Suggest alternative implementation of transform interface in LatentDirichletAllocation.

### Describe the workflow you want to enable

The current transform method in LatentDirichletAllocation normalizes the $\gamma$ parameter from the paper. However, this normalization is not supported by the [original LDA paper](https://dl.acm.org/doi/pdf/10.5555/944919.944937), the [paper the current code is based on](https://proceedings.neurips.cc/paper/2010/file/71f6278d140af599e06ad9bf1ba03cb0-Paper.pdf), or [related paper](https://arxiv.org/pdf/1206.7051). Furthermore, $\gamma$ is a parameter for the Dirichlet distribution of $\theta$ (the per-document topic weight), and normalizing it lacks any natural interpretation.

### Describe your proposed solution

Proposing the topic (assignment) frequency $\bar{z}$ according to the paper [Supervised Topic Models by David M. Blei,](https://arxiv.org/pdf/1003.0783) as a metric to measure topic assignment to a document. 
$$\bar{z}:=(1/N)\sum_{n=1}^Nz_n$$
where $z_n$ is the topic assignment, $N$ is the number of words in the documents, the sum is over all words in the document (see equation 1 in the paper). By taking expectation over the variational
distribution $q$, one can compute such metric, 
$$\mathbb{E}(\bar{Z}) = \bar\phi=(1/N)\sum_{n=1}^N\phi_n$$
(see equation 12 in the paper)

Using the relation between $\gamma$ and $\phi_n$, one can see that 
$$\mathbb{E}(\bar{Z}) = (1/N)(\gamma-\alpha)$$
where $\alpha$ is the Dirichlet parameter of the prior document topic distribution (Attribute `doc_topic_pior_`)
This is the proposed implementation of `transform`. 

### Reasons for such implementation:

1. Having natural interpretation: 
- as suggested by David M. Blei, the first author of the original paper of Latent Dirichlet allocation, this metrics can be seen as (expected) topic (assignment) frequency of the document (see paper Supervised Topic Models). By averaging the topic assignment distribution over all every words in the documents, this metric give a better insight to the overall topic distribution in the document. 

- The [original paper of LDA, ](https://dl.acm.org/doi/pdf/10.5555/944919.944937) suggests an interpretation of  $\gamma-\alpha$ 

> Recall that the $i$ th posterior Dirichlet parameter $\gamma_i$ is approximately the $i$ th prior Dirichlet parameter $\alpha_i$ plus the expected number of words which were generated by the $i$ th topic.

therefore normalizing it by total number of words $N$ give a natural interpretation of 'fraction of document generated by $i$ th topic'

2. Naturally normalized: 
- as the paper suggested, this metric naturally sum to 1, one can also check this is indeed true with model trained with the current sklearn Latent Dirichlet allocation implementation (as long as `_e_step` converge). 

3. Independence from Model Variations:
-  Even though the paper isn't about the original Latent Dirichlet allocation implementation, it is closely related, and the interpretation of such metrics is independent of the model variations. 

4. Credibility:
- The paper received >2000 citation, published more than 10 years, should pass the bar. 

### Describe alternatives you've considered, if relevant

_No response_

### Additional context

P.S. This is the first time I try to raise a issue on a large public repo, please go easy on me :pra

COMMENT:
Yea a choice to have an unnormalised transform would be great, this is also suggested in another issues (https://github.com/scikit-learn/scikit-learn/issues/16643). 

I think changing the transform would make a bit different when doc_topic_prior is large, (say >0.5). I agree it won't make much a difference when doc_topic_prior is small (<=0.1) which is most of the case. 

Thanks for the feedback.

Labels: New Feature