I’ve been very reticent to comment on the current AI hype train, mostly because it’s very easy to be wrong in a bull market. But you will be relieved to know I am now ready to put my hat in the ring.
First I would like to say that, like everyone else, I’m deeply impressed by recent advances in AI, particularly large language models (LLMs). They can give you flawless explanations of topological quantum field theory in one breath, then seamlesly switch to a nuanced discussion of the history of cartography. Tailor-made answers to your burning questions on heraldry, hermeneutics, object-oriented programming, epistemology, and just about any other domain of expert human knowledge your heart desires. They’re really quite amazing. I would be extraordinarily proud if I had been part of the teams that brought LLMs to the world.
Case control studies are a staple in the epidemiologist’s toolbox. The basic idea idea is that instead of carrying out an analysis on a random sample of a population, you instead sample from the population conditional on occurrence of an outcome of interest. Typically this will involve selecting all individuals who have the outcome (the cases), and only some people who don’t have the outcome (the controls). The motivation for doing this is that when the outcome is rare, most of the precision in estimates is driven by the cases, and adding a large number of controls often has little effect. Case control studies can thus be more efficient in terms of data collection/processing compared to e.g. a cohort study.
Imagine I offer you the chance to play the following game. The pot starts at \( $ 2\). I flip a coin repeatedly. Each time it comes up tails, the pot doubles. The first time a heads appears, the game ends and you leave with the pot. How much would you be willing to pay to play this game?
In this post, I’m going to give a brief critical overview of loop quantum gravity, the field that I worked in during my time in physics. A fairly strong understanding of physics is assumed.
Loop quantum gravity is an attempt at canonical quantisation of Einstein’s general relativity. Thus the starting point is constructing a Hamiltonian. In a flat spacetime that’s easy, but on a general spacetime manifold it’s not so easy because it requires one to define a time coordinate. The ADM formalism addresses this by assuming that spacetime is foliated. Roughly this means that our spacetime manifold \( M \) is decomposed into a family of spacelike hypersurfaces \( \Sigma_t \) labelled by a timelike coordinate \( t\). Spacelike means that any pair of points on \( \Sigma_t \) are spacelike separated - their spatial and temporal separation is such that light could not travel between them. There are in general many possible foliations of \( M \) that should all be ‘equally as good’ as each other.
Robust standard errors are frequently used in statistics, often in an unthinking way. They are certainly valuable and useful in the context of linear regression; however, they do not serve a similar purpose in maximum likelihood estimation of non-linear models. The basic issue is that the kind of mis-specification that robust standard errors can address in linear regresion does not spoil consistency results for the parameter estimates. On the other hand, it does spoil those results in the context of maximum likelihood estimation of non-linear models. Thus at best you end up with a consistent estimator for the variance of a parameter estimate that is itself inconsistent, which isn’t really of interest except as a diagnostic tool for detecting bad models. I’ll explore this in more detail below.