This post is from Bob.

I’ve been planning for at least a couple years to generate a case study around Galielo’s use of an inclined plane instrumented with water clocks to estimate the terrestrial gravitational constant. Here are some photographs of a replica in the Museo Galileo (click to blow them up). And here’s a video simulation of the experiment. We replace his clever pendulum apparatus explained in the video and the web page with simple Bayesian statistics so we can actually estimate the gravitational constant.

The case study

Here is a draft.

Bob Carpenter. 2026. Estimating g from Galileo’s Water Clock: A scientific Bayesian inverse problem with Stan and CmdStanPy. GitHub.

I list myself as the author here because I’m responsible and AIs can’t own copyright in the U.S., but 100% of the text and code was written by Claude Opus 4.8 (medium or high effort, but I can’t recall which). I used the desktop app, which doesn’t allow sharing, but you can try it yourself.

The prompt

Here’s the sloppy prompt I used, which I just typed in without much thought in a couple minutes to get a feel for what it could do on its own.

I would like to generate a case study written in Quarto and using CmdStanPy to demonstrate solving scientific Bayesian inverse problems. I want to use a simulation of Galileo’s water clock experiment, which can be used to estimate the gravitational constant. I would like you to start by generating the mathematical model description in LaTeX, the model code in Stan to solve the inverse problem, and a simulation driver in Python using CmdStanPy and plotnine for plotting. Please just `import plotnine as pn` and use `pn.geom…`, etc. All I need in the output now is a call to `.summary()` on the fit returned by `.sample()`. Wrap this all up in a quarto document for me from which I can generate HTML by calling `quarto render galileo.qmd`.

It was done before I got back to my desk with a cup of coffee (well under five minutes). So not quite the several hours Andrew said it took him to write his case study on the New York Knicks basketball team, which he posted earlier today. Of course, this was much simpler and I didn’t have to think through any details before generating it.

Is it right?

What Claude produced looks really good to me. If a student had done this, I’d given them an A. I can’t object to the way it described Galileo’s experiment, wrote the math, wrote the Stan code, wrote the Python simulation, or plotted the raw data as Andrew is always urging us to do.^*

The source

You can find the source .qmd file on my GitHub:

https://github.com/bob-carpenter/case-studies/tree/master/galileo-gravity

It’s short, so I would have just included it, but the blog software blocked my post after considering it an attack on the site. To get it to render with resources embedded, I had to ask Claude a follow-up question and manually insert a single line of config into the .yaml header for the markdown document.

Putting this blog post together took longer than writing the prompt and checking the results.

^*   Maybe Claude runs a little simulation of Andrew like I do. Andrew himself claims to run a simulation of Jennifer Hill—it’s the basis of his
handy statistical lexicon entry for “WWJD,” which he told me stands for “What would Jennifer do?” Unfortunately, neither the lexicon entry nor its underlying link explains the acronym.