Compare a sequence of INLA approximations to confirm accuracy.

[tbenthompson]

On a basic level, INLA is a scarier algorithm than MCMC. This is because an INLA user has one fewer tool to check correctness. With MCMC, you can jack up the sample count really high to check convergence. This is not possible with INLA. Obviously there are MCMC problems that still cause failures even with millions of samples, but such convergence tests help identify many classes of errors/convergence failures.

So, to feel more comfortable using INLA, we'd like to have better tools for predicting error. In an ideal world, there would even be an iterative method for improving the INLA posterior... but that doesn't currently exist.

An ideal error predictor would be less expensive than INLA itself while simultaneously providing a useful upper bound on error.

Ideas here:

• Compare the hessian with the tensor of third derivatives at the mode. Perhaps only looking at the diagonal of each in order to keep the comparison tractable? How expensive is it to compute higher order derivatives using automatic differentiation? Could we even compute 4th or 5th derivatives?
• The quadratic approximation at the mode provides predictions for the density under approximation at other points. Compare this prediction with the reality. Is there some kind of very cheap sampling we could with this idea that would provide a useful error estimate/integral?

[tbenthompson]