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.

Another exciting workshop to be held at Warwick Statistics this coming June: The ProbAI Theory of Scaling Laws Workshop from 22-24 June! Website of the workshop: warwick.ac.uk/fac/sc... Registration open until 31 March on a first-come-first-serve basis: warwick.ac.uk/fac/sc....

New open source: cuthbert 🐛 State space models with all the hotness: (temporally) parallelisable, JAX, Kalman, SMC

At a glance, these look decent, even in terms of being quite suitable for analysts / probabilists (as opposed to just number theorists): people.math.ethz.ch/~kowalski/pr...

Some notes on the Central Limit Theorem (following my revisiting of some of Devroye's book on Random Variate Generation): Often, in handling random variables for which 'direct' simulation is challenging, we try to 'reduce' to sampling of something tractable, and then post-process / transform / etc.

some hits from recent reading: arxiv.org/abs/2511.08111 'On the Kantorovich contraction of Markov semigroups' - Pierre Del Moral, Mathieu Gerber www.arxiv.org/abs/2601.19710 'On randomized step sizes in Metropolis-Hastings algorithms' - Sebastiano Grazzi, Samuel Livingstone, Lionel Riou-Durand

Exciting stuff: next week at the Online Monte Carlo Seminar, we will finally have a bit of Randomised Numerical Linear Algebra, courtesy of Ethan Epperly (Berkeley). See our website (sites.google.com/view/monte-c...) for details about the talk, taking place next Tuesday (3 February).

Currently watching a probability seminar about deposition models, KPZ, etc.; meanwhile, an audience member has Tetris open on their phone. There are many ways to capture an audience's attention, I suppose!

powerful response on MO

In the end, the answer: the lectures (from the first day) _were_ indeed recorded! Find them linked here: sites.google.com/view/lpd-tnn....

Itching to have the space in my calendar to process this in detail: arxiv.org/abs/2601.12633 'New Trends in the Stability of Sinkhorn Semigroups' - Pierre Del Moral, Ajay Jasra

Kind of funny to read this and have the reaction that "that's not algebra, that's statistical physics".

Recent downloads: "Distribution-uniform anytime-valid inference and the Robbins-Siegmund distributions" (arxiv.org/abs/2311.033...).

idle thought: if one thinks of the h-index as an estimator, then what 'population-level' quantity (if any) is it (or some simple transformation thereof) an estimator of?

Lovely new work from my friend (and former office mate!) Sven: arxiv.org/abs/2601.09007 'Global polynomial-time estimation in statistical nonlinear inverse problems via generalized stability' - Sven Wang

Have seen this news (cims.nyu.edu/dynamic/news...) about NYU professor Robert Kohn. As good a time as any to share a piece of his work which I found cool: when studying deterministic control problems, one can still end up with second-order PDEs for their solution: math.nyu.edu/~kohn/papers....

If only

lol I have such strong opinions about this (like you should legally be allowed to dunk anyone who says a penalty is a lot prior in a vat of very very cold water)

a favourite

And heading back from it now - great couple of days.

commencing imminently!

Very neat stuff: arxiv.org/abs/2601.05444 'What Functions Does XGBoost Learn?' - Dohyeong Ki, Adityanand Guntuboyina

On my way to this now! Should be fun - always exciting to spend time at events outside of my own usual community, get a sense of what is capturing their attention, etc.

Fun times with the triangular lattice:

One of the reasons for taking a bit of time to reflect on this system relates to my explorations of LLM tools for mathematics research. On the whole, I've found online discussions of this topic to be quite uninformative, in the sense that I rarely get a sense of what is really working well, and why.

Excited for the return of our seminar for another season! Link with details: sites.google.com/view/monte-c...

Stray calculations of limited importance: 1. Let X have the standard Laplace distribution on R, i.e. drawn from Q with density exp(-|x|)/2. 2. For m > 0, define Yₘ by X = Yₘ + m⋅Yₘ⋅|Yₘ| / 2. Then, for 0 < m ≤ 1, the density of Yₘ is strongly log-concave, but for m = 0 and m > 1, it is not.

some fun ones: arxiv.org/abs/2512.21300 'Closed-form empirical Bernstein confidence sequences for scalars and matrices' - Ben Chugg, Aaditya Ramdas www.arxiv.org/abs/2601.03518 'Sharp concentration inequality for the sum of random variables' - Cosme Louart, Sicheng Tan

big fan of alphabet notation

An analogy which I came across a couple of years ago: structural causal models are to 'usual' probabilistic graphical models (/joint distributions) as computational graphs are to functions - not just "what you do" but "how you do it", and so on. Not so complicated in this form.

Resonant tales of the importance of work-life balance can arise in the most unexpected places:

Thrilled for this genius, but would it have killed you to write down this proof of yours?! (written by me in November...)

Along a different axis, I do still have a system for filing my research ideas, although more according to tiers of "credibility", coarsely speaking. Let me sketch out how I think about it (bearing in mind that it's not particularly innovative):

It's quite well-known that Poincaré inequalities imply exponential concentration inequalities for Lipschitz functions, and also that such inequalities imply transport-entropy inequalities for the associated probability measure. You can prove this all in one swoop: drive.google.com/file/d/1T0dJ...

Somewhat entranced by reading about Catoni's robust M-estimator for the mean (projecteuclid.org/journals/ann...). The estimator itself is pretty easy to describe, and the core theoretical analysis is also pretty elementary (i.e. teachably so). Should have learned this all sooner!

nice slides (www.stat.berkeley.edu/~ryantibs/ta...) for a book which I didn't know was a book:

It's been quite an interesting exercise to skim through the (ordered) list of papers I've downloaded this year. It doesn't perfectly map onto the mathematical topics which I've spent time time thinking about this year, largely because one often thinks about questions which are yet to be answered.

PoTC meets RMT meets Wikipedia

A proper write-up of this, accessible from my website, on "Convex Ordering with Two-Point Random Variables": drive.google.com/file/d/1gksU...

A satisfying read! I must concede that I am fairly ignorant of the broader causal literature, and am a priori particularly suggestible to the claims herein. arxiv.org/abs/2512.23408 'Probabilistic Modelling is Sufficient for Causal Inference' - Bruno Mlodozeniec, David Krueger, Richard E. Turner

Doing the treble w/ Big Al

I have developed this into a bit of a write-up, including also an additional (minor) result which was mostly not proved by me: 'On ‘Sub-Gamma’ / ‘Bernstein-type’ Concentration Bounds' drive.google.com/file/d/1tEie...

Quite fun (for me): In the theory of concentration inequalities, a useful tool is the 'sub-Gamma' or 'Bernstein-type' MGF bound, which takes the form log(E[exp(tX)]) ≤ vt^2 / [2(1 - bt)] with v = var(X), b > 0. If X admits any absolute exponential moment, then there is a b for which this holds.

Lots of fun! Can certainly recommend.

Very clever approach to affine-equivariant, robust estimation of a location parameter: study the typical volume of a certain random simplex. (from a talk of Davy Paindaveine)

Talk recording is up on YouTube! youtu.be/9qebotKehfE?...