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> I’ve never personally worked on a problem that I felt wasn’t adequately approached with frequentist methods

Multilevel models are one example of problem were Bayesian methods are hard to avoid as otherwise inference is unstable, particularly when available observations are not abundant. Multilevel models should be used more often as shrinking of effect sizes is important to make robust estimates.

Lots of flashy results published in Nature Medicine and similar journals turn out to be statistical noise when you look at them from a rigorous perspective with adequate shrinking. I often review for these journals, and it's a constant struggle to try to inject some rigor.

From a more general perspective, many frequentist methods fall prey to Lindley's Paradox. In simple terms, their inference is poorly calibrated for large sample sizes. They often mistake a negligible deviation from the null for a "statistically significant" discovery, even when the evidence actually supports the null. This is quite typical in clinical trials. (Spiegelhalter et al, 2003) is a great read to learn more even if you are not interested in medical statistics [1].

[1] https://onlinelibrary.wiley.com/doi/book/10.1002/0470092602

I agree Bayesian approaches to multilevel modeling situations are clearly quite useful and popular.

Ironically this has been one of the primary examples of, in my personal experience, with the problems I have worked on, frequentist mixed & random effects models have worked just fine. On rare occasions I have encountered a situation where the data was particularly complex or I wanted to use an unusual compound probability distribution and thought Bayesian approaches would save me. Instead, I have routinely ended up with models that never converge or take unpractical amounts of time to run. Maybe it’s my lack of experience jumping into Bayesian methods only on super hard problems. That’s totally possible.

But I have found many frequentist approaches to multilevel modeling perfectly adequate. That does not, of course, mean that will hold true for everyone or all problems.

One of my hot takes is that people seriously underestimate the diversity of data problems such that many people can just have totally different experiences with methods depending on the problems they work on.

These days, the advantage is that a generative model can be cleanly decoupled from inference. With probabilistic languages such as Stan, Turing or Pyro it is possible to encode a model and then perform maximum likelihood, variational Bayes, approximate Bayesian inference, as well as other more specialized approaches, depending on the problem at hand.

If you have experienced problems with convergence, give Stan a try. Stan is really robust, polished, and simple. Besides, models are statically typed and it warns you when you do something odd.

Personally, I think once you start doing multilevel modeling to shrink estimates, there's no way back. At least in my case, I now see it everywhere. Thanks to efficient variational Bayes methods built on top of JAX, it is doable even on high-dimensional models.

The evidence "actually supports the null" over what alternative?

In a Bayesian analysis, the result of an inference, e.g. about the fairness of a coin as in Lindley's paradox, depends completely on the distribution of the alternative specified in the analysis. The frequentist analysis, for better and worse, doesn't need to specify a distribution for the alternative.

The classic Lindley's paradox uses a uniform alternative, but there is no justification for this at all. It's not as though a coin is either perfectly fair or has a totally random heads probability. A realistic bias will be subtle and the prior should reflect that. Something like this is often true of real-world applicaitons too.

Thank you. The main problem with Bayesian statistics is that if the outcome depends on your priors, your priors, not the data determine the outcome.

Bayesian supporters often like to say they are just using more information by coding them in priors, but if they had data to support their priors, they are frequentists.

If they were doing frequentist inference they wouldn’t be using priors at all and there is nothing frequentist in using previous data to construct prior distributions.

Not true. In frequentist statistics, from the perspective of Bayesians, your prior is a point distribution derived empirically. It doesn't have the same confidence / uncertainty intervals but it does have an unnecessarily overconfident assumption of the nature of the data generating process.

Not true. In frequentist statistics, from the perspective of Bayesians and non-Bayesians alike, there are no priors.

—-

Dear ChatGPT, are there priors in frequentist statistics? (Please answer with a single sentence.)

No — unlike Bayesian statistics, frequentist statistics do not use priors, as they treat parameters as fixed and rely solely on the likelihood derived from the observed data.

There's always priors, they're just "flat", uniform priors (for maximum likelihood methods). But what "flat" means is determined by the parameterization you pick for your model. which is more or less arbitrary. Bayesians would call this an uninformative prior. And you can most likely account for stronger, more informative priors within frequentist statistics by resorting to so-called "robust" methods.

First, there is not such thing as a ‘uninformative’ prior; it’s a misnomer. They can change drastically based on your paramerization (cf change of variables in integration).

Second, I think the nod to robust methods is what’s often called regularization in frequentist statistics. There are cases where regularization and priors lead to the same methodology (cf L1 regularized fits and exponential priors) but the interpretation of the results is different. Bayesian claim they get stronger results but that’s because they make what are ultimately unjustified assumptions. My point is that if they were fully justified, they have to use frequentist methods.

One standard way to get uninformative priors is to make them invariant under the transformation groups which are relevant given the symmetries in the problem.

It’s not true that “there are always priors”. There are no priors when you calculate the area of a triangle, because priors are not a thing in geometry. Priors are not a thing in frequentist inference either.

You may do a Bayesian calculation that looks similar to a frequentist calculation but it will be conceptually different. The result is not really comparable: a frequentist confidence interval and a Bayesian credible interval are completely different things even if the numerical values of the limits coincide.

Frequentist confidence intervals as generally interpreted are not even compatible with the likelihood principle. There's really not much of a proper foundation for that interpretation of the "numerical values".

What does “as generally interpreted” mean? There is one valid way to interpret confidence intervals. The point is that it’s not based on a posterior probability and there is no prior probability there either.

If you want to say that when you do a frequentist analysis which doesn’t include any concept of prior you get a result that has a similar form to the result of a completely different conceptually Bayesian analysis which uses a flat prior (definitely not “a point distribution derived empirically”) that may be correct. It remains true that there is no prior in the frequentist analysis because they are not part of frequentist inference at all.

Priors are not used in construction of frequentist approaches, but that does not mean that the analyses aren't isomorphic in theory.

Point distribution <=> point estimate as a sample from an initially flat distribution. A priori vs a posteriori perspectives, which are equivocal if we are to take your description of frequentist statistics into account ;)

It’s not my description of frequentist statistics. It’s the frequentist statisticians’ description. This is from Wasserman’s All of Statistics:

The statistical methods that we have discussed so far are known as frequentist (or classical) methods. The frequentist point of view is based on the following postulates:

F1 […]

F2 Parameters are fixed, unknown constants. Because they are not fluctuating, no useful probability statements can be made about parameters.

F3 […]

Thank you for Lindley's paradox! TIL

Curious what you might consider “adequate shrinking”?

Horshoe priors, partial pooling, something more?

I realize that might be highly subject

I guess this depends on the problem at hand.

But I was thinking about a typical hierarchical model with partial pooling and standard weakly informative priors.

In clinical settings and situations where probabilities really matter, its a better fit.

I studied stats at Duke which is a Bayesian academy. Almost every problems come from regimes with small sample sizes. Given that Duke houses the largest academic clinical research organization globally, having a stats and biostats department with this bent is useful: samples are tiny in clinical trials compared to most big data settings.

The biggest problem with the whole Bayesian regime IMO is that as the data gets larger its selling point vanishes. If your data is big or is normal (mean-based statistics), a frequentest/bootstrapped CI approximates the Bayesian CI anyway.

Furthermore, many us work in settings where we're trying to sell toothpaste: we don't need the Bayesian guarantees that an insurer might.

I’m not sure what your professional experience is in, but as a counterpoint, I’ve never been in a situation where I hadn’t wished for a system I’m working with to already be in a Bayesian framework. Having said that, I only occasionally am building things from scratch instead of modifying existing systems, so I’m not always lucky enough to be able to work with them.

The pain points around getting a sampler/model pairing working in a reasonable timeframe is definitely a valid complaint. In my experience, inference methods in Bayesian stats are much less forgiving of poorly specified models (or said another way, don’t let you get away with ignoring important structural components of the phenomena of interest). A poorly performing model (in terms of sampler speed/mixing) is often a sign of a problem with the geometry of the parameter space. Frustratingly this can sometimes be a result of conceptually equivalent, but computationally different parameterizations (e.g. centered vs non-centered multi level effects).

The struggles are worth it IMO because it is helpful feedback that helps guide design, and the ease with which I can compute meaningful uncertainty bounds on pretty much any quantity of interest is invaluable.

A large portion of generative AI is based on Bayesian statistics, like stable diffusion, regularization, LLM as a learned prior (though trained with frequentist MLE), variational autoencoders etc. Chain-of-thought and self-consistency can be viewed as Bayesian as well.

Bayesian methods are not better than frequentist methods and vice versa. I use both, but mostly Bayesian.

Bayesian approaches take a long time thinking, making models, choosing priors, simulations etc. but they provide a better estimate and understanding the parameters. I hate point estimates and decision based arbitrary p-value. Whenever possible I use Bayesian methods.

I feel like I'm a polyglot here but primarily a native frequentist thinker.

I've found Bayesian methods shine in cases of an "intractible partition function".

Cases such as language models, where the cardinality of your discrete probability distribution is extremely large, to the point of intractability.

Bayesians tend to immediately go to things like Monte Carlo estimation. Is that fundamentally Bayesian and anti-frequentist? Not really... it's just that being open to Bayesian ways of thinking leads you towards that more.

Reinforcement learning also feels much more naturally Bayesian. I mean Thompson sampling, the granddaddy of RL, was developed through a frequentist lens. But it also feels very Bayesian as well.

In the modern era, we have Stein's paradox, and it all feels the same.

Hardcore Bayesians that seem to deeply hate the Kolmogorov measure theoretic approach to probability are always interesting to me as some of the last true radicals.

I feel like 99% of the world today is these are all just tools and we use them where they're useful.

When you are using something like Monte Carlo you’re probably using some method that’s more advanced than the Naïve Bayes, is that right?

I'm talking about, for something simple, the negative sampling in word2vec.

Or the temperature setting for an LLM etc.

Given your bias, why bother making this point on a thread about using Bayesian methods where they are applicable? Just seems like unconstructive negativity.