# Should Bayesian estimated error smaller than MLE?

I am dealing with a fitting problem. Specifically, I am fitting a Lorentzian profile to the power spectrum of an solar-like oscillating star. Three parameters in the Lorentzian profile characterize one oscillation mode, i.e. amplitude $A$, linewidth $\Gamma$, and frequency centroid $\nu_0$ (See Anderson et al. 1990, Astrophysical Journal v.364, p.699 for a detailed desciption of the model). The model writes as

$M(\nu)=\frac{A^2/(\pi\Gamma)}{1+4(\nu-\nu_0)^2/\Gamma^2}+B$,

where $B$ is a background constant that accounts for a baseline.

One can obtain parameter estimations from a classical Maximum Likelihood Estimator by maximizing the likelihood function (Ballot et al. 2008, Astronomy and Astrophysics, 486, 867-875, section 2.2),

$\ln L(\theta)=-\sum\left[\ln M(\theta;\nu) + \frac{D(\nu)}{M(\theta;\nu)}\right]$,

where $D$ is the power spectrum and the sum is over all frequency bins of the power spectrum. To obtain MLE errors (in this case, I only care about frequency centroid), one can compute the covariance matrix $\bf C=H^{-1}$, the inverse of the Hessian matrix and its diagonal elements give the errors $c_{jj}=\sigma_j^2$. Ballot et al. (2008, equation 8-10) gives an analytical form of this error.

I also calculated the errors in Bayesian framework by using posterior distribution obtained by MCMC to estimate parameters (medians) and their associated uncertainties (lower and upper credible). The priors of all free parameters are flat uniform distributions around the true values.

And I found the following quote from Lund et al. (2017, the Astrophysical Journal, 835:172, p13, para. 1).

It should be noted that the ML estimator (if unbiased) reaches the minimum variance bound, in accordance with the Cramér-Rao theorem (Cramér 1946; Rao 1945). Thus one should expect uncertainties at least as large as those from the ML estimator (MLE).

My question is, is this a true statement? In this article, the authors claim that their Bayesian MCMC errors are generally 1.3 times larger than those estimated by MLE, which I found puzzling because the posterior probability is generally “thinner” than the likelihood and this should give a smaller variance. In my practice, my Bayesian errors are generally 1.3 times smaller than the MLE errors and I don't know what goes wrong.

Thanks in advance for any help.

• Would give a few more details? For example, what is "the" ML estimator? Do you mean any ML estimator or just one particular estimator? Are only uninformative priors considered? The link to the article is iopscience.iop.org/article/10.3847/1538-4357/835/2/172/pdf. – JimB Apr 22 '18 at 4:26