LDA collapsed Gibbs topic conditional

The topic distribution for one word, given all other assignments

Latent Dirichlet Allocation (Blei, Ng and Jordan, 2003) explains each document as a mixture of topics and each topic as a distribution over words. Collapsed Gibbs sampling (Griffiths and Steyvers, 2004) resamples the topic of one word token at a time. For that token, with its own assignment already removed from the counts, the conditional is

p(z = k | rest) proportional to (n_dk + alpha) * (n_kw + beta) / (n_k + V beta).

Here n_dk counts the other tokens in the document assigned to topic k, n_kw counts occurrences of word w in topic k, n_k is the total count in topic k, and V is the vocabulary size.

Implement topic_conditional(doc_counts, topic_word, topic_totals, alpha, beta, word).

• doc_counts is (K,). topic_word is (K, V). topic_totals is (K,). word is an index in 0..V-1.
• Return the normalized probability vector of length K.

Constraints

• Shapes must be consistent. Otherwise raise ValueError.
• alpha > 0 and beta > 0. Otherwise raise ValueError.
• Counts must be nonnegative. Otherwise raise ValueError.
• word must be in range. Otherwise raise ValueError.
• Do not modify the inputs.

Hints