(Summary in Finnish: Oppimispäiväkirjamerkinnät jatkuvat hitaasti. Tässä lyhyt lukusuosituslinkki Bayes-päättelystä.)
Context: This is a quick interlude note in a series of learning diary notes, where I track my thoughts, comments, and (hopefully) learning process while reading How to Measure Anything by D. W. Hubbard together with some friends in a book club setting. Previous parts in the series: vol.0, vol.1., vol. 2. All installments in the series can be identified by tag “Measure Anything” series on this blog.
Despite the radio silence, the reading club has been marching on steadily but quite slowly. I have work in progress drafts for notes vol. 3, 4 and 5! Unfortunately other life has intervened with finishing the drafts, so the next installments of reading log entries will come up online here on the blog … sometime later.
However, as the book discusses in several places “Bayesian probability”, I thought it would be prudent to share some links to articles that actually explain what it means. (As I am a bit too busy to write a thorough lecture on myself, I will rather defer to experts.)
Without further ado, here are the links:
Difference between Bayesian and frequentist inference
Very shortly described: The frequentist inference is concerned with interpretation of probability, where probability is understood as property of repeated, independent events (“frequency”). Bayesian inference builds on Bayesian interpretation of probability, where “probability” is taken to be a thing that exists for anything, interpreted as quantification of knowledge about many different things. This kind of interpretation makes it possible to sensibly interpret and use Bayes theorem for inference about various random variables.
This is a succinct definition by a person who has had some years of experience working on this stuff, and it might not much sense if you are not already familiar with it.
While looking for something else entirely, I noticed this five-part series of blog posts by Jake VanderPlas. It illustrates the above brief statement in more detail. I recommend the first part (which I have actually read), as it is quite practical example. However, as a word of warning, the author is an astronomer, so for them “practical” includes use of some mathematical notation and calculations.
For a discussion about implications of these concepts, here is a nice pdf of class notes from Orloff and Bloom, MIT.
This Stat.StackExchange answer by Keith Winstein is great explanation how the difference works out between frequentist confidence intervals and Bayesian credible intervals. It involves chocolate chip cookie jars!
Bio-statistician Frank Harrell has a blog post titled My Journey From Frequentist to Bayesian Statistics. It also collects further links at the end.
Use of Bayesian statistics is not always very Bayesian in practice
Have you read all of the above?
Good! Here are some thoughts related to the real-life applications of Bayesian inference.
In addition to all of the above, there is a certain internet crowd who likes to use words like “prior”, “Bayesian belief” and “Bayesian update” for many things because “rational agents are Bayesians”. I do not say it is not useful to have such a concept and thus word for inductive reasoning (or, as one may say, “Bayesian update”): if you have a prior state of belief, and then obtain some new information, and if one can quantify the prior and the likelihood of data with with parametric distributions or probabilistic statements, the Bayes’ theorem will tell you what is the mathematically correct probabilistic state of belief (the posterior). (And if you skip the step of quantifying the numbers, one could still argue the procedure of obtaining the posterior belief should look like application of Bayes’ Theorem if one were to put numbers on it, which maybe gives some intuition about reconciling ones beliefs about some matter with new evidence.)
However, more one works with explicit, quantitative Bayesian statistical models (like presented in the VanderPlas blog series) it starts to sound a bit weird to talk about “updating ones belief” without making calculations with any models or probabilities.
It gets even more weird when practicing statisticians (who write authoritative textbooks on Bayesian data analysis) explain that actually, in real life, the way they do Bayesian statistics does not resemble an inductive series of Bayesian belief updates (pdf):
A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypothetico‐deductivism. We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science. Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework.Gelman and Shalizi, “Philosophy and the Practice of Bayesian Statistics.” British Journal of Mathematical and Statistical Psychology 66, no. 1 (2013): 8–38. https://doi.org/10.1111/j.2044-8317.2011.02037.x.
If you feel like reading 30 page of philosophy of statistics, read the whole article. The way I read it, looking at the whole of knowledge-making produce of successful practical statistics, which includes the part where (1) one formulates the Bayesian model and priors about some phenomenon, (2) fits the model to data and obtains the posterior inferences with math and algorithms, and (3) then checks if it really works with various other methods, only (2) is really about making Bayesian updates. In combination with parts (1) and (3), the whole procedure is more hypothetico-deductive than inductive, and model checks that have some affinity with Popperian falsifications.
If you want to read more about this kind ” statistician’s way of doing” Bayesian inference, you can read a more recent article “Bayesian Workflow” by Gelman et al. 2020 (arxiv) which presents a comprehensive and quite technical 77-page step by step tutorial into it, or less comprehensive but also quite mathematical essay by Michael Betancourt (2020), “Towards A Principled Bayesian Workflow” (html).