The quickest possible introduction to Bayesian decision theory, pt. 1 : Probability
i.e. what you should have been taught in high school
Hello! First post just for my beloved paid subscribers, we’ll see if that does anything. I feel my conscience is clean putting this one behind a paywall, since it’s not like I’m putting any New and Important Ideas here like in my other posts. The goal, here, is to take you from little to no knowledge of Bayesianism to being able to basically follow these sorts of discussions if you’re willing to look more complicated ideas and terms up.
(My conscience would not be clean if someone got a paid subscription and then felt they didn’t get what they’re paying for. So I remind you you can get a 7-day free trial.)
You may have heard of “maximizing expected utility.” What this means is, when making a choice, you look at all the possibilities that might result from it, put a number on how good they are, multiply them by their probabilities, and add that up, and that’s the expected utility of the act. You supposedly want to choose the act that maximizes expected utility. For example, say someone offers you to pay $1 to take part in a bet that has a 50/50 chance if giving you $10, and otherwise you get nothing. The expected utility of not taking the bet is 0 (nothing will happen). Assuming you like getting $10 ten times more than you dislike losing $1, the expected utility of taking the bet is
And five is, famously, bigger than zero, so you should take the bet. Obvious, right? Decision theorists claim that this kind of reasoning can be extended, at least in principle, to all possible choices you might face. Say, for example, you’re thinking about whether to take a walk, when it might rain. If there is, say, a 10% chance of rain, the utility of staying home is 0, the utility of having a nice walk is 10, and the utility of taking a walk in the rain is -20, then the expected utility of taking a walk is
With seven being well-known for being larger than zero, so you should take a walk.
“But Flo,” I hear you saying, “those numbers seem pretty fake. Sure, it might work for gambles and money, but probability of rain? The utility of taking a walk? You can’t put exact numbers on that.” Fortunately, if you read this series of post, you will learn that the numbers are not totally fake. Arguably, insofar as you are rational, we should be able to come up with a probability function (which represents how confident you are that various possibilities might obtain) and a utility function (which represents how relatively valuable various states of affairs are to you) such that you’ll prefer one action to another if it has greater expected utility.
Now, it would be extremely difficult to actually determine someone’s precise probability and utility functions. More than likely, our actions are not totally consistent and so no one pair of functions would accurately describe a given person. But the arguments that a rational agent should be able to be so represented are forceful, and the machinery and rules implied by the formulas are unironically useful to apply to everyday life. I might not know my exact probability function, but I still sometimes notice patterns in my thinking that are inconsistent with any realistic probability function, so I know I’ve screwed up somewhere.
This is the first in a series. I meant for it to be one post, but it turns out the quickest possible introduction requires 2-3, and this part is 6,600 words. So today we’re just covering probability. Overall, the series will cover the following:
§1 Basic probability theory and “Dutch book” theorems
§1.1 Definitions and basic theorems
§1.2 More on updating (+base rates)
§1.3 Dutch book theorems
§2 Expected utility and representation theorems
§3 Suppositional decision theories
Keep reading with a 7-day free trial
Subscribe to Moral Law Within to keep reading this post and get 7 days of free access to the full post archives.