  The inherent bias pathology of the maximum likelihood (ML) estimation method
is confirmed for models with unknown parameters $\theta$ and $\psi$ when MLE
$\hat \psi$ is function of MLE $\hat \theta.$ To reduce $\hat \psi$'s bias the
likelihood equation to be solved for $\psi$ is updated using the model for the
data $Y$ in it. Model updated (MU) MLE, $\hat \psi_{MU},$ often reduces either
totally or partially $\hat \psi$'s bias when estimating shape parameter $\psi.$
For the Pareto model $\hat \psi_{MU}$ reduces also $\hat \psi$'s variance. The
results explain the difference that puzzled R. A. Fisher, between biased $\hat
\psi$ and the unbiased estimate he obtained for two models with the "2-stage
procedure". MUMLE's implementation is equivalent to the abandoned 2-stage
procedure thus justifying its use. MUMLE and Firth's bias correcting likelihood
are also obtained with the Minimum Message Length method thus motivating its
use in frequentist inference and, more generally, model updating with a prior
distribution.
