LDA point estimates for topic mixtures and word distributions

Turn Gibbs counts into document and topic distributions

After Gibbs sampling, the count matrices hold the final topic assignments. The usual point estimates add the Dirichlet priors back in:

• Document-topic mixture: theta[d, k] = (n_dk + alpha) / (sum_k' n_dk' + K alpha).
• Topic-word distribution: phi[k, w] = (n_kw + beta) / (sum_w' n_kw' + V beta).

Implement lda_estimates(doc_topic, topic_word, alpha, beta), which returns (theta, phi).

• doc_topic is (D, K) with nonnegative counts. topic_word is (K, V) with nonnegative counts.
• theta has shape (D, K), and phi has shape (K, V). Each row of both sums to 1.

Constraints

• Shapes must be consistent (doc_topic has K columns and topic_word has K rows). Otherwise raise ValueError.
• alpha > 0 and beta > 0. Otherwise raise ValueError.
• Counts must be nonnegative. Otherwise raise ValueError.
• Do not modify the inputs.

Hints