of“complexity” of inequality-constrained Bayesian models as the admissible probability mass]. For this reason, as the variance of the estimates increases, so does the the probability of so-called “inadmissible” sample estimates of the parameter vector. This holds regardless of whether the “population parameter vector” – the solution to which Equation \[eq:objective\] convergences as $n$ grows without bound – is not itself “inadmissible”. A separate case is misspecification of the model, which is the most widely-recognized cause of “inadmissible” population parameter vectors [@chen_improper_2001]. Here we will ignore such cases, and assume the model is correctly specified. We will see that even in this idealized situation, severe problems with estimation can occur when there are rank-deficient points in the Jacobian.

Under the assumption of a correctly specified model, variance of SEM estimates $\hat{\boldsymbol{\theta}}_n$ is obtained, through standard likelihood theory, as the inverse Fisher information, $$\label{eq:variance}
    \text{Asy.}\text{Var}(\hat{\boldsymbol{\theta}}_n) = \left(\boldsymbol{\Delta}^T \mathbf{V} \boldsymbol{\Delta}\right)^{-1}.$$ Relevant theory and extensions to more general settings can be found in [@satorra_alternative_1989]. Again, since $\mathbf{V}$ is positive-definite by assumption, the inverse in Equation \[eq:variance\] will grow without bound as the Jacobian $\boldsymbol{\Delta}$ approaches singularity, causing bad performance in terms of MSE, as well as “inadmissible” estimates.

Some authors [e.g. @rindskopf_parameterizing_1983] have suggested 