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Model and Properties
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Let $y_i$ denote a continuous health response for individual $i$, and $X_i = (x_{i1},\cdots,x_{ip})^T$ denote a vector of exposure measurements. We propose a latent factor joint model, which includes shared factors in both the predictor and response components while assuming conditional independence. We include interactions among latent variables in the response component. We also assume that, given the latent variables, the explanatory variables and the response are continuous and normally distributed. We assume that the data have been normalized prior to the analysis so that we omit the intercept. The model is as follows: $$\begin{aligned}
& y_i = \eta_i^T \omega + \eta_i^T \Omega \eta_i +\epsilon_{y,i},  \quad \epsilon_{y,i} \sim N(0,\sigma^2) \nonumber,  \\ 
& X_i = \Lambda \eta_i+ \epsilon_i,   \quad  \epsilon_i \sim N_p(0,\Psi), \\
& \eta_i \sim N_k(0,I), \nonumber\end{aligned}$$ where $\Psi = diag(\sigma_1^2,\cdots,\sigma^2_p)$. In a Bayesian fashion, we assume a prior for the parameters $\Theta = (\omega,\Omega,\Lambda,\Psi,\sigma^2)$ that will be specified in *Section 2.2*. Model $(1)$ is equivalent to classical latent factor regression models; refer, for example, to [@Mike03], except for the $\eta_i^T \Omega \eta_i$ term. Here, $\Omega$ is a $k \times k$ symmetric matrix inducing a quadratic latent variable regression that characterizes interactions among the latent 