Source
Blei, Griffiths and Jordan (2010), LIT-1542, §6.1 and Fig. 6, read in full in NOTE-1176 from arXiv 0710.0845v3.
What was actually shown
Ten data sets were drawn from hLDA, each with 100 documents of 250 words, a vocabulary of 100, η = 0.005 and γ = 1, with the levels truncated at three and a Dirichlet over the three level proportions. For each, the posterior mode over trees and allocations was approximated by the highest-probability state of the collapsed Gibbs sampler (§5.5). In the paper’s words: “We exactly recover the correct hierarchies, with only two errors. In one case, the error is a single wrongly allocated path. In the other case, the inferred mode has higher posterior probability than the true tree structure (due to finite data).”
What could have come out otherwise: the sampler could have stuck in local modes, merged or split branches, or misallocated documents widely. In eight of ten it did none of these.
The second failure is of a different kind from the first. A sampler that finds a state of higher posterior probability than the truth has done its job. The posterior itself prefers the wrong tree, so with 100 documents the tree is not identified by the posterior mode in that case.
The authors give the condition in words: “In general we cannot expect to always find the exact tree. This is dependent on the size of the data set, and how identifiable the topics are. Our choice of small η yields topics that are relatively sparse and (probably) very different from each other. Trees will not be as easy to identify in data sets which exhibit polysemy and similarity between topics.”
What this does not say
- Not that hLDA’s trees are identified in general. The test used sparse, well-separated topics chosen to make recovery easy, and three levels. No condition is stated under which the tree is unique.
- Not that a tree found in real data is real. Data generated under the model’s own prior test the inference, not whether the structure exists in data the model did not generate.
- Not about deep trees, other hyperparameters, or larger corpora. One setting was run, ten times.
- Not a rate. Two errors in ten is a count, not an error rate with an interval.
Connections
- QUESTION-276. Its list of what would not answer it includes “Recovering the prior’s structure from data generated under that prior, which tests the inference, not the structure”. This is that test, carried out well. The question needs a world that did not assume the prior, which is T23’s “Option A” (CASE-165).
- CLAIM-203. Its note of 2026-10-26 draws the same lesson for latent-structure recovery.
- CLAIM-135. This is evidence for the first achievement on simulated data, and for the second only within the model and at this size. The case where the posterior prefers a wrong tree shows the second can fail while the sampler succeeds.