X_i \Omega_0 X_i , \\
& X_i \sim N(0,\Lambda_0 \Lambda_0 ^T + \Psi_0), \end{aligned}$$ where $\beta_0$ and $\Omega_0 $ are functions of $\Theta_0$ as in *Proposition 1*, and the true number of factors is $k_0$. When $k_0 = p$, we can write any covariance matrix as $\Lambda_0 \Lambda_0^T + \Psi_0$. We take an “over-fitted” factor modeling approach, related to [@Bhattacharya2013], [@rousseau2011asymptotic], and choose $k$ to correspond to an upper bound on the number of factors. In practice, we recommend the rule of thumb that chooses $k$ such that $\frac{\sum_{j = 1}^k v_j}{\sum_{j = 1}^p v_j} > 0.9$, where $v_j$ is the $j^{th}$ largest singular value of the correlation matrix of $X$. We have found this choice to have good performance in a wide variety of simulation cases. However, there is nonetheless a potential concern that $k$ may be less than $k_0$ in some cases. *Proposition 2* quantifies the distance in terms of Kullback-Leibler divergence between the true data generating model and the likelihood under model miss-specification as $n$ approaches infinity.

Fix $\Lambda_0,\Psi_0 = s_0 I_p$, $k_0$, and assume that $k<k_0$. As $n$ increases the posterior distribution of $\Lambda$ and $\Psi = s I_p$ concentrates around $\Lambda^*$ and $\Psi^*$, satisfying: $$\begin{aligned}
