Contrasting Bayesian Probability With Humean Probability

A Non-Comprehensive Overview

In a previous post, I wrote about how John Earman got Hume wrong and how miracle apologists need to stop citing Earman to rebut Hume. I closed that post by saying:

“To the extent that Hume is wrong, it’s because it’s difficult to update his probability into Bayesian terms without revision. In fact, one of the best Hume scholars is Peter Millican, who has a lot of interesting criticisms of Hume’s maxim on miracles on this very point.”

In an upcoming post, I’m going to discuss that. But before we can do that, we have to understand key differences between Bayesian numerical probability and Humean factual probability. This post will be a rough guide. It’s not a comprehensive overview, but the key points you’ll need to understand Millican’s criticism of Hume’s maxim.

Language Of Uncertainty And Impossibility

To start, Bayesian probability is numerical in nature, while Hume’s probability is factual in nature. This affects how the two methods conceptualize uncertainty and possibility.

Namely, Bayesian probability will give you a numerical answer to a probability question, which will guide one to label their confidence as low, moderate, or high. Meanwhile, the Humean will define their confidence as something along the lines of “undetermined,” “insufficient evidence,” “unlikely,” “likely,” “morally certain,” and so on.

The actual math of Bayes’ Theorem doesn’t tell you if two hypotheses are mutually exclusive or contradictory. And so, for a Bayesian, nothing is seen as impossible unless one of your priors specifically says so. If nothing is considered impossible a priori (with a prior probability of 0), then nothing will be judged impossible a posteriori.

Hume is similar in that he does not a priori consider anything impossible, so long as its truth does not imply a contradiction. But when he is formulating a posteriori probability, he is looking for moral certainty, which is the certainty to act. In the abstract, the Humean may never be certain that something is impossible or that something is certain. Still, he can be confident enough that something is certain or impossible to act as if it were.

And so, for Hume, if there is no good evidence supporting a proposition, we can be morally certain it is untrue. What’s more, we can be morally certain something is impossible because it lacks evidence supporting it and because there’s abundant evidence against it.

Low Credence vs. A Posteriori Impossibility

Bayesian probability functions something like a sliding scale, where at one end our credence in something is at 0 or impossible, while at the other end, it’s at 1 or certain. Having said that, unless we define certain priors at 0 or 1, our credence will never reach that level of certainty or impossibility. And so, functionally, Bayesians almost never outright reject a proposition, so much as they give it a very low credence.

Meanwhile, Hume’s probability is more like an algorithm based on the information one actually possesses. Humeans will reject a proposition if it lacks evidence of a sufficient quality, especially if there’s strong evidence against that proposition. That is, the Humean will often judge a proposition as a posteriori impossible. Sure, he could be mistaken, but until he has evidence that he is mistaken, he is morally certain of the impossibility of a proposition.

To use an illustration, a Bayesian and Humean may hear of a body builder bench-pressing 2000 pounds. The Bayesian puts low credence in this rumor, because of prior knowledge they have about human anatomy, physiology and physics. He may dismiss the rumor, but he doesn’t consider it impossible. The Humean, meanwhile, believes this rumor is a posteriori impossible, given the contrary evidence we have to the possibility of humans lifting 2000 pounds, and the low quality evidence for the proposition, as that evidence is merely a rumor.

Some may say that this makes the Humean more dogmatic than the Bayesian, but this is untrue. The Humean’s confidence can be subverted with future evidence of sufficient quality, increasing the probability of the previously impossible proposition (it just so happens to be the case that rumors never amount to this quality evidence).

The key difference between the two is Bayesians will maintain some credence in extremely improbable propositions, even without good evidence. Meanwhile, the Humean will deny a sufficiently improbable proposition until he has sufficient evidence to subvert that confidence.

Weighing Probability And Suspending Judgment

The other important difference between the two probabilities is that Humean probability is judged by relative evidential quality, while Bayesian probability precisely calculates credences in propositions.

This may seem like a distinction without a difference, but the key point is that with relative probabilities, you don’t need precision as you do with calculated probabilities. For instance, you don’t need to know the exact temperature of three cups of tea to know which cup is the hottest between a near-frozen cup, a room-temperature cup, and a boiling cup. It would be great to know the precise temperatures of each cup of tea, but in life we often don’t have that level of precise information.

What’s more, Humean probability usually weighs two mutually exclusive probabilities, like a court case does, with different evidence having different “weight” when considering the verdict. It’s algorithmic, where some forms of evidence naturally carry more weight and are judged more probable than others. The “algorithm” of probability looks something like this, where the “>” stands for “is more probable”:

A Law of Nature > Proof > Probability > Impossibility.

The reason why this is important is that a Bayesian may calculate two hypotheses as equally probable, have no credence about them being mutually exclusive, and thus believe both of them, and look to somehow harmonize them together.

In contrast, the Humean will suspend judgment on the truth of the matter until he can find more evidence supporting one hypothesis over another. That’s not to say that Humeans wouldn’t try to harmonize two hypotheses, but as a general methodology, he’s trying to affirm one hypothesis over another.

In this way, the Humean methodology is extremely useful when assessing two disparate, mutually exclusive probabilities, like a one-off anomalous observation against a law of nature. But it’s much less certain when assessing uncertain probabilities.

And so, the Humean won’t confidently side with one probability over another just because he weighs it slightly more. Rather, he will tepidly affirm that hypothesis with minimal confidence, with as many caveats as possible. More likely, he will suspend judgment until a better explanation fits the data.

All of this is in contrast with the Bayesian, who is not judging different hypotheses as mutually exclusive, and will ultimately affirm the hypothesis that is the most probable relative to other hypotheses, oftentimes even when the probability of that hypothesis is still very low.

They Aren’t Mutually Exclusive Methodologies

To wrap up, I want to make clear that Humean and Bayesian methodologies are not entirely mutually exclusive. Indeed, I think they enhance one another.

For the Humean, Bayesian reasoning is an extremely useful tool to assess the data we have for causal relationships that we are familiar with. Bayes’ Theorem is extremely helpful for evaluating murder mysteries, election forecasts, sporting events, and other events that we have lots of data for.

Indeed, the Humean would go a step further and say that much of the data that informs these Bayesian formulas have a quasi-objective quality, and so subjective credences aren’t necessary.

That is, in theory, we can measure the probability of something like a political party winning an election, given various indicators, given how those indicators varied in the past and were correlated with various election outcomes. To learn more of this I highly recommend everyone interested in probability read The Signal and the Noise by Nate Silver. Silver is a Bayesian, but in my opinion, he’s grounded in strong Humean instincts. I learned and internalized much of what I know about probability from that book about a decade ago.

For the Bayesian, Humean factual probability maps on closer to how most people practically understand probabilistic reasoning. There’s a joke in election modeling that the general public only understands three probabilities: 0 percent, 50 percent, and 100 percent. In my opinion, this reflects the intuitive appeal of Humean reasoning which has impossibilities (0 percent), probabilities (such as 50 percent or so), and proofs (100 percent). Similarly, when a piece of evidence seems fraudulent or bad, most people don’t discount it near 0, they operationalize it as 0. For example, if we’re trying to piece together what happened with an event, and all we have are witnesses, and one of those witnesses is a proven liar, most people just disregard what that person has to say completely instead of greatly discounting that testimony.

And so, a Bayesian may reject Humean language of a posteriori impossibility or moral certainty, but they should have an understanding of these terms and heuristics, so they can communicate Bayesian probability to non-Bayesian audiences.

In all, Bayes and Hume are more harmonious with each other than disharmonious. To the extent that they disagree, they bring up deep, interesting, and important philosophical disagreements (which I don’t have time to go into here).

Okay! So that’s a rough contrast between Humean and Bayesian probability. In an upcoming post, I’ll explain Peter Millican’s criticism of Hume’s maxim on miracles, and why it doesn’t easily translate into Bayesian probability.