Today's clip is from Episode 158 featuring Stefan Radev. In this conversation, Alex Andorra and Stefan break down a core argument from their paper: Bayesian statistics has never been more computational than it is now, and simulation is the thread that ties the whole workflow together.

Stefan parcellates the Bayesian workflow into four stages, and this clip covers the first two. Stage one is model specification, where the workflow community has long recommended prior predictive checks. You can do this informally, just running simulations from your model and eyeballing whether the output meets your expectations, or formally, à la Michael Betancourt, by pushing your model's high-dimensional output through a transformation into a low-dimensional, interpretable space and checking it against reality.

The punchline: a surprising number of models can be discarded before you've even seen real data, yet Stefan notes these checks remain underused in practice.

Stage two is model verification, where the question shifts to whether your inferences are well calibrated. This is the territory of simulation-based calibration and parameter recovery studies, classic tools that have always carried a steep computational price. You simulate thousands of synthetic datasets and run inference on every single one, which is exactly why these checks are so often skipped in papers, even though doing one well can be a contribution in its own right.

Here's where amortized simulation-based inference changes the math entirely. Checks that used to take days now take seconds, and instead of laboriously running inference dataset by dataset, you get millions of posterior samples essentially for free. The calibration checks that the field has always known it should be doing finally become cheap enough to actually do.

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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.

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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

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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
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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 simulation-based inference and what does "sim-to-real" mean?
A: Simulation-based inference (SBI) uses a mechanistic simulator as an epistemic tool: you train a neural network on a large number of labeled simulations and then deploy it on real, unlabeled data. The "sim-to-real" framing captures the key asymmetry -- your network never sees real data during training, only simulations, but it generalizes to real observations at inference time. This is the opposite of the more common "synthetic-for-ML" approach, where fake data is used purely to augment real training data.

Q: What is the amortized inference agent skill and what does it do?
A: It's an open-source AI agent skill, co-developed by Stefan and Alexandre, that teaches an AI coding agent to run a complete, state-of-the-art amortized inference workflow. Because amortized inference is recent enough that it's underrepresented in LLM training data, vanilla agents tend to get it wrong. The skill injects the right methodology: it guides the agent to set up the simulator, choose the right network architecture, run a pilot, train with appropriate diagnostics, and produce an actionable report -- without the user needing to know the details.

Q: What is calibration coverage and why should you never skip it?
A: Calibration coverage tells you whether your posterior uncertainty is honest -- whether your credible intervals actually contain the true parameter at the right frequency. A model can show poor parameter recovery yet still be well-calibrated (because it's falling back on the prior), or it can appear to recover parameters while being poorly calibrated. Running calibration diagnostics both in-sample and out-of-sample is especially revealing for hierarchical models, which often appear to underfit in-sample but generalize much better out-of-sample thanks to shrinkage.

Full takeaways here

Chapters:
00:00:00 How does amortized inference fit into the Bayesian workflow?
00:12:03 What does "sim-to-real" mean in simulation-based inference?
00:15:57 Why is amortized inference particularly suited to psychology and neuroscience?
00:21:51 What is the amortized inference agent skill?
00:39:00 What is calibration coverage and how do you interpret it?
00:41:50 How do you decide what to do next after your first training run?
00:44:53 How do actionable insights make Bayesian workflows more usable?
00:49:08 What are the unique challenges of hierarchical models in amortized inference?
01:00:51 What is the current state of BayesFlow's support for hierarchical models?
01:05:00 What are the main failure modes of amortized inference and how do you handle model misspecification?

Thank you to my Patrons for making this episode possible!

Links from the show

Today's clip is from Episode 156 featuring Adam Foster. In this conversation, Adam explains Expected Information Gain (EIG) -the scoring function at the heart of optimal Bayesian experimental design.

The core idea: when designing an experiment, you need a way to compare possible designs and pick the best one. EIG is that score - it tells you how much information you expect to gain about your model parameters from a given design. The higher the EIG, the better the design.

Adam builds intuition for EIG from two directions that sound completely different but lead to the same place. First, the Bayesian angle: simulate datasets from your prior predictive distribution, run inference on each, measure how much uncertainty dropped, and average across datasets. Second, a classic puzzle - the 12 prisoners balance scale problem - where the best weighing strategy turns out to be the one that makes all three outcomes (tip left, tip right, balance) equally likely. This maximizes outcome entropy, which is exactly what EIG does: it steers you toward designs where every possible result narrows down your hypotheses as fast as possible.

The takeaway: good experimental design isn't about intuition or convention - it's about making your data work as hard as possible, and EIG gives you a rigorous way to do that.

Get the full discussion here

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→ Support the show on Patreon
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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 Bayesian experimental design and what problem does it solve?
A: It's the practice of using a Bayesian model to decide how to collect data before you collect it. Most statistical thinking starts with a fixed dataset. Bayesian experimental design sits upstream -- you have control over experimental parameters (which questions to ask, which reagents to mix, which conditions to test) and you want to choose them optimally. The Bayesian angle is to ask: what new data would most reduce my current uncertainty?

Q: When should you actually use Bayesian experimental design?
A: When two conditions hold: you have active control over how data is collected (not just passive observation), and you have a Bayesian model whose prior predictive distribution gives a reasonable picture of what typical data might look like. It's especially valuable when data collection is expensive or irreversible -- when the "committal step" of running an experiment has real cost, it's worth doing the analysis first.

Q: What is expected information gain (EIG) and why is it central to Bayesian experimental design?
A: EIG is the score you assign to a candidate experimental design -- the amount of information you expect to gain about your model parameters by running an experiment with that design. You compute it by simulating datasets from your prior predictive, doing Bayesian inference on each, and averaging how much the uncertainty decreased. What's remarkable is that you can derive the same quantity from two completely different starting points -- reducing parameter uncertainty, or maximizing outcome uncertainty while correcting for noise - and arrive at the same formula. That convergence is why EIG keeps being re-discovered independently across fields.

Full takeaways here

Chapters:

00:00 What is Bayesian experimental design and why does it matter?

00:06:02 What problem does Bayesian experimental design actually solve?

00:08:54 When should practitioners use Bayesian experimental design?

00:12:00 Is Bayesian experimental design changing how scientists work in practice?

00:15:04 What are the limitations of Bayesian experimental design?

00:17:55 What is expected information gain (EIG) and how does it work?

00:21:05 How do you compute expected information gain in practice?

00:23:48 What is active learning and how does it connect to Bayesian experimental design?

00:41:02 What is active learning by disagreement?

00:48:57 What is deep adaptive design and when should you00: use it?

00:56:02 How is Bayesian experimental design applied in protein dynamics and quantum chemistry?

01:01:58 What does a practical Bayesian experimental design workflow look like?

Thank you to my Patrons for making this episode possible!

Links from the show

Today's clip is from Episode 152 of the podcast, featuring Daniel Saunders. In this conversation, Daniel explores how Bayesian decision theory handles real-world risk aversion beyond the textbook maximum expected utility framework.

The key insight: classical Bayesian decision theory assumes risk neutrality, but in practice, people and businesses are risk-averse. Using a pricing optimization example, Daniel shows how uncertainty varies dramatically across price points—lower prices have predictable demand, while higher prices create wide uncertainty in profits. This asymmetry matters when you want safer decisions.

Daniel introduces exponential utility functions—a technique from economics that models diminishing returns on money. By adjusting a risk-aversion parameter, you can see how increasing risk aversion shifts optimal decisions away from high-uncertainty, high-profit scenarios toward more predictable outcomes.

The broader lesson: optimal decision-making requires separating the modeling process from the decision process, allowing you to build in constraints and risk adjustments that pure expected utility maximization would miss.

Get the full discussion here

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Our theme music is « Good Bayesian », by Baba Brinkman (feat MC Lars and Mega Ran). Check out his awesome work at https://bababrinkman.com/ !