For a two-state marker system, both additive effects β _j , j = 1, ⋯, p _β , and epistatic effects γ _j , j = 1, ⋯, p _γ , are the primary focus of QTL mapping. As is often the case p = (p _β +p _γ ) ≫ n, many of these coefficients are zero, either because the variation of the trait can be explained by only a few QTL or because the limited sample size precludes selecting too many variables (otherwise the constructed model is not reliable as shown in the previous section). It is also possible that the number and/or scale of the positive coefficients may be different from those of the negative ones. To account for these properties, a three-component mixture prior is specified for each coefficient β _j or γ _j . More specifically,

{ β j ~ i i d ( 1 − w β + − w β − ) δ ( ⋅ ) + w β + N + ( μ β + , σ β + 2 ) + w β − N − ( μ β − , σ β − 2 ) , γ j ~ i i d ( 1 − w γ + − w γ − ) δ ( ⋅ ) + w γ + N + ( μ γ + , σ γ + 2 ) + w γ − N − ( μ γ − , σ γ − 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@BE0D@

where δ (·) is a Dirac function with mass one at zero; N ₊(μ, σ ²) and N _-(μ, σ ²) positively and negatively truncate the normal distribution, i.e., N(μ, σ ²), respectively. Therefore, w _β+ (or w _β-) is the probability for any single marker, and w _γ+ (or w _γ-) is the probability for any pair of markers in I MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8heHKeaaa@3691@ {X}, to have positive (or negative) interactive effect on the trait.

The hyperparameters, σ β + 2 , σ β − 2 , σ γ + 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabek7aIjabgUcaRaqaaiabikdaYaaakiabcYcaSiabeo8aZnaaDaaaleaacqaHYoGycqGHsislaeaacqaIYaGmaaGccqGGSaalcqaHdpWCdaqhaaWcbaGaeq4SdCMaey4kaScabaGaeGOmaidaaaaa@3DE9@ and σ γ − 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabeo7aNjabgkHiTaqaaiabikdaYaaaaaa@314B@ , are assumed to have priors as inverse gamma distributions, that is, IG(θ _β+, φ _β+), IG(θ _β-, φ _β-), IG(θ _γ+, φ _γ+), and IG(θ _γ-, φ _γ-), respectively (e.g., setting θ _β+ = θ _β- = θ _γ+ = θ _γ- = 0.1 and φ _β+ = φ _β- = φ _γ+ = φ _γ- = 10). As a result, the prior on β (and γ) is essentially a mixture of a point mass at zero and some truncated t-distributions, which shrinks the smaller effects towards zero and allows sufficient flexibility for non-zero effects. Furthermore, t-type prior distributions yield Bayes rules with desirable decision-theoretic frequentist properties 47 . The hyperparameters, μ _β+, μ _γ+, μ _β- and μ _γ-, are assumed to have diffuse priors, and the prior distribution for σ ε 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabew7aLbqaaiabikdaYaaaaaa@305E@ is proportional to 1/ σ ε 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabew7aLbqaaiabikdaYaaaaaa@305E@ .

As suggested by the predictability of a size n data set, we expect to select at most p _n = O( n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ ) out of the p variables for the final model. Therefore, we specify the priors for (w _β+, w _β-) and (w _γ+,w _γ-) as

w β + + w β − ~ U ( 0 , c n / p β ) , w γ + + w γ − ~ U ( 0 , c n / p γ ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5B08@

that is, expecting at most c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ significant additive effects and epistatic effects, respectively. Gaffney 48 and Yi et al. 17 , among others, employed similar ideas to rescale the priors based on the number of possible effects. Apparently, when n ≪ (p _β + p _γ ), either c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ /p _β or c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ /p _γ is very small, which implies a restrictive prior on each corresponding coefficient. Therefore, we usually select c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ additive effects and c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ epistatic effects during the first run of Bayesian analysis. We then apply the same Bayesian analysis to these pre-selected variables. The second run of Bayesian analysis has both w _β+ + w _β- and w _γ+ + w _γ-, a priori, uniformly distributed on [0, 1].

Let Y _n be the column vector including the trait values of all strains under investigation, let X _i be the vector of all marker values of the i-th strain and X n = ( X 1 T , ⋯ , X n T ) T MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacbmGae8hwaG1aaSbaaSqaaiabd6gaUbqabaGccqGH9aqpcqGGOaakcqWGybawdaqhaaWcbaGaeGymaedabaGaemivaqfaaOGaeiilaWIaeS47IWKaeiilaWIaemiwaG1aa0baaSqaaiabd6gaUbqaaiabdsfaubaakiabcMcaPmaaCaaaleqabaGaemivaqfaaaaa@3E0C@ , and let Z _i be the vector of all epistatic candidate values of the i-th strain. Denote the marginal distribution of A as [A], and the conditional distribution of A given B as [A|B]. With data (Y _n , X _n ) and the prior specification in Section 3.1, we have the likelihood function, that is, the joint distribution function of the data ( Y _n , X _n ), the parameters (μ, β , γ ), σ ε 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabew7aLbqaaiabikdaYaaaaaa@305E@ , and all hyperparameters

( w β + , w β − , w γ + , w γ − , μ β + , μ γ + , σ β + 2 , σ γ + 2 , μ γ + , μ γ − , σ β − 2 , σ γ − 2 ) , L ∝ [ Y n | X n , μ , β , γ , σ ε 2 ] × [ μ ] × [ μ β + ] × [ σ β + 2 ] × [ μ β − ] × [ σ β − 2 ] × [ w β + , w β − ] × [ β | w β + , w β − , μ β + , σ β + 2 , μ β − , σ β − 2 ] × [ μ γ + ] × [ σ γ + 2 ] × [ μ γ − ] × [ σ γ − 2 ] × [ w γ + , w γ − ] × [ γ | w γ + , w γ − , μ γ + , σ γ + 2 , μ γ − , σ γ − 2 ] × [ σ ε 2 ] × [ X n ] ∝ σ ε − n − 2 exp ⁡ { − ∑ i = 1 n ( Y i − μ − X i β − Z i γ ) 2 2 σ ε 2 } × exp ⁡ ( − σ β + 2 φ β + − σ β − 2 φ β − ) × ( σ β + 2 ) − θ β + − 1 × ( σ β − 2 ) − θ β − − 1 × exp ⁡ ( − σ γ + 2 φ γ + − σ γ − 2 φ γ − ) × ( σ γ + 2 ) − θ γ + − 1 × ( σ γ − 2 ) − θ γ − − 1 × [ β | w β + , w β − , μ β + , σ β + 2 , μ β − , σ β − 2 ] × I [ w β + + w β − ≤ c n p β ] × [ γ | w γ + , w γ − , μ γ + , σ γ + 2 , μ γ − , σ γ − 2 ] × I [ w γ + + w γ − ≤ c n p γ ] × [ X n ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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The distribution of X _n can be specified based on the available linkage map information 2 . The conditional distribution of [ β | w β + , w β − , μ β + , σ β + 2 , μ β − , σ β − 2 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaei4waSfccmGae8NSdiMaeiiFaWNaem4DaC3aaSbaaSqaaiabek7aIjabgUcaRaqabaGccqGGSaalcqWG3bWDdaWgaaWcbaGaeqOSdiMaeyOeI0cabeaakiabcYcaSiabeY7aTnaaBaaaleaacqaHYoGycqGHRaWkaeqaaOGaeiilaWIaeq4Wdm3aa0baaSqaaiabek7aIjabgUcaRaqaaiabikdaYaaakiabcYcaSiabeY7aTnaaBaaaleaacqaHYoGycqGHsislaeqaaOGaeiilaWIaeq4Wdm3aa0baaSqaaiabek7aIjabgkHiTaqaaiabikdaYaaakiabc2faDbaa@521B@ is a product of the prior distribution for each β _j . Similarly, the conditional distribution of [ γ | w γ + , w γ − , μ γ + , σ γ + 2 , μ γ − , σ γ − 2 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaei4waSfccmGae83SdCMaeiiFaWNaem4DaC3aaSbaaSqaaiabeo7aNjabgUcaRaqabaGccqGGSaalcqWG3bWDdaWgaaWcbaGaeq4SdCMaeyOeI0cabeaakiabcYcaSiabeY7aTnaaBaaaleaacqaHZoWzcqGHRaWkaeqaaOGaeiilaWIaeq4Wdm3aa0baaSqaaiabeo7aNjabgUcaRaqaaiabikdaYaaakiabcYcaSiabeY7aTnaaBaaaleaacqaHZoWzcqGHsislaeqaaOGaeiilaWIaeq4Wdm3aa0baaSqaaiabeo7aNjabgkHiTaqaaiabikdaYaaakiabc2faDbaa@5245@ is a product of the prior distribution for each γ _j . The priors of the hyperparameters, θ _β+, θ _γ+, φ _β+, φ _γ+, θ _β-, θ _γ-, φ _β- and φ _γ-, are specified to be as noninformative as possible.

Since the specified priors are conditionally conjugate, Bayesian variable selection can be implemented with a Gibbs sampling algorithm. We initialize the algorithm by imputing missing genotypic values based on the observed genotypes and linkage information. The initial value of μ is set as the mean of the observed trait values. Then, with individuals having fully observed trait values, each component of β and γ is initially estimated using recursive univariate regression. Other parameters, w β + , w β − , μ β + , σ β + 2 , μ β − MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4DaC3aaSbaaSqaaiabek7aIjabgUcaRaqabaGccqGGSaalcqWG3bWDdaWgaaWcbaGaeqOSdiMaeyOeI0cabeaakiabcYcaSiabeY7aTnaaBaaaleaacqaHYoGycqGHRaWkaeqaaOGaeiilaWIaeq4Wdm3aa0baaSqaaiabek7aIjabgUcaRaqaaiabikdaYaaakiabcYcaSiabeY7aTnaaBaaaleaacqaHYoGycqGHsislaeqaaaaa@460E@ and σ β − 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabek7aIjabgkHiTaqaaiabikdaYaaaaaa@3145@ , are simply initialized based on the initial value of β , and similarly, the initial values for w γ + , w γ − , μ γ + , σ γ + 2 , μ γ − MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4DaC3aaSbaaSqaaiabeo7aNjabgUcaRaqabaGccqGGSaalcqWG3bWDdaWgaaWcbaGaeq4SdCMaeyOeI0cabeaakiabcYcaSiabeY7aTnaaBaaaleaacqaHZoWzcqGHRaWkaeqaaOGaeiilaWIaeq4Wdm3aa0baaSqaaiabeo7aNjabgUcaRaqaaiabikdaYaaakiabcYcaSiabeY7aTnaaBaaaleaacqaHZoWzcqGHsislaeqaaaaa@462C@ , and σ γ − 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabeo7aNjabgkHiTaqaaiabikdaYaaaaaa@314B@ can be specified using the information from γ . For example, we can initialize σ β + 2 = σ β − 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabek7aIjabgUcaRaqaaiabikdaYaaakiabg2da9iabeo8aZnaaDaaaleaacqaHYoGycqGHsislaeaacqaIYaGmaaaaaa@37BA@ with an estimate from the initial value β, and then use max{#{j : β _j > 2σ _β+}, 1}/p _β to initialize w _β+.

Let X _i,-j be X _i excluding the j-th component, and define β _-j and γ _-j similarly. Based on the likelihood function in (4), the Gibbs sampler can be developed by recursively drawing the missing genotypic values, the missing trait values, and the model parameters from their full conditional posterior distributions as follows.

Sample missing values: Sample each missing genotypic value X _ij from its full conditional posterior distribution,

[ X i j | Y i , X i , − j , μ , β , γ , σ ε 2 ] ∝ [ Y i | X i , − j , X i j , μ , β , γ , σ ε 2 ] × [ X i j | X i , j − 1 , X i , j + 1 ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaei4waSLaemiwaG1aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGG8baFcqWGzbqwdaWgaaWcbaGaemyAaKgabeaakiabcYcaSGqadiab=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@8507@

and then sample each missing trait value Y _i from its full conditional posterior distribution [Y _i |X _i , μ, β, γ, σ ε 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabew7aLbqaaiabikdaYaaaaaa@305E@ ].

Sample μ: Sample μ from its full conditional posterior distribution,

μ | Y n , X n , β , γ , σ ε 2 ~ N ( 1 n ∑ i = 1 n ( Y i − X i β − Z i γ ) , σ ε 2 n ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6605@

Sample β and γ: Sample each β _j and γ _j from their full conditional posterior distributions,

[ β j | Y n , X n , μ , β − j , γ , w β + , w β − , σ ε 2 , σ β + 2 , σ β − 2 ] ~ ( 1 − w ˜ β j + − w ˜ β j − ) δ ( β j ) + w ˜ β j + N + ( μ ˜ β j + , σ ˜ β j + 2 ) + w ˜ β j − N − ( μ ˜ β j − , σ ˜ β j − 2 ) , [ γ j | Y n , X n , μ , β , γ − j , w γ + , w γ − , σ ε 2 , σ γ + 2 , σ γ − 2 ] ~ ( 1 − w ˜ γ j + − w ˜ γ j − ) δ ( γ j ) + w ˜ γ j + N + ( μ ˜ γ j + , σ ˜ γ j + 2 ) + w ˜ γ j − N − ( μ ˜ γ j − , σ ˜ γ j − 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@2699@

where w ˜ β j + , w ˜ β j − , μ ˜ β j + , σ ˜ β j + 2 , μ ˜ β j − MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4DaCNbaGaadaWgaaWcbaGaeqOSdiMaemOAaOMaey4kaScabeaakiabcYcaSiqbdEha3zaaiaWaaSbaaSqaaiabek7aIjabdQgaQjabgkHiTaqabaGccqGGSaalcuaH8oqBgaacamaaBaaaleaacqaHYoGycqWGQbGAcqGHRaWkaeqaaOGaeiilaWIafq4WdmNbaGaadaqhaaWcbaGaeqOSdiMaemOAaOMaey4kaScabaGaeGOmaidaaOGaeiilaWIafqiVd0MbaGaadaWgaaWcbaGaeqOSdiMaemOAaOMaeyOeI0cabeaaaaa@4D2A@ , and σ ˜ β j − 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4WdmNbaGaadaqhaaWcbaGaeqOSdiMaemOAaOMaeyOeI0cabaGaeGOmaidaaaaa@32B1@ are specified in the APPENDIX. In addition, w ˜ γ j + , w ˜ γ j − , μ ˜ γ j + , σ ˜ γ j + 2 , μ ˜ γ j − MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4DaCNbaGaadaWgaaWcbaGaeq4SdCMaemOAaOMaey4kaScabeaakiabcYcaSiqbdEha3zaaiaWaaSbaaSqaaiabeo7aNjabdQgaQjabgkHiTaqabaGccqGGSaalcuaH8oqBgaacamaaBaaaleaacqaHZoWzcqWGQbGAcqGHRaWkaeqaaOGaeiilaWIafq4WdmNbaGaadaqhaaWcbaGaeq4SdCMaemOAaOMaey4kaScabaGaeGOmaidaaOGaeiilaWIafqiVd0MbaGaadaWgaaWcbaGaeq4SdCMaemOAaOMaeyOeI0cabeaaaaa@4D48@ , and σ ˜ γ j − 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4WdmNbaGaadaqhaaWcbaGaeq4SdCMaemOAaOMaeyOeI0cabaGaeGOmaidaaaaa@32B7@ can be obtained similarly.

Sample w _β+, w _γ+, w _β-, and w _γ-: These parameters can be sampled from the conditional posterior distributions,

( w β + , w β − , 1 − w β + − w β − ) | β ~ D i r i c h l e t ( p ˜ β + + 1 , p ˜ β − + 1 , p β − p ˜ β + − p ˜ β − + 1 ) , w β + + w β − ≤ c n p β , ( w γ + , w γ − , 1 − w γ + − w γ − ) | γ ~ D i r i c h l e t ( p ˜ γ + + 1 , p ˜ γ − + 1 , p γ − p ˜ γ + − p ˜ γ − + 1 ) , w γ + + w γ − ≤ c n p γ , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@E1B0@

where p ˜ β + MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiCaaNbaGaadaWgaaWcbaGaeqOSdiMaey4kaScabeaaaaa@2FFC@ = #{β _j > 0 : 1 ≤ j ≤ p _β } and p ˜ β − MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiCaaNbaGaadaWgaaWcbaGaeqOSdiMaeyOeI0cabeaaaaa@3007@ = #{β _j < 0 : 1 ≤ j ≤ p _β }; p ˜ γ + MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiCaaNbaGaadaWgaaWcbaGaeq4SdCMaey4kaScabeaaaaa@3002@ = #{γ _j > 0 : 1 ≤ j ≤ p _γ } and p ˜ γ − MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiCaaNbaGaadaWgaaWcbaGaeq4SdCMaeyOeI0cabeaaaaa@300D@ = #{γ _j < 0 : 1 ≤ j ≤ p _γ }.

Sample σ ε 2 , σ β + 2 , σ γ + 2 , σ β − 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabew7aLbqaaiabikdaYaaakiabcYcaSiabeo8aZnaaDaaaleaacqaHYoGycqGHRaWkaeaacqaIYaGmaaGccqGGSaalcqaHdpWCdaqhaaWcbaGaeq4SdCMaey4kaScabaGaeGOmaidaaOGaeiilaWIaeq4Wdm3aa0baaSqaaiabek7aIjabgkHiTaqaaiabikdaYaaaaaa@435C@ , and σ γ − 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabeo7aNjabgkHiTaqaaiabikdaYaaaaaa@314B@ : With conditionally conjugate priors, the posterior for all variance parameters are still inverse gamma distributions. Specifically,

σ ε 2 | Y n , X n , μ , β , γ ~ I G ( n 2 , 2 ∑ i = 1 n ( Y i − μ − X i β − Z i γ ) 2 ) , σ β + 2 | β ~ I G ( θ β + + p ˜ β + 2 , 2 2 φ β + + ∑ j = 1 p β β j 2 I [ β j > 0 ] ) , σ γ + 2 | γ ~ I G ( θ γ + + p ˜ γ + 2 , 2 2 φ γ + + ∑ j = 1 p γ γ j 2 I [ γ j > 0 ] ) , σ β − 2 | β ~ I G ( θ β − + p ˜ β − 2 , 2 2 φ β − + ∑ j = 1 p β β j 2 I [ β j < 0 ] ) , σ γ − 2 | γ ~ I G ( θ γ − + p ˜ γ − 2 , 2 2 φ γ − + ∑ j = 1 p γ γ j 2 I [ γ j < 0 ] ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabuqaaaaabaGaeq4Wdm3aa0baaSqaaiabew7aLbqaaiabikdaYaaakiabcYha8Hqadiab=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@4464@

For each variable in model (1), one pair of parameters is used to select the corresponding variable. They are, for the j-th additive effect, the posterior probabilities w _{β j+} = P (β _j > 0|Y _n , X _n ) and w _{β j-} = P (β _j < 0|Y _n , X _n ). With the full conditional posterior distribution of β _j and all the notations in the APPENDIX, we have

w β j + = E [ w ˜ β j + | Y n , X n ] , w β j − = E [ w ˜ β j − | Y n , X n ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5DB2@

Therefore, the two parameters w _{β j+} and w _{β j-} can be estimated with the Markov chains of w ˜ β j + MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4DaCNbaGaadaWgaaWcbaGaeqOSdiMaemOAaOMaey4kaScabeaaaaa@3167@ and w ˜ β j − MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4DaCNbaGaadaWgaaWcbaGaeqOSdiMaemOAaOMaeyOeI0cabeaaaaa@3172@ drawn from the above Gibbs sampler. If and only if both w _{β j+} and w _{β j-} are less than 0.5, the median of the posterior distribution of β _j is zero. Similarly, the posterior probabilities w _{γ j+} = P (γ _j > 0| Y _n , X _n ) and w _{γ j-} = P (γ _j < 0| Y _n , X _n ) can be estimated with the Markov chains of w ˜ γ j + MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4DaCNbaGaadaWgaaWcbaGaeq4SdCMaemOAaOMaey4kaScabeaaaaa@316D@ and w ˜ γ j − MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4DaCNbaGaadaWgaaWcbaGaeq4SdCMaemOAaOMaeyOeI0cabeaaaaa@3178@ drawn from the above Gibbs sampler.

We propose to select variables twice under the above Bayesian framework for model (1). At the first step, we use a restrictive prior for each coefficient to ensure an identifiable Bayesian model and enforce to stochastically search for an optimal low-dimensional parameter subspace. We then rank the j-th additive effect based on max{w _{β j+}, w _{β j-}}, and rank the j-th epistatic effect based on max{w _{γ j+}, w _{γ j-}}. The top c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ out of p _β additive effects, and the top c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ out of p _γ epistatic effects are selected, respectively. At the second step, we select variables out of those selected c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ additive effects and c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGUbGBaSqabaaaaa@2D55@ epistatic effects, under the above Bayesian framework for model (1). Obviously, we have a non-restrictive prior for each coefficient at the second step, and therefore avoid possible over-penalization due to restrictive priors.

Following Jeffreys 49 50 , we test the hypothesis H ₀ : β _j = 0 vs. H ₁: β _j ≠ 0 on the basis of the Bayes factor, which was defined as

B 10 ( β j ) = P ( D a t a | β j ≠ 0 ) P ( D a t a | β j = 0 ) = P ( β j ≠ 0 | D a t a ) P ( β j = 0 | D a t a ) × π ( β j = 0 ) π ( β j ≠ 0 ) = w β j + + w β j − 1 − w β j + − w β j − , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@A202@

where π (β _j = 0) and π (β _j ≠ 0) are the a priori probabilities, and the last equality follows the fact that π (β _j = 0) = π (β _j ≠ 0) at the second step of our Bayesian Classification. As suggested by Jeffreys 50 , a B ₁₀ (β _j ) with value between 1 and 10 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqaIXaqmcqaIWaamaSqabaaaaa@2DCE@ ≈ 3.2 provides "not worth more than a bare mention" evidence against H ₀; a B ₁₀ (β _j ) with value from 10 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqaIXaqmcqaIWaamaSqabaaaaa@2DCE@ to 10 provides "substantial" evidence against H ₀; a B ₁₀ (β _j ) with value from 10 to 100 provides "strong" evidence against H ₀; and a B ₁₀ (β _j ) with value larger than 100 provides "decisive" evidence against H ₀. Similarly, we can test the hypothesis H ₀: γ _j = 0 vs. H ₁: j ≠ 0 using the following Bayes factor

B 10 ( γ j ) = w γ j + + w γ j − 1 − w γ j + − w γ j − . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOqai0aaSbaaSqaaiabigdaXiabicdaWaqabaGccqGGOaakcqaHZoWzdaWgaaWcbaGaemOAaOgabeaakiabcMcaPiabg2da9KqbaoaalaaabaGaem4DaC3aaSbaaeaacqaHZoWzcqWGQbGAcqGHRaWkaeqaaiabgUcaRiabdEha3naaBaaabaGaeq4SdCMaemOAaOMaeyOeI0cabeaaaeaacqaIXaqmcqGHsislcqWG3bWDdaWgaaqaaiabeo7aNjabdQgaQjabgUcaRaqabaGaeyOeI0Iaem4DaC3aaSbaaeaacqaHZoWzcqWGQbGAcqGHsislaeqaaaaacqGGUaGlaaa@5072@