Sam Power

Somehow, theorems about P_{\hat{\theta}} should 'always' be viable, not depend on parameterisations, etc., which seems ideal on the theory side.

Ah, sure; this isn't really one of those. This is coming out of teaching a statistical theory course, and trying to sound out what sort of theorems are "natural". Increasingly, I see that I want to prove things about e.g. P_{\hat{\theta}} rather than just \hat{\theta}, which is more usual.

*- any functional of the data-generating process is a parameter

A byproduct of this framing is that it (not necessarily beneficially) precludes the existence of certain types of non-identifiability.

My relationship with "parameters" in statistical models has passed through approximately three phases:

1) parameters are convenient ways of thinking about models
2) parameters are a red herring; focus directly on the data-generating process
3) parameters are good, because anything* is a parameter.

Let me be explicit in highlighting Rocco as leading the work on this project and doing an excellent job - very talented junior researcher, and well worth keeping on your radar for all the expected reasons.

A little part of the paper which I like a lot is to express the main algorithm (MTM) as an approximation to an approximation of a certain 'ideal' algorithm. Among other things, this 'twice-approximated' perspective helps to pin down which of the approximations is more problematic / delicate.

Some work freshly published at EJS: projecteuclid.org/journals/ele...

'Analysis of Multiple-try Metropolis via Poincaré inequalities'
- Rocco Caprio, Sam Power, Andi Q. Wang

We conduct a convergence analysis of a specific class of MCMC procedures based on multiple-proposal strategies.

Splendid!

It has a name (Cayley transform), but relevance to this problem is not a priori clear, e.g. doesn't clearly extend or generalise to cube roots.

Generalisations come relatively easily when they are also expressible as an LP / convex program (so e.g. maximising worst-case power over some set is good, while maximising best-case power is not as straightforward; adding in extra significance-type constraints is usually fine, etc.).

Basically, write the test in terms of an f mapping to [0, 1]; significance is an expectation of that f, as is the power. Hence, optimising power s.t. significance constraint (and constraining f to map to [0, 1]) is an LP, and the form of the optimiser is informed by those constraints.

Or maybe just linear programming by itself; need to think more carefully about whether the duality aspect is key. I guess it often is with LPs.

Fun thing to clarify for myself: the Neyman-Pearson Lemma is, at its core, an application of linear programming duality. Once this perspective clicks, it becomes clearer why certain extensions are and are not possible.

A fun thing which is not so widely known: consider the 'Newton ODE' for convex minimisation. Then g_t := grad f (x_t) satisfies a very simple universal ODE.

Very cool!

(Save yourself time in the calculations by taking a = 1!)

A simple calculation with a neat consequence:

Consider using Newton's method to solve

x^2 = a

for some a > 0.

The iterates x_n don't obviously lead to a solvable recurrence.

However, upon setting

w = (x - sqrt(a)) / (x + sqrt(a)),

the recursion for w_n becomes remarkably simple.

Haven't checked the first yet, second is a classic (but equally slightly out of date), I think the third is associated to a one-off workshop / grant / similar.

The office library expands ... (courtesy of Peter Green)

Maybe the more serious limitations would be seen to kick in when you move to the nonparametric setting.

I guess you can levy the critique that likelihood ratios restrict you to the setting in which things are absolutely continuous with respect to one another, though in fairness, some parts of statistics get appreciably easier when that fails.

Which is to say, lots of statements about the score and information are shadows or limits of more general results about likelihood ratios, information-theoretic divergences, etc.

It is interesting just how much can be squeezed out of, or reduced to, the likelihood ratio. The score and information operators are important, but are essentially derived quantities, which are convenient to work with when they exist and are regular enough.

Chopin and Papaspiliopoulos!

I can see the case for this, but somehow I still maintain that one ought to start by emphasising the loss that one "actually means".

(not that the paper is bad or anything)

bsky.app/profile/stem...

boooo

(Hopefully this doesn't read as too much of a non sequitur!)

I may have made this point on here in the past, but I often end up thinking about YouTube: in the early days, I didn't really get it, because I was already adept at finding media to watch, and so on. Eventually, it became clear that some useful things are disproportionately easy to find on YouTube.