## Problem 069 — rank:131

**Title:** Horizontal-gene-transfer reconstruction

**Added binding assumptions (LaTeX):**

```latex
The factorization assumes gene-family conditional independence given the dated species history and rate model. Specify substitution branch lengths and uncertainty in the species tree; incomplete lineage sorting and unsampled donors are alternative causes of discordance. Posterior support depends on the stated prior and is not automatically a frequentist confidence guarantee. Report donor/time equivalence classes when the likelihood cannot separate them.
```

**Direct edits:** 0. **Fresh external checks:** None; supplied sources retained as provenance.