Frequentist Model Averaging and Applications to Bernoulli Trials

In several instances of statistical practice, it is not uncommon to use the same data for both model selection and inference, without taking account of the variability induced by model selection step. This is usually referred to as post-model selection inference. The shortcomings of such practice are widely recognized, finding a general solution is extremely challenging. We propose a model averaging alternative consisting on taking into account model selection probability and the likelihood in assigning the weights. The approach is applied to Bernoulli trials and outperforms Akaike weights model averaging and post-model selection estimators.


Introduction
In statistical modeling practice, it is typical to ignore the variability of the model selection step, which can result in inaccurate post-selection inference (Berk et al. ([1] [2]), Belloni et al. ([3] [4]), Tibshirani et al. [5], and Chernozhukov et al. [6]).The model selection step is often a complex decision process and can involve collecting expert opinions, preprocessing, applying a variable selection rule, data-driven choice of one or more tuning parameters, among others.Except in simple cases, it is hard to explicitly characterize the form of the post-selection of interest while incorporating the variability of model selection.References for model selection include e.g.Zucchini [7] and Zucchini et al. [8].An alternative to selecting a single model for estimation purposes is to use a weighted average of the estimates resulting from each of the models under consideration.This leads to the class of model averaging estimators.Model averaging can be done either in Bayesian and frequentist approaches.The most common Bayesian approach is Bayesian model averaging (BMA) and its variants, using Bayesian information criterion (BIC) as approximation (Schwarz [9]).The seminal paper of Hoeting et al. [10] fully describes the basic of BMA.BMA and its applications can be found in Nguefack-Tsague ( [11] [12]), Nguefack-Tsague and Ingo [13], Nguefack-Tsague and Zucchini ([14] [15]).Several options are available for specifying the weights in frequentist approaches; references on least squares regression types and like include Hansen ([16]- [21]), Hansen and Racine [22], Cheng and Hansen [23], Charkhi et al. [24], and Wan et al. [25].The aforementioned weighting schemes perform model averaging on a set of nested candidate models with the weights vector chosen such that a specific criterion is minimized.
References using Akaike's information criterion, AIC (Akaike [26]) include Burnham and Anderson [27], Nguefack-Zucchini [28], Nguefack-Tsague ([29]- [32]).The R package [33] MuMIn is used to perform model averaging based on Burnham and Anderson [27].Schomaker and Heumann [34], and Schomaker [35] developes model averaging schemes based on multiple imputation and shrinkage; the R package MAMI is used for practical implementations.This paper is organized as follows: In Section 2, we develop model averaging based on information criterion while, in Section 3, we propose a new approach for computing the weights for the competing models, one that takes both account the selection probability and the likelihood of each model.Section 4 illustrates with applications to Bernoulli trials.The paper ends with concluding remarks.

Frequentist Model Averaging Based on Information Criterion
be a set of K plausible models to estimate µ , the quantity of interest.Denote by ˆk µ the estimator of µ obtained when using model k M .Model averaging involves finding non-negative weights, 1 , , K w w  , that sum to one, and then estimating µ by 1 ˆˆ.
In model selection, the model selection criterion determines which model is to be assigned weight one, i.e. which model is selected and subsequently used to estimate the parameter of interest.We note that, since the value of the selection criterion depends on the data, the index, k , of the selected model is a random variable.We therefore denote the selected model by k M , and the corresponding estimator of the quantity of interest, µ , by k µ .In terms of the notation introduced above, we may write Clearly, the selected model depends on the set of candidate models,  , and on the selection procedure, which we denote by S.However, it is important to realize that, even if the same  and S, are used, different samples can lead to different models being selected; k M is a "randomly selected model".In this section we focus attention on post-model selection estimators (PMSEs), which is the special case of model averaging estimators with zero/one weights only.
Some classical model averaging weights base the weights on penalized likelihood values.Let k IC denote an "information criterion"of the form 2 log , where k s is a penalty term, and k L is the maximized likelihood value for the model k M .The Akaike infor- mation criterion (AIC, Akaike [26]) is the special case with 2 , where k q is the number of parameters of model k M .Buckland et al. [36] proposed using weights of the form: "Akaike weights" (denoted by , aic k w ) refer to the case with . Numerous applications of Akaike weights are given in Burnham and Anderson [27].

Likelihood and Selection Probability in Assigning the Weights
Since the selection procedure (S) and likelihood are important for model selection, we therefore suggest estimating µ by a weighted average of the ˆk µ in which the weights take account of S, specifically where they depend on estimators ( )  ( ) The likelihoods are taken into account because they quantify the relative plausibility of the data under each competing model; the estimated selection probability ( ) p M S adjusts the weights for the selection procedure.Both of these components are required.If one were to use only the likelihoods to determine the weights then complex models (i.e.models having many parameters) would automatically be assigned larger weights.The weights , al k w are similar to the weights k w defined in (3) but they differ in the way the likelihood is adjusted.With the proposed method a "bad" model will be penalized by any reasonable selection procedure through the probability ( ) , even if it is complex in terms of the number of parameters.We let the selection procedure determine in how far a model is penalized.
If the selection probabilities depend on some parameter for which a closed form expression exists, and if one can find an estimator of the parameter, then it is possible to obtain estimators for these probabilities.

Applications to Bernoulli Trials
is the number of successes; Y-binomial (n, θ), θ unknown.Inference will be based on Y, since the likelihood function of the X i 's is = −   be the probabilities of choosing models 1 and 2, respectively.
, n y a θ θ > and 2 θ otherwise.The properties of θ  are given by ( ) showing different values of the parameters space used for weighting.Figure 2 compares PMSE to estimators based on Akaike weights and adjusted weights using true model selection probabilities.It can be seen that adjusted likelihood is always better than PMSE and Akaike weights estimators.However, for some values of the true parameter, the risk of Akaike weight tends to be slightly bigger than that of PMSEs.Maxima occur at 0.5 θ = while minima occur at 0.4 and 0.6.(b) Consider now a choice between the following two models: : ~binomial , M Y n θ and ( ) ˆy n θ = , for illustration, we choose 1 0.    .
Figure 3 displays model selection probabilities with both curves crossing at 0.6 and 0.4.At 0.5, while Model 2 is at the minimum, Model 1 is at maximum. Figure 4 displays risks performance of estimators.It can be seen that Akaike weighting does not perform better than PMSEs when the true parameter is between ( ) 0, 0.3 and between ( ) 0.7,1 .However, the adjusted weights perform better than both.

Multi-Model Choice
Consider also a choice between the following models: ( ) For a choice using AIC criterion, since there is no unknown parameter, this is the same as selecting the model with higher likelihood.regions of the parameter space, but the adjusted likelihood weights are better than both.

Concluding Remarks
In this paper, we have considered model averaging in frequentist perspective; and proposed an approach of assigning weights to competing models taking account model selection probability and likelihood.The method appears to perform well for Bernoulli trials.The method needs to be applied in variety of situations before it can be adopted.

Figure 1
Figure 1 shows model selection probabilities for 1 0.6 θ = and ( ) 2 | p M AIC are obtained by replacing θ by 2 ˆy n

Figure 2 .
Figure 2. Risk of two simple proportions comparing PMSEs, Akaike weights estimators and adjusted estimators as a function of θ.

Figure 4 .
Figure 4. Risk of two proportions comparing PMSEs, Akaike weights estimators and adjusted estimators as a function of θ.

Figure 5 .
Figure 5. Risk of 30 models comparing PMSEs, Akaike weights estimators and adjusted estimators as a function of θ.