docs/numpy_ml.lda.rst
########################### Latent Dirichlet allocation ###########################
Latent Dirichlet allocation_ (LDA, commonly known as a topic model) is a
generative model for bags of words_.
.. _Latent Dirichlet allocation : https://en.wikipedia.org/wiki/Latent_Dirichlet_allocation
.. _bags of words : https://en.wikipedia.org/wiki/Bag-of-words_model
.. figure:: img/lda_model_smoothed.png :scale: 25 % :align: center
The smoothed LDA model with `T` topics, `D` documents, and :math:`N_d` words per document.
In LDA, each word in a piece of text is associated with one of T latent
topics. A document is an unordered collection (bag) of words. During
inference, the goal is to estimate probability of each word token under each
topic, along with the per-document topic mixture weights, using only the
observed text.
The parameters of the LDA model are:
- :math:`\theta`, the document-topic distribution. We use
:math:`\theta^{(i)}` to denote the parameters of the `categorical`_
distribution over topics associated with document :math:`i`.
- :math:`\phi`, the topic-word distribution. We use :math:`\phi^{(j)}` to
denote the parameters of the `categorical`_ distribution over words
associated with topic :math:`j`.
.. _categorical : https://en.wikipedia.org/wiki/Categorical_distribution
The standard LDA model [1]_ places a Dirichlet_ prior on :math:\theta:
.. math:: \theta^{(d)} \sim \text{Dir}(\alpha)
The smoothed/fully-Bayesian LDA model [2]_ adds an additional Dirichlet_ prior on :math:\phi:
.. math:: \phi^{(j)} \sim \text{Dir}(\beta)
.. _Dirichlet : https://en.wikipedia.org/wiki/Dirichlet_distribution
To generate a document with the smoothed LDA model, we:
1. Sample the parameters for the distribution over topics,
:math:`\theta \sim \text{Dir}(\alpha)`.
2. Sample a topic, :math:`z \sim \text{Cat}(\theta)`.
3. If we haven't already, sample the parameters for topic `z`'s categorical
distribution over words, :math:`\phi^{(z)} \sim \text{Dir}(\beta)`.
4. Sample a word, :math:`w \sim \text{Cat}(\phi^{(z)})`.
5. Repeat steps 2 through 4 until we have a bag of `N` words.
The joint distribution over words, topics, :math:\theta, and :math:\phi
under the smoothed LDA model is:
.. math::
P(w, z, \phi, \theta \mid \alpha, \beta) = \left( \prod_{t=1}^T \text{Dir}(\phi^{(t)}; \beta) \right) \prod_{d=1}^D \text{Dir}(\theta^{(d)}; \alpha) \prod_{n=1}^{N_d} P(z_n \mid \theta^{(d)}) P(w_n \mid \phi^{(z_n)})
The parameters of the LDA model can be learned using variational expectation maximization_ or Markov chain Monte Carlo (e.g., collapsed Gibbs sampling_).
.. _variational expectation maximization: https://en.wikipedia.org/wiki/Variational_Bayesian_methods
.. _collapsed Gibbs sampling: https://en.wikipedia.org/wiki/Gibbs_sampling#Collapsed_Gibbs_sampler
Models
~numpy_ml.lda.LDA~numpy_ml.lda.SmoothedLDAReferences
.. [1] Blei, D., Ng, A., & Jordan, M. (2003). "Latent Dirichlet allocation". Journal of Machine Learning Research, 3, 993–1022. .. [2] Griffiths, T. & Steyvers, M. (2004). "Finding scientific topics". PNAS, 101(1), 5228-5235.
.. toctree:: :maxdepth: 3 :hidden:
numpy_ml.lda.lda numpy_ml.lda.smoothed_lda