3 Parameter Estimation Let θ = ( B , Φ , π ) denote the parameters of interest where π = ( π$_{α}$ : α ∈ { 0 , 1 } $^{K}$). The current study does not impose any particular form on π$_{α}$ ; instead, it regards π as one of the model parame- ters to be estimated together with B and Φ . Let θ ( t ) = ( B ( t $^{)}$, Φ ( t $^{)}$, π ( t $^{)}$) be the provisional estimate of θ obtained at the t -th iteration. The likelihood function of the unknown parameters is obtained by integrating out the nuisance parameter from the joint probability distribution f ( x , α | θ ). The corresponding objective function is expressed as where x is the observation from a single subject. The expectation of the complete-data log-likelihood with respect to the conditional distribution of α given x is obtained as where L ( θ | x , α ) is the complete-data likelihood for a single subject. The support set in the summation over α runs through { 0 , 1 } K and is omitted for notational simplicity. As alluded to above, evaluating f ( x , α | θ ) is computationally demanding due to the intractable normalizing constant. Hence, we consider the pseudo-likelihood function for optimization: Q ( θ | θ ( t $^{)}$) = E $^{[}$log f ( x , α | θ ) ∣ $_{∣}$x , θ ( t ) $^{]}$, E $^{[}$log L ( θ | x , α ) ∣ $_{∣}$x , θ ( t ) ] = ∑ α f ( α | x , θ ( t $^{)}$) log f ( x , α | θ ) , (5) L ( θ | x , α ) ≜ J ∏ j =1 f ( x$_{j}$, α | x$_{−}$$_{j}$ , θ ) = J ∏ j =1 f ( x$_{j}$ | x$_{−}$$_{j}$ , α , θ ) f ( α | x$_{−}$$_{j}$ , θ ) , (6) f ( x$_{j}$ | x$_{−}$$_{j}$ , α , θ ) = exp $^{(}$x$_{j}$ β ⊤ j α + ∑ j $^{′}$̸ = j φ$_{jj}$ ′ x$_{j}$x$_{j}$ ′ ) 1 + exp $^{(}$β ⊤ j α + ∑ j $^{′}$̸ = j φ$_{jj}$ ′ x$_{j}$ ′ ) , f ( α | x$_{−}$$_{j}$ , θ ) = π$_{α}$ exp ( ∑ j $^{′}$̸ = j x$_{j}$ ′ β ⊤ j $_{′}$α $^{) (}$1 + exp $^{(}$β ⊤ j α + ∑ j $^{′}$̸ = j φ$_{jj}$ ′ x$_{j}$ ′ )) ∑ ˜ α π$_{˜}$ $_{α}$exp ( ∑ j $^{′}$̸ = j x$_{j}$ ′ β ⊤ j ′ ˜ α $^{) (}$1 + exp $^{(}$β ⊤ j ˜ α + ∑ j $^{′}$̸ = j φ$_{jj}$ ′ x$_{j}$ ′ $_{))}$. where and The summation over j ′ in both the expressions runs through 1 ≤ j ′ ≤ J where j ′ = j . 9