Q:

Mixture of Poisson Additive Regression Model

This is for a personal project (Just learnt EM algorithm and GAM model; and thinking about combine them together)
Believing that these data belongs to different clusters (where the labels are missing), and I want to do regression on them. Similar to mixture of expert that each regression take up its own data space
Now, consider we have a much more complicated nonlinear data, instead of doing mix of simple linear regression, I'm thinking doing mix of Poisson regression with additive model
So I'm thinking about mixture of regression using EM.The latent variable is the missing labels. So if I choose Poisson Regression with monomial basis functions
Latent variable is Z, poisson counts is Y, input data is X, d is dimension of each data.
So the likelihood would be $P(x) = \sum_k \alpha_k Poi(\lambda_k)$ where $\alpha_k$ is the weight for k-th cluster density
==============Updated================
Initialization: uniform outer weight $\alpha_k$ for each of GAM;  naive guess for each GAM coefficient $\beta$; where $GAM_k = \alpha_k(\beta_0 + \sum_{d=1}^D \beta_d \times f_{d,k}(x)$) 
Repeat:
(1) for each data point, I compute the posterior of responsibility $p(Z=z|X) = \frac{\alpha_z Poi(\lambda_z)}{\sum_k \alpha_k Poi(\lambda_k)}$
(2) update outer weight 
$\alpha_z = \frac{\sum_x P(Z=z|x)}{N}$ 
(3) for each GAM: backfitting algorithm update coefficient to get a new $GAM_k'$. Then $GAM_k \leftarrow P(Z=z|x) \times GAM_k'$
========================================
Is this the approach you suggested?
Any suggestions

A:

First, there should be a separate GAM for each Poisson mean. You can then fit each GAM using the latent probabilities as weights, or simply fit the GAM on the subset of the data $$S=\{x\in X: \mathop{\mathrm{argmax}}_i P(Z=i|X=x)=z\}$$ Rinse and repeat.

