"proposal" starting points: pool prior draws until enough land in support

A single batch of n_chains prior draws is finite only with probability acceptance^n_chains, so a posterior that is entirely ordinary but excludes a third of the prior's support (acceptance 0.65) already fails almost every batch: 0.65^20 is about 3e-4. Requiring one batch to succeed whole conflates that low-but-fine acceptance rate with true degeneracy. Drawing a pool per attempt and keeping every finite draw across attempts, the way "resample" already pools, fixes this: the chance of collecting n_chains finite draws now grows with the total number sampled, not with the size of one lucky batch.

Usage

mcmc_init_proposal(prior, log_prob_fn, n_chains, n_pool = 1000L)

Details

Up to 20 pools of max(n_pool, n_chains) prior draws are drawn (n_pool defaults to 1000, so 20000 draws by default), stopping as soon as enough finite ones have accumulated. If the budget runs out first, the error distinguishes two cases: no draw at all landed in support, which is consistent with a degenerate surrogate, versus some draws did but not enough, which is just a low acceptance rate and should not be reported as degeneracy.