Priors as Mathematical Objects
Followup to: "Inductive Bias"
What exactly is a "prior", as a mathematical object? Suppose you're looking at an urn filled with red and white balls. When you draw the very first ball, you haven't yet had a chance to gather much evidence, so you start out with a rather vague and fuzzy expectation of what might happen - you might say "fifty/fifty, even odds" for the chance of getting a red or white ball. But you're ready to revise that estimate for future balls as soon as you've drawn a few samples. So then this initial probability estimate, 0.5, is not repeat not a "prior".
An introduction to Bayes's Rule for confused students might refer to the population frequency of breast cancer as the "prior probability of breast cancer", and the revised probability after a mammography as the "posterior probability". But in the scriptures of Deep Bayesianism, such as Probability Theory: The Logic of Science, one finds a quite different concept - that of prior information, which includes e.g. our beliefs about the sensitivity and specificity of mammography exams. Our belief about the population frequency of breast cancer is only one small element of our prior information.
In my earlier post on inductive bias, I discussed three possible beliefs we might have about an urn of red and white balls, which will be sampled without replacement:
- Case 1: The urn contains 5 red balls and 5 white balls;
- Case 2: A random number was generated between 0 and 1, and each ball was selected to be red (or white) at this probability;
- Case 3: A monkey threw balls into the urn, each with a 50% chance of being red or white.
In each case, if you ask me - before I draw any balls - to estimate my marginal probability that the fourth ball drawn will be red, I will respond "50%". And yet, once I begin observing balls drawn from the urn, I reason from the evidence in three different ways:
- Case 1: Each red ball drawn makes it less likely that future balls will be red, because I believe there are fewer red balls left in the urn.
- Case 2: Each red ball drawn makes it more plausible that future balls will be red, because I will reason that the random number was probably higher, and that the urn is hence more likely to contain mostly red balls.
- Case 3: Observing a red or white ball has no effect on my future estimates, because each ball was independently selected to be red or white at a fixed, known probability.
Suppose I write a Python program to reproduce my reasoning in each of these scenarios. The program will take in a record of balls observed so far, and output an estimate of the probability that the next ball drawn will be red. It turns out that the only necessary information is the count of red balls seen and white balls seen, which we will respectively call R and W. So each program accepts inputs R and W, and outputs the probability that the next ball drawn is red:
- Case 1: return (5 - R)/(10 - R - W) # Number of red balls remaining / total balls remaining
- Case 2: return (R + 1)/(R + W + 2) # Laplace's Law of Succession
- Case 3: return 0.5
These programs are correct so far as they go. But unfortunately, probability theory does not operate on Python programs. Probability theory is an algebra of uncertainty, a calculus of credibility, and Python programs are not allowed in the formulas. It is like trying to add 3 to a toaster oven.
To use these programs in the probability calculus, we must figure out how to convert a Python program into a more convenient mathematical object - say, a probability distribution.
Suppose I want to know the combined probability that the sequence observed will be RWWRR, according to program 2 above. Program 2 does not have a direct faculty for returning the joint or combined probability of a sequence, but it is easy to extract anyway. First, I ask what probability program 2 assigns to observing R, given that no balls have been observed. Program 2 replies "1/2". Then I ask the probability that the next ball is R, given that one red ball has been observed; program 2 replies "2/3". The second ball is actually white, so the joint probability so far is 1/2 * 1/3 = 1/6. Next I ask for the probability that the third ball is red, given that the previous observation is RW; this is summarized as "one red and one white ball", and the answer is 1/2. The third ball is white, so the joint probability for RWW is 1/12. For the fourth ball, given the previous observation RWW, the probability of redness is 2/5, and the joint probability goes to 1/30. We can write this as p(RWWR|RWW) = 2/5, which means that if the sequence so far is RWW, the probability assigned by program 2 to the sequence continuing with R and forming RWWR equals 2/5. And then p(RWWRR|RWWR) = 1/2, and the combined probability is 1/60.
We can do this with every possible sequence of ten balls, and end up with a table of 1024 entries. This table of 1024 entries constitutes a probability distribution over sequences of observations of length 10, and it says everything the Python program had to say (about 10 or fewer observations, anyway). Suppose I have only this probability table, and I want to know the probability that the third ball is red, given that the first two balls drawn were white. I need only sum over the probability of all entries beginning with WWR, and divide by the probability of all entries beginning with WW.
We have thus transformed a program that computes the probability of future events given past experiences, into a probability distribution over sequences of observations.
You wouldn't want to do this in real life, because the Python program is ever so much more compact than a table with 1024 entries. The point is not that we can turn an efficient and compact computer program into a bigger and less efficient giant lookup table; the point is that we can view an inductive learner as a mathematical object, a distribution over sequences, which readily fits into standard probability calculus. We can take a computer program that reasons from experience and think about it using probability theory.
Why might this be convenient? Say that I'm not sure which of these three scenarios best describes the urn - I think it's about equally likely that each of the three cases holds true. How should I reason from my actual observations of the urn? If you think about the problem from the perspective of constructing a computer program that imitates my inferences, it looks complicated - we have to juggle the relative probabilities of each hypothesis, and also the probabilities within each hypothesis. If you think about it from the perspective of probability theory, the obvious thing to do is to add up all three distributions with weightings of 1/3 apiece, yielding a new distribution (which is in fact correct). Then the task is just to turn this new distribution into a computer program, which turns out not to be difficult.
So that is what a prior really is - a mathematical object that represents all of your starting information plus the way you learn from experience.
Comments (8)
I'm confused when you say that the prior represents all your starting information plus the way you learn from experience. Isn't the way you learn from experience fixed, in this framework? Given that you are using Bayesian methods, so that the idea of a prior is well defined, then doesn't that already tell how you will learn from experience?
Hal, with a poor prior, "Bayesian updating" can lead to learning in the wrong direction or to no learning at all. Bayesian updating guarantees a certain kind of consistency, but not correctness. (If you have five city maps that agree with each other, they might still disagree with the city.) You might think of Bayesian updating as a kind of lower level of organization - like a computer chip that runs programs, or the laws of physics that run the computer chip - underneath the activity of learning. If you start with a maxentropy prior that assigns equal probability to every sequence of observations, and carry out strict Bayesian updating, you'll still never learn anything; your marginal probabilities will never change as a result of the Bayesian updates. Conversely, if you somehow had a good prior but no Bayesian engine to update it, you would stay frozen in time and no learning would take place. To learn you need a good prior and an updating engine. Taking a picture requires a camera, light - and also time.
This probably deserves its own post.
Another thing I don't fully understand is the process of "updating" a prior. I've seen different flavors of Bayesian reasoning described. In some, we start with a prior, get some information and update the probabilities. This new probability distribution now serves as our prior for interpreting the next incoming piece of information, which then causes us to further update the prior. In other interpretations, the priors never change; they are always considered the initial probability distribution. We then use those prior probabilities plus our sequence of observations since then to make new interpretations and predictions. I gather that these can be considered mathematically identical, but do you think one or the other is a more useful or helpful way to think of it?
In this example, you start off with uncertainty about which process put in the balls, so we give 1/3 probability to each. But then as we observe balls coming out, we can update this prior. Once we see 6 red balls for example, we can completely eliminate Case 1 which put in 5 red and 5 white. We can think of our prior as our information about the ball-filling process plus the current state of the urn, and this can be updated after each ball is drawn.
Hal,
You are being a bad boy. In his earlier discussion Eliezer made it clear that he did not approve of this terminology of "updating priors." One has posterior probability distributions. The prior is what one starts with. However, Eliezer has also been a bit confusing with his occasional use of such language as a "prior learning." I repeat, agents learn, not priors, although in his view of the post-human computerized future, maybe it will be computerized priors that do the learning.
The only way one is going to get "wrong learning" at least somewhat asymptotically is if the dimensionality is high and the support is disconnected. Eliezer is right that if one starts off with a prior that is far enough off, one might well have "wrong learning," at least for awhile. But, unless the conditions I just listed hold, eventually the learning will move in the right direction and head towards the correct answer, or probability distribution, at least that is what Bayes' Theorem asserts.
OTOH, the reference to "deep Bayesianism" raises another issue, that of fundamental subjectivism. There is this deep divide among Bayesians between the ones that are ultimately classical frequentists but who argue that Bayesian methods are a superior way of getting to the true objective distribution, and the deep subjectivist Bayesians. For the latter, there are no ultimately "true" probability distributions. We are always estimating something derived out of our subjective priors as updated by more recent information, wherever those priors came from.
Also, saying a prior should the known probability distribution, say of cancer victims, assumes that this probability is somehow known. The prior is always subject to how much information the assumer of a prior has when they being their process of estimation.
Eliezer ,
Just to be clear . . . going back to your first paragraph, that 0.5 is a prior probability for the outcome of one draw from the urn (that is, for the random variable that equals 1 if the ball is red and 0 if the ball is white). But, as you point out, 0.5 is not a prior probability for the series of ten draws. What you're calling a "prior" would typically be called a "model" by statisticians. Bayesians traditionally divide a model into likelihood, prior, and hyperprior, but as you implicitly point out, the dividing line between these is not clear: ultimately, they're all part of the big model.
Barkley, I think you may be regarding likelihood distributions as fixed properties held in common by all agents, whereas I am regarding them as variables folded into the prior - if you have a probability distribution over sequences of observables, it implicitly includes beliefs about parameters and likelihoods. Where agents disagree about prior likelihood functions, not just prior parameter probabilities, their beliefs may trivially fail to converge.
Andrew's point may be particularly relevant here - it may indeed be that statisticians call what I am talking about a "model". (Although in some cases, like the Laplace's Law of Succession inductor, I think they might call it a "model class"?) Jaynes, however, would have called it our "prior information" and he would have written "the probability of A, given that we observe B" as p(A|B,I) where I stands for all our prior beliefs including parameter distributions and likelihood distributions. While we may often want to discriminate between different models and model classes, it makes no sense to talk about discriminating between "prior informations" - your prior information is everything you start out with.
Eliezer, I am very interested in the Bayesian approach to reasoning you've outlined on this site, it's one of the more elegant ideas I've ever run into.
I am a bit confused, though, about to what extent you are using math directly when assessing truth claims. If I asked you for example "what probability do you assign to the proposition 'global warming is anthropogenic' ?" (say), would you tell me a number?
Or is this mostly about conceptually understanding that P(effect|~cause) needs to be taken into account?
If it's a number, what's your heuristic for getting there (i.e., deciding on a prior probability & all the other probabilities)?
If there's a post that goes into that much detail, I haven't seen it yet, though your explanations of Bayes theorem generally are brilliant.
See When (Not) To Use Probabilities.