content_image/327033.jpg content_image/327034.jpg content_image/327035.jpg content_image/327037.jpg content_image/327036.jpg
## 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 

<img src='content_image/327033.jpg'>

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 

<img src='content_image/327034.jpg'>

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: 

<img src='content_image/327035.jpg'>

where 

and 

<img src='content_image/327037.jpg'>

<img src='content_image/327036.jpg'>

The summation over j ′ in both the expressions runs through 1 ≤ j ′ ≤ J where j ′ ̸ = j . 

