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Evan Miller
Machine Learning @ Anthropic
Evan Miller is currently a Member of Technical Staff at Anthropic, a company focused on AI safety. He previously worked as a Statistics Engineer at Eppo, where he developed various statistical tools and methodologies, including A/B testing techniques.
Miller holds a bachelor's degree in physics from Williams College and has experience in software development and statistics, with his work recognized in top medical journals. He has also contributed to discussions in the AI community, particularly regarding the mechanics of transformer models and their attention mechanisms, proposing potential improvements to existing algorithms.123
In addition to his professional work, Miller maintains a personal website where he shares insights on statistics, programming, and AI, further showcasing his expertise in these fields.1
Highlights
Hey this is neat! Nine years ago, I lifted an A/B testing idea from Ben Tilly (who deserves an @-mention, but I can't find his LinkedIn) and dressed it up in some math. The basic idea is that you can conduct a sequential A/B test just by keeping count of success events from control and treatment – if the difference in counts ever exceeds a certain amount, the treatment wins. Even better – you get to go home early. (Or at least call the test early...)
By now, that's old news. But a couple of months ago, a team of researchers at Zalando (viz. Alexey Kurennoy, Majed Dodin, Tural Gurbanov, PhD, and Ana Peleteiro Ramallo) took the idea further and figured out the math for the continuous case.
The key to the lock was the Lévy inequality – if you multiply the incoming metric values from the control group by –1, then under the null hypothesis, the incoming metric values from both treatment and control must come from a symmetric distribution centered about zero. Paul Lévy, writing in 1937, gives us a nice equation relating the distribution of a maximum partial sum of a symmetric random variable to the distribution of its final sum. So if we can estimate the variance in the metric data, we can estimate the variance in the sums – and if we know the variance of the final sums, we know how rarely (under the null hypothesis) that a given sum boundary should be crossed before the finish line. No need for arbitrary alternative hypotheses, or defining group sizes in advance!
The researchers call their new method YEAST – YEt Another Sequential Test. I assume it is so named because it is destined to make the practitioner's bread 💵 rise 📈. That warm fragrance that you smell is their arXiV paper emerging from the oven:
https://lnkd.in/edz4__2Q
Two additional improvements the authors make over my 2015 blog post are 1) they do real math instead of my 18th century aristocratic hand-waving and 2) they work out formulas for a "staircase boundary" that loosens up over the course of the test. Bravo!
Two shortcomings in the paper worth noting: they lack a satisfying solution when treatment and control allocations are unequal (I never could figure this out!), and they don't offer guidance on choosing N for a desired amount of statistical power. Clearly leaving a gap in the market for someone to design and publish a continuous-metric, sequential sample-size calculator...
The paper is relatively accessible – give it a read. I suspect that over time, YEAST will prove to be a popular ingredient in the ever-growing kitchen of sequential techniques. But for now, enjoy the warm and buttery taste of a fresh statistics paper! 🍞
I'm back – with some news. But first – a paper! By me! My first!
Usually we run A/B tests on averages... Average time, average clicks, average profit. But what if the median makes more sense? Or the 90th/95th/99th percentile? Many engineering metrics, such as latency, benefit from quantile analysis – where a few extreme outliers can severely skew the mean.
An analytic solution has evaded statisticians for… decades? The best anyone has done is bootstrap both data sets and measure the differences in quantiles across the two groups. But this is computationally expensive. Recently Mårten Schultzberg's group at Spotify proposed some analytic tricks to speed up the bootstraps. A couple years ago, I showed that one of their conjectures, that quantile indexes from a Poisson bootstrap converge to a binomial distribution, wasn't quite right…
This got me thinking about what a proper two-sample quantile test ought to look like. In the one-sample problem, you can form a likelihood function using the binomial distribution – essentially, the likelihood that a quantile value is equal to X is the probability that you'd see L values less than X, and (N – L) values greater than X.
As I closed my eyes and drifted into a dreamy trance, I realized that in an A/B test, treatment and control are independent – so you can make a joint likelihood function by multiplying the two binomial distributions. Then, if you add the constraint that the quantile of the control group equals the same quantile of the treatment group, you have everything you need for a likelihood ratio test. Wilks' theorem is all you need!
In the (mini-)paper – "Likelihood-ratio inference on differences in quantiles" – I work out the large sample behavior and propose a simple two-step estimator. Those binomials I mentioned converge to a normal distribution, and with some mild assumptions about the shape of the treatment effect, you can form a confidence interval using exactly four order statistics from each sample. This reduces the computational complexity of the problem to O(N) – the same as for mean metrics. Bootstraps, begone!
Details are in the paper... I was unable to "break" the algorithm when I tried it with every generating distribution I could think of. Can you? 💎
https://lnkd.in/e-fwQ-_t
…And finally, a bit of promised news. This was one of my favorite projects that I undertook at Eppo. Because it was my first arXiV submission, it took several months to get approved and published – so by the time the email arrived, I was no longer researching A/B tests, no longer Statistics Engineering…
Which is to say: I'm now a Member of Technical Staff at Anthropic, the safety-focused A.I. company (👋 to 🐜s). After consulting with many high-end marketing gurus, A/B Evan is now A.I. Evan. But there's no escaping the iron laws of statistics; more on that later. If you want to read what I'm up to in A.I.-land, flick that Follow button… And if you're in New York, and want to chat stats, shoot me a DM!
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