An insight from Hammersley-Chapman-Robbins is that while the usual Cramér-Rao bound is typically framed as an ordering between PSD matrices, there is something to be gained by thinking of it as a containment between two ellipsoids.

One implication of this of which I'm fond of that the Cramér-Rao inequality is not really a statement about unbiased estimators of a parameter ("there's no such thing as a parameter", etc.), but rather about estimators of linear functionals of the data-generating process (that is, p |-> E_p [f(x)]).

A curiosity: one view on parameterisation-agnostic procedures in statistical inference is that you pass from thinking about a 'big' space of parameters inside some finite-dimensional space to thinking about a 'thin' manifold of probability measures within the space of all probability measures.

The first one here is a pretty quick read and a cute idea, though I think that there's a potential condensation of what they do, namely if you want to approximate the distribution of a sum, then approximate the generating function of the sum, by the generating function of something simpler.

fun bits and pieces today: arxiv.org/abs/2603.23923 'Elements of Conformal Prediction for Statisticians' - Matteo Sesia, Stefano Favaro arxiv.org/abs/2603.24421 'E-values as statistical evidence: A comparison to Bayes factors, likelihoods, and p-values' - Ben Chugg, Aaditya Ramdas, Peter Grünwald

It's been said plenty of times in plenty of different ways, but it is striking to emphasise thar if you've fit a linear model to data, your eventual plot of the fitted model need not be linear (since you might be linear in features of x rather than in x, etc.).

I am on a poster!

Getting curious about MMD minimization problems. Seems to be nice if you're just fiddling with weights, and suddenly gets much harder to say nice things once you enter the (arguably most interesting) setting of moving the locations (or other parameters). A bit like mixtures in that regard, I guess.

Some more-or-less equivalent facts about probabilities: 1) for random variables of a given mean and variance, the Chebyshev-Cantelli inequality provides an upper bound on tail probabilities 2) for random variables of a given variance, the mean and quantile (at a given level) can't differ too much

SVD in Action

A little analysis exercise: fix t > 0, and consider the recursion X_n = X_{n - 1} + 1 + t / X_{n - 1}, and how it grows for large n. Simplest: show that X_n ~ n. A bit harder: show that X_n - n ~ t * log n. Harder again: show that X_n - n - t * log n has a limit L = L(X_0, t).

Reading about a notion from (Online) Prediction called "Shtarkov's Normalised Maximum Likelihood" (Method?), which I'm only partially understanding at present. One perspective seems to be the following: to quantify the complexity of a dataset relative to some parametric family, compute its maximal

Toying with the idea of contrasting K-Means and PCA by "where should you build hospitals?" and "where should you build roads?" (bearing in mind that both are very imperfect cartoon analogies in any case).

Slightly missed opportunity to not name "Reversible Jump MCMC" as "Multiverse Monte Carlo"; so it goes.

Another little example which I finally got around to looking at: let ε∈(0,1), and consider F(x, y) = |x - y| + ε⋅{|x| + |y|} . F is convex, coercive, and bounded below, with unique global minimiser at the origin. However, from generic initialisation, alternating minimisation gets stuck!

One way of deriving this is to observe that under the stated assumptions, the function F_M (z) = sqrt(z)^T M sqrt(z) (where sqrt is applied elementwise) is concave in z, and hence can be majorised by a tangent (followed by a bit of bookkeeping).

A nice little trick that I hadn't revisited for a bit: Let M be a PSD matrix with nonnegative entries, and let x be a vector with positive entries. Define then y = (Mx)/x (element-wise) and let Y be the diagonal matrix with entries lifted from y. Then, M is dominated by Y in the Loewner ordering.

Something a bit cool which I hadn't clocked until today: In classical (oblivious) randomised dimension reduction (following e.g. Johnson-Lindenstrauss), your stochastic power (informally) comes from blending together coordinates of your high-dimensional base vector(s).

do you ever write an insane math sentence? like i just wrote "is a convex path component a path component that is convex?"

Quite neat - answers some questions I had (at least in one dimension): arxiv.org/abs/2603.08939 'Shape-constrained density estimation with Wasserstein projection' - Takeru Matsuda, Ting-Kam Leonard Wong

In somebody else's words:

Seeing the sights at the Bristol Art Gallery:

Didn't know that this one had a name, but it's a cute one. There's a nice "polynomial-style" proof (cf. SoS hierarchies etc.), a nice "martingale-style" proof, and probably some others too. Neat!

An odd miscellaneum: viewed as a functional of the data-generating process, the Bayes risk is concave. Corollary: variance and mean absolute deviation are concave functions of the measure in question.

Something which I finally got around to: figuring out the origins of "the Holley-Stroock perturbation lemma" (in the context of functional inequalities). This seems self-explanatory enough, since the relevant 1987 paper of Holley and Stroock is relatively well-documented. Still, some surprises:

do mine eyes deceive me? a release date in the present year?

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.

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.

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 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.

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

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.

boooo

Something rather related: having already bemoaned that I don't really know what it looks like when people use LLMs while doing mathematics research, it bears mentioning that in many ways, I only vaguely know what it looks like when people do mathematics research of any form.

An elegant approach to { "out of scope", "incomplete", etc. } from Bol'shev:

A first-year probability problem which I was tutoring gave me cause to think about the following problem: given iid samples { Xᵢ : 1 ≤ i ≤ N }, upon observing their sum, one anticipates that E [X₁ | sumᵢ Xᵢ = S ] = S / N. There's a simple symmetry argument for this - **if** the expectation exists.

Incredible track record of writing short papers with strong results and clean proofs; still amazed each time: arxiv.org/abs/2602.17305 'Hyper-contractivity and entropy decay in discrete time' - Justin Salez

Fun times: a paper of ours is finally "properly published" in AAP: 'Weak Poincaré inequalities for Markov chains: Theory and applications' - Christophe Andrieu, Anthony Lee, Sam Power, Andi Q. Wang This was the final part of a trilogy of joint works during Andi's and my postdoc.

something nice about writing papers where you're not concerned enough about the outcome to stop having fun:

Do it for the (histo)gram:

Reminded today of the Chapman-Robbins variance lower bound, which extends the Cramér-Rao lower bound to mildly non-regular settings (and is also pretty easy to prove!). The likelihood ratio remains a gem.

New preprint 📢 arxiv.org/abs/2602.09161 Lead by Sherman Khoo (PhD student at Compass CDT), with Song Liu and Mark Beaumont. We adjust SBI summaries at test time to improve robustness.

Another clash with my Bristol rival (the Sam Power in the Politics department): we both attended the same graduation ceremony (as staff) and ended up in one another's academic gown!

sweet recent papes: arxiv.org/abs/2602.09842 'Step-Size Stability in Stochastic Optimization: A Theoretical Perspective' - Fabian Schaipp, Robert M. Gower, Adrien Taylor

My reaction to hallucinated references in LLM chats is that they are kind of funny, in as much as they often involve papers which I would like to exist, and often by authors who could very credibly write them. There's also a slight aspect of fanfic to them that I can sort of get behind.

A deterministic expansion for the error in the MLE (under a bit of smoothness and an implied uniqueness assumption), which gives one perspective on where approximate normality comes from (even non-asymptotically).

Reminded once again of my frustration with stopgrad being included in the definition of a loss function. Boring and grumpy behaviour from me, but the stopgrad operation is not really about the loss.

Eclectic research interests

How do I get Diaconis-Holmes-Neal as my loading screen?

My most recent attempt at visualising what spectral / L2 convergence of Markov chains looks like. Not fully satisfied with the outcome (would somehow be better if I could relate these waves to the distribution of X_t over time), but I suppose that it does illustrate the global character a bit.