# What’s the Deal With Bayesian Statistics?

The Rev. Thomas Bayes, whose simple and fairly narrow formula eventually inspired a vast new field of statistics.

It is really, really hard to find stuff to write about other than the C19 pandemic. So instead, how about a complicated post about Bayesian statistics? The underlying study that prompted this post is, unfortunately, about coronavirus testing, but it’s really about Bayesian statistics. Honest.

Bad news first: most of you probably don’t know what Bayesian statistics is, and it would take way too long to explain properly. I’m not even sure I could do a fair job of it, actually, so just skip this whole post if it’s not your thing. OTOH, if you’re really bored, go ahead and read it and then use the internet machine to enlighten yourself about the whole subject.

The whole thing starts with that Santa Clara study of C19 antibody testing. The authors concluded that the infection rate of the entire population was between 2.5 percent and 4.2 percent, which includes even those who had no symptoms and never knew they were infected. Unfortunately, the antibody test had a false positive rate of 1.5 percent, which means that the true range of the infection rate was about 0 to 5 percent.

So were the authors justified in using their narrower range of 2.5 to 4.2 percent? The loud and clear voice of the statistics community was no. But today, Andrew Gelman, who was quite critical of the study, writes this:

It seems clear to me that the authors of that study had reasons for believing their claims, even before the data came in. They viewed their study as confirmation of their existing beliefs. They had good reasons, from their perspective.

….Itâ€™s a Bayesian thing. Part of Bayesian reasoning is to think like a Bayesian; another part is to assess other peopleâ€™s conclusions as if they are Bayesians and use this to deduce their priors. Iâ€™m not saying that other researchers are Bayesianâ€”indeed Iâ€™m not always so Bayesian myselfâ€”rather, Iâ€™m arguing that looking at inferences from this implicit Bayesian perspective can be helpful, in the same way that economists can look at peopleâ€™s decisions and deduce their implicit utilities.

This has always been my big problem with Bayesian statistics and Gelman makes it very clear in this post. Bayesiansim requires you to start with “prior beliefs” as part of the methodology. That can be a “flat prior,” in which you assume nothing, but then you end up with little more than you’d get from ordinary frequentist statistics. On the other hand, if you assume a meaningful prior, it means that your results can be stretched into almost anything you want. You can simply explain, in whatever detail you want, that “previous studies have shown X,” and when you plug that in as your prior you get a new result of X Â± Î”.

But as Gelman acknowledges, this makes “prior” just a nicer word for “bias.” That’s not a very good selling point. At the same time, without priors it’s not clear to me how useful Bayesian statistics is in the first place. For now, then, I remain confused.

### DONALD TRUMP & DEMOCRACY

Mother Jones was founded to do things differently in the aftermath of a political crisis: Watergate. We stand for justice and democracy. We reject false equivalence. We go after, and go deep on, stories others donâ€™t. And weâ€™re a nonprofit newsroom because we knew corporations and billionaires would never fund the journalism we do. Our reporting makes a difference in policies and peopleâ€™s lives changed.

And we need your support like never before to vigorously fight back against the existential threats American democracy and journalism face. Weâ€™re running behind our online fundraising targets and urgently need all hands on deck right now. We canâ€™t afford to come up shortâ€”we have no cushion; we leave it all on the field.

Please help with a donation today if you canâ€”even just a few bucks helps. Not ready to donate but interested in our work? Sign up for our Daily newsletter to stay well-informedâ€”and see what makes our people-powered, not profit-driven, journalism special.

### DONALD TRUMP & DEMOCRACY

Mother Jones was founded to do things differently in the aftermath of a political crisis: Watergate. We stand for justice and democracy. We reject false equivalence. We go after, and go deep on, stories others donâ€™t. And weâ€™re a nonprofit newsroom because we knew corporations and billionaires would never fund the journalism we do. Our reporting makes a difference in policies and peopleâ€™s lives changed.

And we need your support like never before to vigorously fight back against the existential threats American democracy and journalism face. Weâ€™re running behind our online fundraising targets and urgently need all hands on deck right now. We canâ€™t afford to come up shortâ€”we have no cushion; we leave it all on the field.

Please help with a donation today if you canâ€”even just a few bucks helps. Not ready to donate but interested in our work? Sign up for our Daily newsletter to stay well-informedâ€”and see what makes our people-powered, not profit-driven, journalism special.