Are degrees of beliefs represented as probabilities just emotions?

Question

After reading up on probability theory, it seems that there are two camps: objective and subjective probability theory.

Objective probability can refer to something like the probability of a dice roll being 1/6. Often, these are frequentist and what they really mean is “if I was to toss this dice many times, about 1/6 of them will land on 1.” Okay, so in this case, the probability just means historical frequency. So ultimately we already have a term for this: frequency, that we then rephrase as probability for god knows what reason.

Subjective probability, common in Bayesianism, recognizes that probability is a matter of how strongly you believe something. But why on earth is this relevant, important, significant, or much less even talked about? It came to my surprise that entire books have been written about this concept and how to systematize your “degrees of belief” when coming across evidence. But since when did how strongly you believe something have anything to say about what’s true? Why would how many people strongly believe that the earth is flat matter in philosophy? If it doesn’t, as subjective probability proponents will hopefully recognize, why bother systematizing it?

If philosophy is about seeking truth and not bookkeeping your feelings, why is this part of it?

Answer 1

Rather than objective and subjective, it might be better to say that broadly speaking accounts of probability divide into physical or epistemic. Bayesians and others fall into the epistemic camp and treat probabilities as a way to represent degrees of rational belief.

This is valuable in philosophy because we are interested not only in what things are true but also in questions like: How do we know things are true? Is there such a thing as a justification for a belief? Are some beliefs more strongly justified than others? If so, how? Are there criteria for determining what combinations of beliefs are inconsistent? Is it possible to devise a formal system for describing how we confirm and disconfirm beliefs? All of these questions form part of epistemology.

If you proceed to ask why these questions are important, then it is because life is full of uncertainty. Much of the information we have is inaccurate and imprecise. Arguably, all of our information is incomplete. We are constantly compelled as a matter of practical necessity to form uncertain beliefs and make decisions under this uncertainty. Being able to quantify uncertainty helps greatly if we wish to make good decisions and avoid bad decisions. Probability theory, understood epistemically, is a step in the direction of quantifying uncertainty. Not perfect, but a good approximation.

The fact that probability theory is useful for this purpose can be justified theoretically in at least two ways. One was worked out by Richard Cox in a series of articles published as The Algebra of Probable Inference. Cox shows how probability theory arises as a way of describing how degrees of rational belief are conserved in valid arguments. Another approach was taken by Bruno de Finetti starting from decision theory and using Dutch book arguments. De Finetti shows how probability theory can be derived from assumptions about how to avoid irrational combinations of beliefs, where irrationality is characterised by the criterion that were you to bet on your beliefs you would find yourself in a position where you are bound to lose.

Using probabilities epistemically is highly practical. It is used in risk analysis. In actuarial calculations by insurance companies. In forensic analysis. In evidence based medicine. In machine learning. In cryptography.

One common application is probabilistic information retrieval. Search engines, at least in the early days, expressly use Bayesian methods to return results. When you type a search expression, the engine determines how probable it is that you are interested in a particular document or web page, conditional upon the given search terms, and it ranks the results accordingly. When you click to read a page, the engine updates its assessment to determine how probable it is that you are interested in some other documents, given the ones you have already looked at.

Answer 2

Probabilities are subjective in the way that an estimate might be. If I show you a pile of stones and ask how many stones you think are in the pile, you will come up with your guess, which is likely be different from mine, based on your personal assessment.

Clearly, the guess is only as good as the guesser. If you show me a pile of thousands of stones and I stupidly guess there are a hundred, my ridiculous guess doesn't increase the likelihood of there being a hundred. Probability theory is subject to the 'garbage in, garbage out' rule in a big way.

Leaving aside stupidity (as much as I am able!), suppose I go on to ask you how confident you are that your guess about the number of stones in the pile is correct to within ten percent- how would you answer that? Suppose I also ask how confident are you that your guess is correct to within fifty percent- how would you answer that? And suppose I ask you how confident you are that your guess is correct to within 99.999 percent- how would you answer that?

If you are rational, your degree of confidence will increase in line with the size of the margin for error I give you. To take another analogy, if I give you a dart, you are probably more confident in hitting the board than in hitting the bullseye. If the board is within touching distance, you will probably say you are certain that you can put the dart in it. If the board is a mile away, you will be certain that you can't hit the bullseye. Between those extremes, you are likely to have intermediate degrees of confidence in the various outcomes. Probabilities are just a way of comparing different degrees of confidence. They should line up with frequentist probabilities in the sense that if I judge that I have a 0.5% chance of hitting the bullseye, I should find that I hit it five times in a thousand throws, or thereabouts. It's not an exact science, but it is still meaningful. If I judge that my probability of hitting a twenty is higher than that of hitting a bull, it is a meaningful assessment of the likely outcomes, and will be born out in tests when I try to hit those targets. So you are going too far if you write off subjective probabilities as 'just' emotions.

Answer 3

The bottom line, the origin of probability, is an engineering design problem. We humans are interested in how to design a system of reasoning that produces judgments that tend to yield better outcomes.

It just so happens that probabilistic reasoning tends to yield better outcomes in many domains. If we are designing robots, a robot capable of weighing the probabilities of different outcomes has a tool it can use to outperform a robot that only focuses on a single outcome and assumes that will definitely happen. If we are evaluating stocks on the stock market, an analyst capable of estimating the numerical risk of different stocks will outperform an analyst who can't do that. If we are predicting where photons will be observed in a quantum physics experiment, someone incapable of probabilistic reasoning simply can't do the job.

Precisely which probabilities an agent ought to assign for maximum performance will depend on many factors, and sometimes there might be no way to pin down the exact best set of probabilities. (Or maybe there is, in theory - see the Aumann Agreement Theorem). But we do find that agents that use probabilistic reasoning tend to outperform agents who don't use it. So it's important. Finding the right probabilities - or at least good ones, reasonable ones - is often key to an agent's success.