Asking for help with a probability question for "learning dyadic data"

Question

I am reading a paper "Learning from dyadic data" written by Thomas Hofmann, Jan Puzicha and Michael I. Jordan, which lays a foundation for a later paper that proposes the famous probabilistic latent semantic analysis model. However, I find myself very confused with one of the formulas in the paper. There is a public version of this paper which can be found here.

Dyadic data analysis can be simplified as observing the co-occurrence between two sets $X = ({x_1},{x_2},...,{x_n})$ and $Y = ({y_1},{y_2},...,{y_m})$, which forms a sample $S = \{ ({x_{i_1}},{y_{k_1}}),({x_{i_2}},{y_{k_2}}),...,\}$ where ${i_1},{i_2},... \in \{ 1,...,n\} $ and ${k_1},{k_2},... \in \{ 1,...,m\} $. Then we define $c$ as the unobserved classification of $X \times Y$, which is mathematically a mapping from $X \times Y \to \{ {c_1},...,{c_K}\} $, where ${c_1},...,{c_K}$ are classes.

There are two assumptions:

1. Observations of sample $S$ are i.i.d., which means $P(S) = P({x_{{i_1}}},{y_{{k_1}}}) \times P({x_{{i_2}}},{y_{{k_2}}}) \times .... = \prod\limits_{i,k} {{{[P({x_i},{y_k})]}^{n({x_i},{y_k})}}} $ where ${n({x_i},{y_k})}$ is the number of occurrences of $({x_i},{y_k})$

2. Given a class ${c_\alpha }$, ${x_i},{y_k}$ are conditionally independent, which means $P({x_i},{y_k}|{c_\alpha }) = P({x_i}|{c_\alpha }) \times P({y_j}|{c_\alpha })$.

Now here comes the problem. The paper presents the following formula without providing detailed reason. $P(S,c) = \prod\limits_{i,k} {{{[P({c_{ik}})P({x_i}|{c_{ik}})P({y_k}|{c_{ik}})]}^{n({x_i},{y_k})}}} $ (1) where $c$ is a classification (mapping from $X \times Y$ to a class in $\{ {c_1},...,{c_K}\} $) and ${c_{ik}} = c({x_i},{y_k})$

By my calculation, $P(S,c) = P(S|c)P(c)$ (2) where $P(S|c) = \prod\limits_{i,k} {{{[P({x_i},{y_k}|c)]}^{n({x_i},{y_k})}}} = \prod\limits_{i,k} {{{[P({x_i}|c)P({y_k}|c)]}^{n({x_i},{y_k})}}} $. Because only ${c_{ik}}$ is not independent from $({x_i},{y_k})$, and as a result $P(S|c) = \prod\limits_{i,k} {{{[P({x_i}|{c_{ik}})P({y_k}|{c_{ik}})]}^{n({x_i},{y_k})}}} $ (3).

Comparing (1) (2) and (3) leads to $P(c) = \prod\limits_{i,k} {{{[P({c_{ik}})]}^{n({x_i},{y_k})}}} $. That does not make sense because $P(c)$ is depending on the sample!!!!!

Where am I going wrong or I misunderstood something? I have been struggling with this for half a day...

The following is the screen capture of part of the paper relevant to my question.

Answer 1

You are right. The paper is describing the wrong model in (1). In order to get (2), the model should be \begin{align} P(S,c) &= \prod_z P(c = c_z) P(x = x_z | c = c_z) P(y = y_z | c = c_z) \\ P(S) &= \prod_z \sum_\alpha P(c = c_\alpha) P(x = x_z | c = c_\alpha) P(y = y_z | c = c_\alpha) \\ &= \prod_{ik} (\sum_\alpha P(c = c_\alpha) P(x = x_i | c = c_\alpha) P(y = y_k | c = c_\alpha))^{n(x_i, y_k)} \end{align} where $z$ indexes the observations. The footnote of this page hints at the inconsistency.