Consider a random sample $X_i = (U_i,V_i)$ where $i=1,2,...,n$ from a bivariate normal population with mean $(\mu_1,\mu_2)$ and variances $(\sigma_1 ^2, \sigma_2 ^2)$ and correlation $\rho$. Let's consider random variables $C_1,...,C_n$ which take the values $\{0,1,2\}$ and are independent of $X_i$. We observe each $C_i$ Now, if $C_i=0$, we observe $X_i$; if $C_i=1$, we observe only $U_i$; and finally if $C_i=2$, we observe $V_i$. I want to develop an EM algorithm to estimate the parameters of the distribution of $X_i$ given data.
My approach:
Let us define $Z_i = X_i$ if $C_i=0$ say, with probability ($\pi_1$), $Z_i=U_i$ if $C_i=1$ with probability $\pi_2$ and $Z_i=V_i$ if $C_i=2$ with probability $\pi_3$. Since, we observe both $Z_i,C_i$, the observed data likelihood is: $$L = \prod_{i=1}^{n} (\pi_1 f_{X}(x_i) + \pi_2 f_{U}(u_i) +\pi_3 f_{V}(v_i))$$.
In general, for EM algorithm we are required to find for the E-step: $E[\text{unobserved data}|\text{observed data}]$, but I am a bit confused with guessing what that should be in this case. In this case, if we observe $C_i=0$ then both $U_i,V_i$ are observed. However, if we observe $C_i=1$, then only $V_i$ is unobserved and for $C_i=2$, then $U_i$ is unobserved. So, does that mean our E step should deal with $E[V_i|U_i]$ and $E[U_i|V_i]$?
Can anyone help me out with this problem?