Source
- Griffiths & Kalish (2007), LIT-769, read in NOTE-599: Eqs. 25–36 (the prior and the prior predictive are stationary), §4.2 (the Gibbs-sampler identification), §5.1 and Table 1 (the MAP case), §7 (populations).
- Kirby, Cornish & Smith (2008), LIT-771, skimmed in NOTE-596: human diffusion chains, cited for what they do and do not test.
The claim
Let every learner in a chain share a hypothesis space H, a prior P(h) and a likelihood equal to the production distribution P(d|h). Let each learner see data from the previous learner, sample a hypothesis h from P(h|d), and produce data from P(d|h). Then the sequence of (h, d) pairs is a Gibbs sampler for P(d|h)P(h). If the chain is ergodic, the probability that the n-th learner holds hypothesis h converges to P(h), and the distribution of the data it produces converges to the prior predictive Σ_h P(d|h)P(h). The amount of data passed between learners affects only the speed of convergence, not where it ends.
When learners instead take the maximum a posteriori hypothesis, the chain is stochastic EM with the transmitted data as the latent variable and no observations. Its stationary distribution concentrates near the prior’s mode, and, unlike the sampling case, depends on how much data is passed and how noisy it is. In the two-language case it is identical for every prior strength α between 0.5 and 1 − ε, so a weak bias can give the same outcome as a strong one.
What was actually shown
For sampling, a proof by substitution: the prior satisfies θ = Qθ for the transition matrix on hypotheses, and the prior predictive satisfies ρ = Rρ on data, for any production algorithm. It could have failed: a learning rule other than posterior sampling (MAP, or a likelihood that does not match production) does not, in general, leave the prior stationary, and the paper’s MAP results show such a departure in closed form. In the 260-hypothesis compositionality simulation, long-run frequencies under sampling matched the prior at every data size m from 1 to 10, while MAP chains departed from it (no compositional languages at all with α = 0.01; only compositional ones at α = 0.5 with m ≤ 2). Under equal fitness, the same distribution is the stable equilibrium of an unbounded population.
What this does not say
- It does not say human languages, or human transmission chains, converge to human priors. That is QUESTION-137. That depends on whether people learn by something like posterior sampling, which the source calls unexplored for language, and on all learners sharing one prior and one likelihood matched to production.
- It does not say the bottleneck is irrelevant to the emergence of structure. Under sampling it is irrelevant to the endpoint. Under MAP, which covers the minimum-description-length learners of earlier iterated learning simulations, the amount of data passed on changes the outcome.
- It does not say structure is innate. The prior is whatever makes a hypothesis easier to adopt, from any source; the source declines the innate reading.
- It does not cover selection. With unequal fitness, or a filter on what is transmitted, the prior need not be stationary. Kirby, Cornish & Smith’s Experiment 2, LIT-771, applies exactly such a filter (homonyms removed from training), so its compositional outcome is neither a confirmation nor a refutation of this claim.
- Kirby, Cornish & Smith do not test it. Their unfiltered chains lose colour distinctions in every chain, an outcome shaped by learner bias as the claim would lead one to expect, but no participant’s prior was measured, and the paper does not use the Bayesian analysis.
- It does not extend by itself to stories, norms or beliefs. The source’s closing extrapolation to “legends, religious concepts, and social norms” is not analysed there. Bartlett’s serial reproduction of a folk tale (LIT-768) is the classical case it would have to meet, and the record has not read Bartlett.