Similar, but different.
$$Z \sim \rr{Beta}(\alpha,\beta), \rr{ } Y_i \given Z \sim \rr{Bernoulli}(L_i)$$
$$\begin{align}L(\{ \alpha, & \beta\}\given Y, Z) \\ &\propto \prod_{i=1}^n\frac{1}{\rr{B}(\alpha,\beta)}\, z^{y_i + \alpha-1}(1-z)^{1 - y_i + \beta-1}\end{align}$$
Problems with this model?
$$\begin{align}p(\{ \alpha, & \beta, Z\}\given Y) \\ &\propto \prod_{i=1}^n\frac{1}{\rr{B}(\alpha,\beta)}\, z^{y_i + \alpha-1}(1-z)^{1 - y_i + \beta-1}\end{align}$$
Still suffers from the same problems. Solution: set \\( \alpha, \beta \\) to some values (hyperparameters).
Set \\( \alpha =1, \beta = 1 \\).
$$\begin{align}p(Z\given Y) \propto \prod_{i=1}^n z^{y_i}(1-z)^{1 - y_i}\end{align}$$
This looks like the likelihood function if \\( Z \\) is a parameter instead of a random variable.
$$\begin{align}L(& \{ \alpha, \beta\}\given Y, Z) \\ &\propto \prod_{i=1}^k\prod_{j=1}^{n_i}\frac{1}{\rr{B}(\alpha,\beta)}\, z_i^{y_{ij} + \alpha-1}(1-z_i)^{1 - y_{ij} + \beta-1}\end{align}$$
$$\begin{align}L(\theta & \given Y_{\rr{obs}}, M ) \propto f ( Y_{\rr{obs}}, M \given \theta) \\ & = \prod_{i=1}^r f(y_i, M_i \given \theta) \cdot \prod_{i=r+1}^n f(M_i \given \theta) \\ & = \theta^{-r} exp \left( -\sum_{i=1}^r\frac{y_i}{\theta}\right)exp \left( -\frac{(n-r)c}{\theta}\right)\end{align}$$
Complete-data, or full likelihood $$L(\theta \given Y_{\rr{obs}}, Y_{\rr{mis}}) \propto f_\theta\left(y_{\rr{obs}},y_{\rr{mis}}\right)$$
Observed-data likelihood $$L(\theta \given Y_{\rr{obs}}) \propto \int f_\theta\left(y_{\rr{obs}},y_{\rr{mis}}\right)dy_{\rr{mis}}$$
Ignorable missing data mechanism: can use observed-data likelihood.
Fraction of missing data
Large
Small
Latent variables
Posit a model
Never happens
Missing data
Model-sensitive, area of current research
Sensitive but manageable
$$\begin{align} L(\theta \given Y, Z ) \propto & \left(\rr{Bern}(y_1|p_l)\right)^{z_1}\left(\rr{Bern}(y_1|p_s)\right)^{1-z_1} \\ \prod_{i=2}^n & \left(\rr{Bern}(y_i|p_l)\right)^{z_i}\left(\rr{Bern}(y_i|p_s)\right)^{1-z_i} \\ & \cdot T(z_i \given z_{i-1})\end{align}$$
$$\begin{align} X_i & = \begin{cases}Z_i &\rr{if observation }i\rr{ is missing,} \\ Y_i &\rr{if observation }i\rr{ is present}\end{cases} \\ X_{i} | X_{r(i)} & \sim \rr{N}\left(\rho_i X_{r(i)} + (1 - \rho_i) \mu, (1 - \rho_i^2) \sigma^2 \right)\\ X_i & \sim \rr{N} (\mu, \sigma^2) \rr{ if } i \rr{ is a seed} \end{align}$$
$$\begin{align}L(&\{ \mu, \rho, \sigma^2\} \given Y, Z) \propto \prod_{\{\rr{seeds } j\}} \frac{1}{\sqrt{\sigma^2}}e^{\frac{(x_j - \mu)^2}{2\sigma^2}} \cdot \\ & \prod_{\{\rr{referred }i\}} \frac{1}{\sqrt{(1-\rho^2)\sigma^2}}e^{\frac{(x_i - (\rho x_{r(i)} + (1 - \rho) \mu))^2}{2\sigma^2}}\end{align}$$
Is this complete or observed-data likelihood?
Assumptions?
Issues to consider
Rater \\( i\\), window \\( j \\), winning object \\( k \\).
Number of preference classes \\( N \\), number of objects/window \\( L \\).
$$Z_i \given \vec{p_i} \sim \rr{Multinom}_N(1, \vec{p_i})$$
$$C_{ij} \given Z_i, X \sim \rr{Multinom}_L(1, \vec{w}_{ij})$$
with
$$w_{ijl} \propto \rr{exp} \left\{ X_{jl} \beta_i\right\}$$
$$\begin{align} L& \left(\left\{ \vec{\beta}_n\right\} \given \left\{Z_{in}, C_{ijk}\right\} \right) \\ & \propto \prod_{i=1}^I \prod_{j \in J_i} \prod_{k \in K_{ij}} \prod_{n=1}^N \left( \frac{exp\left\{\vec{x}_{jk}' \vec{\beta}_n \right\}} {\sum_{l = 1}^{L_j} exp \left\{ \vec{x}_{jl}' \vec{\beta}_n \right\}} \right)^{Z_{in}} \end{align}$$