Bayes's Theorem: What's the Big Deal?

Good article.

I'm only a bit disappointed that the author seems not to realize that Bayes' theorem is just a simple consequence of probability theory, and should be attractive not because "maybe the brain is Bayesian", but because it is based on sound set-theoretic and analytic principles. If Bayes' theorem is false, so is probability theory, and so is nearly everything we know about probability.

Edit: Here is a good explanation of the theorem that makes it visually clear how only set theory is involved in deriving it: https://oscarbonilla.com/2009/05/visualizing-bayes-theorem/

Read through most of the article just for "people can abuse priors"? Come on. Anything, used wrongly, can promote superstition and pseudoscience.

> Oh please. You can do plenty of psuedoscience and superstition with good old frequentist statistics.

That's not really the point. The article is simply saying that Bayesian methods are not a silver bullet; it's not saying that other methods of statistics are free from problems.

I prefer to think of it in terms of the statistical inversion problem. That is, we have an event(s) that occur, which we may imperfectly understand. We take noisy measurements of that event. Clearly, the causal relationship is the events cause the measurements - a bad measurement does not cause the event to move.

But, in practice all we have are measurements, and from that we want to find an optimal (or good) estimate for what the events were. Hence, inversion.

Bayes formula expresses P(x|y) in terms of P(Y|x), so you can perform the inversion using bayes.

In many fields establishing the prior is difficult, hence frequentist methods are popular.

There are many techniques for the statistical inversion problem. Trying to track a ballistic object in a vacuum? Fit the measurements to a second order polynomial (parabola) and you are done (well, you have to decide least squares vs robust methods, but it is not such a hard problem in the scheme of things). Trying to track a manuevering jet, stock prices, or disease incidence rates. Now your model of the problem is much less clear.

We model lack of information as random variables. It isn't "random" in the deterministic sense, just in the sense that we don't know. Establish a good probabilistic description of that lack of knowledge in your prior, and you are probably going to get good result: this jet fighter is probabilisticly either moving straight, performing a coordinated turn, or performing an uncoordinated turn. Use a Markov chain to model those likelihoods (e.g.), and you may end up with good results. But if your modeling of the prior is poor, well, good luck to you, your output is probably nonsense.

I think you're missing the broader argument, which is using 'mathy' concepts to dress up poor reasoning. Obviously priors matter, but what matters most of all is how good/complete your evidence is. Using a mathematical formula to lend credence to weak evidence (through liberal use of assumptions) is a hallmark of pseudoscience. The same could be said of many of the abuses of statistics and Bayes theorem is merely one good example of this.