normal-theory standard errors are arbitrarily large or cannot be calculated, so that the user cannot tell that the estimates were accurate.

To illustrate that these problems are not specific to the estimation method, or to maximum-likelihood, I ran the same model in the Hamiltonian Monte Carlo (HMC) sampler `stan`, a popular package for Bayesian modeling. The HMC chains in the bottom panel of Figure \[fig:shapiro\] show poor mixing and exhibit bias due to the constraint that estimates should be positive. In other words, the “admissibility” of solutions has been traded for bias. This demonstrates that Bayesian estimation can solve the problem of inadmissibility only at the cost of a bias that does not go to zero as the sample size grows. In addition, those draws that are actually near the rank-deficient true value are marked by `stan`’s HMC sampler as divergences. Again, the user would be warned off these most accurate estimates.

![Empirical estimates of the parameter involved in Shapiro’s rank deficiency at large samples. Yellow (light color) indicates “inadmissible” estimates ($\hat{\psi}_3 < 0$). Top: maximum-likelihood estimates. Bottom: a single Bayesian trace plot produced by `stan`. []{data-label="fig:shapiro"}](shapiro-ml.pdf "fig:") ![Empirical estimates of the parameter involved in Shapiro’s rank deficiency at large 