Episode 51 of Probable Causation featured professors Amanda Agan and Anna Harvey. They talked with me about our coauthored research on the effect of prosecution for non-violent misdemeanors. Originally published: June 8, 2021.

PosteriorBench: going beyond point reconstructions for scientific inverse-problem to full posterior distributions. Many scientific problems involve indirect or partial observations. Multiple physical fields can explain the same measurements. A solver should capture these possibilities, yet reconstruction accuracy alone is misleading. We spent substantial compute to construct high-fidelity reference posteriors across four tasks: Darcy flow inversion, Poisson source recovery, carbon capture and storage, and light transport material inference. These reference data let researchers directly evaluate their solvers using five complementary metrics. One of our key findings: better reconstruction accuracy can coincide with worse posterior recovery. Even strong generative samplers struggle to get both the mean and variance right, often underestimating the uncertainty. Paper: Code: Thanks to Jiachen Yao, Sean Hsu, Xi Deng, and all our coauthors for making this work possible @Caltech #AIforScience# #InverseProblems# #UncertaintyQuantification#

There is a heated debate right now. Tristan Buckmaster says OpenAI told him its AI had solved the Navier–Stokes Millennium Prize Problem. That claim remains unverified, and the discussions around it have erupted into a dispute over scientific credit. Here is what happened and what we know so far: Several different things are getting mixed together. Here is what the available evidence actually says. Building on work by Córdoba and Martínez-Zoroa, Alpöge and Buckmaster used Claude and OpenAI models to establish how initially smooth fluid flows can develop singularities under smooth external forcing. They released papers and Lean formalizations, and Terence Tao praised the advance. Their results cover Euler and related equations. They do not solve the Navier–Stokes Millennium Problem, which additionally involves viscosity. A suitable extension could resolve it, even with smooth external forcing. Then comes the disputed part. According to Buckmaster, OpenAI told him on September 6 that an internal model had already produced a proof for forced Navier–Stokes, potentially taking that final step. He says OpenAI’s effort began after information about their progress reached the company. He also disputes the initial description of minimal human involvement, reporting that the discussions revealed a team effort, multiple attempts, and substantial compute. Buckmaster further says publication proposals included having him write up OpenAI’s result without Alpöge as a coauthor, with Alpöge’s employment at Anthropic cited as an obstacle. He reports being asked, “Why would you ruin your career?” after saying he would make the circumstances public. Bubeck has publicly rejected allegations against him as “false and inflammatory.” These accounts are contested. The social posts do not establish what happened in the private discussions. There is also no established evidence that OpenAI trained on their private Codex sessions or stole their drafts. Buckmaster raises the question but explicitly writes: “I do not know whether our data was used.” tl;dr Alpöge and Buckmaster used AI to prove new results about how smooth fluid flows can develop singularities, advancing a research program relevant to the Navier–Stokes Millennium Problem. Their published work does not solve that problem, but Buckmaster says OpenAI privately told him its internal model had taken the remaining step. He has not seen that proof, and the claim remains unverified. But rumors are that OpenAI actually fully solved that milleniun problem.