[*Latent class analysis*][1] (LCA) is a discrete [*finite mixture model*][2]. Finite mixture model is a model-based clustering algorithm, that treats the distribution of the data $f$ as a mixture of $k$ distributions $f_k$, each appearing with mixing proportion $\pi_k$,

$$
f(x, \vartheta) = \sum^K_{k=1} \pi_k \, f_k(x, \vartheta_k)
$$ 

where the class assignments (clusters) are unknown and learned from the data. In case of LCA, the variables are discrete, so the aim is to cluster the discrete data into $K$ latent classes, each characterized by different conditional probability distribution. For two discrete variables $A$ and $B$, and latent variable for the class assignment $X$, the distribution may be defined as

$$
P(A=i, B=j) = \sum_{k=1}^K \, \overbrace{P(X=k)}^{\pi_k} \, \overbrace{P(A=i, B=j|X=k)}^{f_k}
$$

where, to simplify the computations, it is often assumed that the variables are independent $P(A=i, B=j|X=k) = P(A=i|X=k)\,P(B=j|X=k)$. What may be confusing, is that the LCA literature quite commonly uses rather peculiar notation, where:

$$
P(A=i, B=j) = \sum_{k=1}^K \, P(X=k) \, P(A=i|X=k)\, P(B=j|X=k)
$$

can be written as something like below, or variants of it:

$$
\pi_{ij} = \sum_{k=1}^K \, \pi^X_k \, \pi^{\bar A X}_{ki} \, \pi^{\bar B X}_{kj} 
$$

For learning more, there is nice introduction with examples in the documentation of *poLCA* R package (Linzer and Lewis, 2011), and brief tutorial by Vermunt and Magidson (2003). There is a big variety of latent class analysis models, you can find extended review in Hagenaars and McCutcheo (2009).

> Hagenaars J.A. and McCutcheon, A.L. (2009). *Applied Latent Class
> Analysis.* Cambridge University Press.
> 
> Vermunt, J.K., and Magidson, J. (2003). [Latent class models for
> classification.][3] *Computational Statistics & Data Analysis, 41*(3),
> 531-537.
> 
> Linzer, D. A., and Lewis, J. B. (2011). [poLCA: An R package for
> polytomous variable latent class analysis.][4] *Journal of statistical
> software, 42*(10), 1-29.


  [1]: https://stats.stackexchange.com/questions/122213/latent-class-analysis-vs-cluster-analysis-differences-in-inferences/134346#134346
  [2]: https://stats.stackexchange.com/questions/130805/are-there-any-non-distance-based-clustering-algorithms/130810#130810
  [3]: https://www.statisticalinnovations.com/wp-content/uploads/Vermunt2003_csda.pdf
  [4]: https://www.jstatsoft.org/article/view/v042i10