How can you use Bayesian inference and modeling to stay ahead of trends?

Introductory material to parametric Bayesian theory usually describe Bayesian posterior density ρ(m|x) as prior-dependent conditional probability enabling one to attribute credible intervals to parameters m after observing x. When dense datasets allow determination of empirical priors, it should be ascertained that the priors correspond to stationary solutions of an iterative procedure before using posteriors to such ends. Need for iterations stems from convex geometry underlying Bayesian theory. Csiszár-Tusnády iterative alternating algorithm can be expressed in terms of alternating Bregman projections. Entropic convex duality in the determination of data-constrained kernel-based Bayes-Jaynes priors. J. Convex Anal. 2022 Jan 1;29:623-47.

Frequentism is the probability theory that everyone is taught in school. It tells us what to expect given some set of probabilities that are absolute, well-defined and known. If I flip a fair coin 10 times, then I expect a 'frequency' of 5 heads and 5 tails. What if you don't know that it's a fair coin? A swindler might weight the coin to come up heads 75% of the time. How could you tell? You answer that question with Bayesian inference. Bayesian inference is a mindset and framework that allows you to figure out the probability of events from your data. In a way, it's the polar opposite to Frequentism. They make a wonderful pair. Bayesian Inference turns data into probabilities, which Frequentism then uses to predict outcomes.

You don't have to use Frequentism to make predictions from Bayesian Inferences (BI). You can do it while still operating in the Bayesian space. That's what Bayesian modelling is. Based on the 'beliefs' in the probabilities of events, which you obtained from BI, you can cleverly combine your beliefs to distil them into probabilities for events. These event probabilities are the steady state for your system. In other words, if you were to use monte carlo methods with a frequentist approach, the long-term average would approach the BM event probabilities. That's the real benefit of Bayesian modelling. You end up with the same fundamental answer as the Frequentist approach, but it's often a lot quicker. Especially in multivariate space.

It's not that Bayesian methods are all-powerful. They're not. What makes them an essential part of your toolkit, is that Bayesian methods allow you to ask the right question. For example, I wanted to know if a vendor could deliver the necessary tolerances on a part. We could have ordered 100 units, measured them and quantified the part variation. That's a lot of time, effort and expense. Using Bayesian methods I was able to answer this question from 5 units. I was able to include our previous experience with this and similar vendors into my prior, and then update it based on the 5 units. Next I marginalised over the posterior to determine the probability that the typical tolerance would be better than X. 2 days work and not 2 months.

It depends on your background. I'm an Astrophysicist and a Cosmologist. Bayesian methods weren't a big leap for me. What I recommend is that rather than focussing specifically on Bayesian methods, you try to learn about statistics in general. Bayesian methods are a great tool. They aren't the only tool though. What you should be doing is acquiring as many tools as you can. That way you can answer as many questions as possible. If you already have a solid background in statistics, then make sure Bayesian methods are in your toolbox. The resources listed to the left are a great suggestion.