is the oe that is cojugate to the beroulli likelihood, i.e. the Beta prior. A similar calculatio as the oe above will show that if use π(p) Beta(α, β), the the posterior distributio will also be a Beta distributio: ad our Bayes estimator would be: f(p X,..., X ) Beta(α + X, β + ( X)), ˆp = α + X α + β +. Example 2: Gaussia Bayes estimator: Here we suppose that our prior belief is that the parameter has distributio N(µ, τ 2 ) ad we observe X,..., X draw from N(θ, σ 2 ). We assume that σ 2, µ, τ 2 are all kow. A fairly ivolved calculatio i this case (see for istace Problem of Chapter of Wasserma) will show that the posterior distributio is also a Gaussia with parameters: ( ) ˆµ = τ 2 σ 2 X σ 2 + τ 2 i + σ 2 + τ (µ) 2 ˆσ 2 = σ2 τ 2 σ 2 + τ 2, which suggests the poit estimate: ˆµ = τ 2 σ 2 + τ 2 ( ) σ 2 X i + σ 2 + τ (µ). 2 Which ca oce agai be see as a covex combiatio of our prior belief ad the MLE, i.e.: ( ) ˆµ = w X i + ( w)µ, for some 0 w.
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