Myriads of model selection criteria (Bayesian and frequentist) have been proposed in the literature aiming at selecting a single model regardless of its intended use. An honorable exception in the frequentist perspective is the “focused information criterion” (FIC) aiming at selecting a model based on the parameter of interest (focus). This paper takes the same view in the Bayesian context; that is, a model may be good for one estimand but bad for another. The proposed method exploits the Bayesian model averaging (BMA) machinery to obtain a new criterion, the
A variety of model selection criteria (Bayesian or frequentist) have been proposed in the literature; most of them aim at selecting a single model for any purposes. For an overview of model selection criteria, see the studies by Leeb and Poetscher [
Consider a situation in which some quantity of interest,
Consider a sample of data,
The prior probability that
Within this framework, classical Bayesian model selection involves selecting the model with the highest posterior. Sometimes Bayes factors are used. From the Bayes factor framework, the Bayes factor (Kass and Raftery [
Model
As one can notice, in classical Bayesian model selection, a single (selected) model is used to explain all aspects of data, that is, regardless of the purpose of the selection, irrespective of the inference to follow (Claeskens and Hjort [
The posterior distribution of
Hoeting et al. [
Implementing BMA is demanding, especially the computation of the integrated likelihood. Software for BMA implementation, as well as some BMA papers, can be found at “
Another problem is the selection of priors for both models and parameters. In most cases, a uniform prior is used for each model; that is,
The purpose of this section is to define the focused Bayesian model averaging (FoBMA).
For a set of
Under the square error loss and the weighted posterior probability of (
Conditioning on all models, that is, under (
In particular, for two models, Corollary
For two models, the selected model is the one with the highest posterior probability.
From Proposition
In this section, we apply the methodology to three models: linear regression, logistic regression, and survival analysis. The three following examples have been widely used in Bayesian model averaging (Raftery et al. [
In this subsection, FoBMA is applied to data of the effect of punishment regimes on crime rates (Ehrlich [
Criminologists are interested in the effect of punishment regimes on crime rates. This is a dataset on per capita crime rates in 47 U.S. states in 1960 originally published in the study by Ehrlich [
Table
Description of variables used for linear regression (Ehrlich [
Label  Description 


Percentage of males aged 14–24 

Indicator variable for a Southern state (South = 1 and others = 0) 

Mean years of schooling 

Police expenditure in 1960 

Police expenditure in 1959 

Labour force participation rate 

Number of males per 1000 females 

State population 

Number of nonwhites per 1000 people 

Unemployment rate of urban males 14–24 

Unemployment rate of urban males 35–39 

Gross domestic product per head 

Income inequality 

Probability of apprehension and imprisonment 

Average time served in state prisons (years) 

Rate of crimes in a particular category per head of population 
Table
Linear regression example: selected (best) model according to the focus parameter (FoBMA) compared to its BIC rank and value, the probability that the focus is not null (100
Focus  Best model (FoBMA)  BIC  Rank 

Mean  SD 



51.6  34  97.5  1.4  0.53 


52.1  30  6.3  0.007  0.04 


52.2  29  100  2.13  0.51 


52.5  23  75.4  0.67  0.42 


51.2  45  24.6  0.22  0.40 


50.9  46  2.1  0.003  0.08 


52.3  27  5.2  −0.07  0.45 


52.9  18  28.9  −0.02  0.04 


52.5  22  89.6  0.09  0.05 


52.3  28  11.9  −0.04  0.14 


54.7  3  83.9  0.27  0.19 


51.4  37  33.8  0.20  0.35 


53.1  14  100.0  1.38  0.33 


52.4  25  98.8  −0.25  0.10 


52.9  18  44.5  −0.13  0.18 
In this subsection, FoBMA is applied to data of risk factors associated with low infants birth weights (Hosmer and Lemeshow [
The “birthwt” data frame has 189 rows and 10 columns. The data were collected at Baystate Medical Center, Springfield, Massachusetts, during 1986.
Table
Description of variables used for logistic regression (Hosmer and Lemeshow [
Label  Description 


Indicator of birth weight (BWT) less than 2.5 kg 

Mother’s age in years 

Mother’s weight in pounds at last menstrual period 

Mother’s race (1 = white, 2 = black, and 3 = other) 

Smoking status during pregnancy 

Number of previous premature labours 

History of hypertension 

Presence of uterine irritability 

Number of physician visits during the first trimester 

Birth weight in grams 
Table
Logistic regression example: selected (best) model according to the focus parameter (FoBMA) compared to its BIC rank and value, the probability that the focus is not null (100
Focus  Best model (FoBMA)  BIC  Rank 

Mean  SD 



749.8  20  12.6  0.008  0.024 


748.6  38  68.7  −0.012  0.010 


748.1  46  9.6  0.115  0.390 


748.2  44  9.6  0.092  0.320 


748.5  40  33.2  0.256  0.43 


748.5  40  35.1  0.62  0.089 


748.5  40  35.1  0.170  0.61 


749.3  29  35.1  −4.91  520 


748.1  46  65.4  1.167  1 


748.5  39  33.8  0.310  0.51 


748.7  33  1.5  −0.001  0.024 
In this subsection, FoBMA is applied to the Mayo Clinic Primary Biliary Cirrhosis Data (Therneau and Grambsch [
The data are from the Mayo Clinic trial on primary biliary cirrhosis (PBC) of the liver conducted between 1974 and 1984. A total of 424 PBC patients, referred to Mayo Clinic during that tenyear interval, met eligibility criteria for the randomized placebo controlled trial of the drug Dpenicillamine. The first 312 cases in the dataset participated in the randomized trial and contain largely complete data. The additional 112 cases did not participate in the clinical trial but consented to have basic measurements recorded and to be followed for survival. Six of those cases were lost to followup shortly after diagnosis, so the data here are on an additional 106 cases as well as the 312 randomized participants.
Table
Description of variables used for logistic regression (Hosmer and Lemeshow [
Label  Description 


Age of patients in years 

Serum albumin (mg/dL) 

Alkaline phosphatase (U/liter) 

Presence of ascites 

Aspartate aminotransferase, once called SGOT (U/mL) 

Serum bilirubin (mg/dL) 

Serum cholesterol (mg/dL) 

Urine copper (ug/day) 

0 no edema, 0.5 untreated or successfully treated edema, and 1 edema despite diuretic therapy 

Presence of hepatomegaly or enlarged liver 

Platelet count 

Standardized blood clotting time 

Male/female 

Blood vessel malformations in the skin 

Histologic stage of disease (which needs biopsy) 

Status at endpoint, 0/1/2 for censored, transplant, or dead 

Number of days between the date of registration and the earlier date of death or transplantation 

1/2/NA for Dpenicillamine, placebo, not randomized 

Triglycerides (mg/dL) 
Table
Survival regression example: selected (best) model according to the focus parameter (FoBMA) compared to its BIC rank and value, the probability that the focus is not null (100
Focus  Best model (FoBMA)  BIC  Rank 

Mean  SD 



174.8  1  100  0.031  0.009 


168.9  34  7.8  0.000  0.000 


168.9  24  1.7  0.003  0.045 


170.8  14  100  0.78  0.13 


170.8  15  84.7  0.74  0.43 


170.7  16  2.8  0.006  0.05 


168.8  39  5.4  0.000  0.000 


171.0  13  78  2.5  1.7 


170.4  18  5.6  −0.018  0.1 


169.6  28  20.9  0.097  0.23 


170.0  27  1.4  0.000  0.026 


169.5  29  36.6  0.1  0.16 


170.2  23  1.6  0.002  0.027 


171.3  11  73.8  0.26  0.2 


172.0  7  100  −2.8  8 
As with any other Bayesian approach, parameter priors and model priors are a source of uncertainty in FoBMA. FoBMA shares the same drawbacks of BMA, namely, the computation of integrated likelihoods of various models and the choice of model space. As model selection criterion, compared to Bayesian likes, it is simpler after BMA estimates have been obtained. It was shown in Corollary
It was shown in Paragraph 4 that, in general, FoBMA is found to select a model different from the one selected without a focus. Hereby, the difference in terms of ranking is sometimes large. For example, in studying the risk factors associated with low infants birth weight, the classical Bayesian model selection approach selects the one with variables
Consider a set of data
The present paper has derived a new model selection criterion (FoBMA), which, in contrast to other Bayesian model selection criteria, focuses on the parameter singled out for interest. The methodology was applied to concrete examples. The method needs to be applied in a variety of conditions; in particular, more works need to be done to find the asymptotic properties of FoBMA. It is expected that this paper brings motivations for more researches on focusrelated criteria in Bayesian model selection.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank three anonymous referees for helpful comments that led to improvement on earlier versions of this paper.