Learning Bayesian Statistics

Today's clip is from episode 159 featuring Matthijs Hollanders. In this conversation, Alex and Matthijs dig into a deceptively practical question: when you're modeling wildlife across space and time with Gaussian Processes, how do you keep the math from becoming computationally unbearable - and what does good engineering actually look like in the field?

Matthijs explains that for most real camera trapping datasets, exact GPs still hold up fine. The reason is less about clever math and more about ecological reality: researchers are usually resource-constrained, so datasets tend to be a few hundred sites, not thousands.

And when datasets do get large, they're rarely one giant connected grid - they're clusters of independent regions. That structure is exploitable. Run a separate, smaller GP per region, share the hyperparameters, and you avoid building the massive covariance matrix that makes exact GPs expensive in the first place.

But the more interesting thread is where this is heading. Alex introduces Hilbert Space Gaussian Processes (HSGPs) - an approximation that makes compute time nearly linear in dataset size, rather than cubic. The catch, as Matthijs points out, is that approximations aren't always better: if your dataset isn't large enough to be in the regime where the approximation accuracy kicks in, you're better off with the exact GP and its mathematical guarantees. The rule of thumb is simple - if you can use the vanilla GP, just do it.

Get the full discussion here

Support & Resources
→ Support the show on Patreon
→ Bayesian Modeling Course (first 2 lessons free)

Our theme music is « Good Bayesian », by Baba Brinkman (feat MC Lars and Mega Ran). Check out his awesome work

Support & Resources
→ Support the show on Patreon
→ Bayesian Modeling Course (first 2 lessons free)

Our theme music is « Good Bayesian », by Baba Brinkman (feat MC Lars and Mega Ran). Check out his awesome work

Takeaways:
Q: What is a Bayesian occupancy model and what problem does it solve?
A: An occupancy model accounts for the fact that you don't always detect a species when surveying for it, especially when the species is rare. A naive count of where you found it underestimates true occupancy. The model adds a repeated-measures component: you visit each site multiple times, and from the pattern of detections vs. non-detections it estimates a detection probability. Matthijs framed it as a zero-inflation structure where the zero-inflation happens at the site level rather than the observation level -- which keeps the model conceptually simple, just a standard GLM with a Bernoulli “is the species here at all?” stacked on top of a detection-rate process.

Q: What are Automated Recording Units and why don't traditional occupancy models handle them well?
A: ARUs are camera traps and acoustic monitors that record continuously over deployment periods of days, weeks, or months. The data they produce isn't a sequence of discrete human-led surveys; it's a continuous-time observation stream. Traditional occupancy models were designed for the discrete case -- a human visits a site, records yes or no, goes home. With ARUs, the question becomes how to bin or threshold the continuous data without losing the richer signal it actually contains.

Q: When should you not reach for occARU?
A: When your dataset is large and your survey interval is fine-grained. The bottleneck is Stan's fitting speed -- years of daily count data across many sites will fit slowly. The workaround is to bin coarser (weekly or monthly), which doesn't hurt occupancy estimation at all and only loses some detection-rate resolution. If you're only interested in occupancy, big grouping windows are fine.

Full takeaways here

Chapters:
00:12:14 What is an occupancy model and what problem does it solve?
00:16:16 What are Automated Recording Units and why do they need different models?
00:18:45 What is the occARU R package and why does it exist?
00:23:55 Why does occARU model counts directly rather than binary detection?
00:26:38 What does multi-species hierarchical modeling with Gaussian processes look like?
00:32:22 How does occARU implement Gaussian processes efficiently?
00:41:01 Why are Gaussian processes such a powerful but tricky modeling tool?
00:44:11 What is variance decomposition with global-local shrinkage priors?
00:49:02 How does occARU leverage recent Stan features for zero-sum constraints?
00:57:37 When does within-chain parallelization actually help?
01:01:30 How does Monte Carlo integration reduce high Pareto-k values?
01:15:27 When does occARU underperform and what's on the roadmap?

Thank you to my Patrons for making this episode possible!
Links from the show here.

Today's clip is from episode 158 featuring Stefan Radev. In this conversation, Alex and Stefan explore a genuinely fascinating problem: how do you turn an expert's intuition into a mathematically valid prior distribution - and can AI help automate that process?

Alex explains that prior elicitation is essentially a translation problem. Experts don't walk around thinking in probability distributions - their knowledge lives in intuitions, rules of thumb, and rough ranges. The challenge is converting that into something a Bayesian model can actually use.

The traditional approach? Ask an expert for quantiles or a mean, then parameterize your prior with hyperparameters and simulate until the model-implied quantities match what the expert described. If your pipeline is differentiable end-to-end, you use gradient descent. If not, you fall back to something like Bayesian optimization. Either way, you're iterating toward a prior that genuinely reflects expert knowledge - not just a convenient assumption.

But the really exciting part is what came next. In a follow-up paper, they pushed this further: instead of optimizing within a fixed parametric family (say, a Gaussian), they replaced the prior entirely with a normalizing flow - a flexible generative network - and ran the same procedure. No assumed distribution family. Just let the data and the expert's knowledge shape the prior from scratch.

The catch? More flexibility means more non-identifiability and stability headaches. But the direction is clear: a fully automated, end-to-end pipeline for building priors from non-probabilistic expert knowledge. And in 2026, that pipeline could theoretically be driven by an agent.

Get the full discussion here

Support & Resources
→ Support the show on Patreon
→ Bayesian Modeling Course (first 2 lessons free)

Our theme music is « Good Bayesian », by Baba Brinkman (feat MC Lars and Mega Ran). Check out his awesome work

Support & Resources
→ Support the show on Patreon
→ Bayesian Modeling Course (first 2 lessons free)

Our theme music is « Good Bayesian », by Baba Brinkman (feat MC Lars and Mega Ran). Check out his awesome work

Takeaways:

Q: Why are prior predictive checks so underused in practice, and how do simulations help?
A: They're underused because researchers don't always think to run them before seeing data -- but also because doing them rigorously (in the style Michael Betancourt advocates, with prior push-forward checks on interpretable summaries) takes effort. Simulations make it cheap to generate thousands of “what-if world” datasets from your model and check whether they look plausible, catching bad priors before you ever touch real data.

Q: How can generative AI help with prior elicitation?
A: Rather than forcing a domain expert to choose a distributional family and parameterize it, you can use a generative model to translate their qualitative knowledge directly into a prior. The expert describes what realistic data should look like; the generative model produces synthetic datasets matching that description; those datasets are used to fit a prior distribution. It removes the assumption that experts can think in terms of parameters and replaces it with the more natural question: does this look like your data?

Q: What would a foundation model for Bayesian inference actually look like?
A: Stefan's bet is that it won't be a fine-tuned general LLM. The right analogy is chess: you don't fine-tune GPT to play chess, you teach it when to call Stockfish. For Bayesian inference, you'd want a semantic layer – an LLM that understands the analysis goal – calling specialized numerical engines (MCMC samplers, amortized inference networks) that do the actual computation. Agent skills are already a step in this direction; the longer-term vision is engines that have been trained from scratch to generalize across large families of models and priors.

Full takeaways here.

Chapters:
00:00 How does amortized inference fit into modern Bayesian workflows?
06:01 What role do simulations play across the full Bayesian workflow?
12:12 How do you elicit priors from a domain expert who doesn't think in distributions?
19:01 What would a foundation model for Bayesian inference actually look like?
35:32 What is self-consistency in amortized inference and why does it matter?
39:22 How does semi-supervised learning improve simulation-based inference?
43:16 Why is sensitivity analysis so important yet so underused in Bayesian practice?
47:40 What is multiverse analysis and how does it change how we report Bayesian results?
51:32 How does amortized inference make sensitivity and multiverse analysis affordable?
01:02:47 How do amortized inference and classical MCMC complement each other?
01:10:08 What are the next major directions for BayesFlow and amortized inference research?

Thank you to my Patrons for making this episode possible!

Links from the show here.

Today's clip is from Episode 157 featuring Stefan Radev. In this conversation, Alex and Stefan dig into one of the hardest open problems in simulation-based inference — hierarchical models.

The core idea: when you move from flat to hierarchical models, you're no longer estimating one set of parameters. You have local parameters that vary by location (or subject, or city) and global parameters that capture what's shared across all of them. And you don't just want each separately — you want the full joint posterior, because that's where the Bayesian magic of shrinkage actually lives.
Stefan builds the problem from the ground up. Start with the simplest hierarchical case: a two-level model. He uses electoral forecasting in France as the example — cities nested inside departments nested inside the whole country.

Now your simulator has to cover all three levels. If that simulator is slow (think: brain emulators, minutes per sample), scaling to hundreds of groups becomes completely intractable. Memory issues, specialized network requirements, the works.

The key insight: this problem has structure you can exploit. The joint posterior factorizes in a particularly nice way — each local parameter depends on its own local data and on the global parameters. That means instead of cramming everything into one giant high-dimensional vector and hoping a neural network figures it out, you can decompose the problem. Estimate local parameters conditioned on local data and the globals. Use composition.

The takeaway: hierarchical models aren't just "harder flat models" - they have a geometry that demands a different architecture. Respecting that structure is what makes amortized inference scale.

Get the full discussion here

Support & Resources
→ Support the show on Patreon
→ Bayesian Modeling Course (first 2 lessons free)

Our theme music is « Good Bayesian », by Baba Brinkman (feat MC Lars and Mega Ran). Check out his awesome work