information matrix with IFA models

> mxComputeReportDeriv() for some reason does not return the gradient in the attached example. Wondering if I did something wrong. Really I want the gradient at particular values of some item parameters that may not necessarily be at the MLE, but I'll worry about how to do that later.

My hunch is that the gradient is not returned because the gradient is only computed in the M-step and not preserved outside of mxComputeEM.

To get the gradient, take a look at inst/models/nightly/ifa-meat-2d.R

> “meat” = empirical cross-product (observed?)

I am confused about the distinction between observed and expected information. But yeah, "meat" is the empirical cross-product. Mislevy (1984) is implemented for the latent distribution.

Methods "bread" and "sandwich" are implemented according to Yuan, Cheng, & Patton (2013).

With respect to the EM related methods, "mr" and "tian" require the convergence history while "oakes1999" and "agile" do not.

You have covered all the options.

> “meat” = empirical cross-product (observed?)
> “bread” = Fisher information matrix (expected?)
> “sandwich” = same as the name implies (but, expected or observed?)
There are two different ways of defining the Fisher information: (1) the variance of the first derivative of the loglikelihood and (2) the expectation of the second derivative of the loglikelihood. The true Fisher information is generally unknown, because it requires knowledge of the true parameter values and the true functional form (i.e., family) of the distribution. If we assume a particular functional form of the distribution, and "plug-in" our MLEs in place of the true parameter values, we can estimate the true Fisher information from data. If the necessary integrals are analytically tractable, then we can estimate the true Fisher information as the expected information matrix. But, sometimes those integrals are not so easy, so we instead use the observed information as an estimator, and calculate it either as a quadratic function of the first derivative loglikelihood (proportional to the "empirical cross-product" or "meat") or as a first-degree function of the second-derivative loglikelihood (proportional to the Hessian matrix, or "bread"). When the distribution is correctly specified, those two ways of calculating the observed information will on average give the same answer in large samples. The "sandwich" is `solve(bread)%*%meat%*%solve(bread)`, and so if the distribution is correctly specified, then asymptotically, the product of the first two factors tends toward an identity matrix, and the sandwich merely tends toward the inverted Hessian. But if the distribution is incorrectly specified, it can be shown that the sandwich nonetheless provides a consistent estimator of the repeated-sampling covariance matrix of the MLEs.