Once you realize this you can easily develop very sophisticated testing models (if necessary) that are also easy to understand and reason about. This dramatically simplifies.
If you're looking for a specific book recommendation Statistical Rethinking does a good job covering this at length and Bayesian Statistics the Fun Way is a more beginner friendly book that covers the basics of Bayesian hypothesis testing.
Edit: Haha I just found the textbook and I’m remembering now that I actually worked through sections of it back when I was working through BDA several years back.
There is an example in his book discussing efficacy trials across seven hospitals. If you stratify the data, you lose a lot of confidence, if you aggregate the data, you end up just modeling the difference between hospitals.
Hierarchical modeling allows you to split your dataset under a single unified model. This is really powerful for extracting signal for noise because you can split your dataset according to potential confounding variables eg the hospital from which the data was collected.
I am writing this on my phone so apologies for the lack of links, but in short the approach in this book is extremely relevant of medical testing.
It’s just a relatively dense book. There’s some other really good suggestions in this thread, most of which I’ve heard good things about. If you have a background in programming, I’d suggest Bayesian Methods for Hackers as a really good starting point. But you can also definitely tackle this book head on, and it will be very rewarding.
After than Statistical Rethinking will take you much deeper into more complex experiment design using linear models and beyond as well as deepening your understanding of other areas of math required.
Statistical Rethinking is a good option too.
That course is a good balance between theory and practice. It gave me a practical intuition understanding why posterior distribution of parameters and data are important and how to compute them.
I took the course in 2016 so a lot could have changed.
It goes through fundamentals of Bayesian ideas in the context of applications in communication and machine learning problems. I find his explanations uncluttered.
First learn some basic probability theory: Peter K. Dunn (2024). The theory of distributions. https://bookdown.org/pkaldunn/DistTheory
Then frequentist statistics: Chester Ismay, Albert Y. Kim, and Arturo Valdivia - https://moderndive.com/v2/ Mine Çetinkaya-Rundel and Johanna Hardin - https://openintrostat.github.io/ims/
Finally Bayesian: Johnson, Ott, Dogucu - https://www.bayesrulesbook.com/ This is a great book, it will teach you everything from very basics to advanced hierachical bayesian modeling and all that by using reproducible code and stan/rstanarm
Once you master this, next level may be using brms and Solomon Kurz has done full Regression and Other Stories Book using tidyerse/brms. His knowledge of tidyverse and brms is impressive and demonstrated in his code. https://github.com/ASKurz/Working-through-Regression-and-oth...
Also, IMHO, his best work has been done describing how to do statistics. He has written somewhere I cannot find now that he sees himself as a user of mathematics, not as a creator of new theories. His book Regression and Other Stories is elementary but exceptionally well written. He describes how great Bayesian statisticians think and work, and this is invaluable.
He is updating Data Analysis Using Regression and Multilevel/Hierarchical Models to the same standard, and I guess BDA will eventually come next. As part of the refresh, I imagine everything will be ported to Stan. Interestingly, Bob Carpenter and others working on Stan are now pursuing ideas on variational inference to scale things further.
[1] https://sites.stat.columbia.edu/gelman/research/unpublished/...
I would say his work with Stan and his writings, along with theorists like Radford Neal, really opened the door to a computational approach to hierarchical modeling. And I think this is a meaningfully different field.
Before Stan existed we used BUGS [1] and then JAGS [2]. And most of the work on computation (by Neal and others) was entirely independent of Gelman.
[1] https://en.wikipedia.org/wiki/Bayesian_inference_using_Gibbs...
[2] https://en.wikipedia.org/wiki/Just_another_Gibbs_sampler
One example is rare class prediction on long form text data eg phone calls, podcasts, transcripts. Other networks including neural networks and LLMs are either not flexible enough or require far too much data to achieve the necessary performance. Structured hierarchical modeling is the balance between those two extremes.
Another example is in genomic analysis. Similarly high dimensional, noisy, low data. Additionally, you don’t actually care about the predictions, you want to understand what genes or sets of genes are driving phenotypic behaviors.
I’d be happy to go into more depth via email or chat if this is something you are interested in (on my profile).
Some useful reads
[1] https://sturdystatistics.com/articles/text-classification
It took me about a year to work through this book on the side (including the exercises) and it provided the foundation for years of fruitful research into hierarchical Bayesian models. It’s a definitely not an introductory read, but for any looking to advance their statistical toolkit, I cannot recommend this book highly enough.
As a starting point, I’d strongly suggest the first 5 chapters for an excellent introduction to Gelman’s modeling philosophy, and then jumping around the table of contents to any topics that look interesting.