Anti-inductive priors

I used to think of Bayesianism as composed of two distinct parts: (1) setting priors and (2) updating by conditionalizing. In my mind, this second part was the crown jewel of Bayesian epistemology, while the first part was a little more philosophically problematic. Conditionalization tells you that for any prior distribution you might have, there is a unique rational set of new credences that you should adopt upon receiving evidence, and tells you how to get it. As to what the right priors are, well, that’s a different story. But we can at least set aside worries about priors with assurances about how even a bad prior will eventually be made up for in the long run after receiving enough evidence.

But now I’m realizing that this framing is pretty far off. It turns out that there aren’t really two independent processes going on, just one (and the philosophically problematic one at that): prior-setting. Your prior fully determines what happens when you update by conditionalization on any future evidence you receive. And the set of priors consistent with the probability axioms is large enough that it allows for this updating process to be extremely irrational.

I’ll illustrate what I’m talking about with an example.

Let’s imagine a really simple universe of discourse, consisting of just two objects and one predicate. We’ll make our predicate “is green” and denote objects a_1 and a_2 . Now, if we are being good Bayesians, then we should treat our credences as a probability distribution over the set of all state descriptions of the universe. These probabilities should all be derivable from some hypothetical prior probability distribution over the state descriptions, such that our credences at any later time are just the result of conditioning that prior on the total evidence we have by that time.

Let’s imagine that we start out knowing nothing (i.e. our starting credences are identical to the hypothetical prior) and then learn that one of the objects (a_1 ) is green. In the absence of any other information, then by induction, we should become more confident that the other object is green as well. Is this guaranteed by just updating?

No! Some priors will allow induction to happen, but others will make you unresponsive to evidence. Still others will make you anti-inductive, becoming more and more confident that the next object is not green the more green things you observe. And all of this is perfectly consistent with the laws of probability theory!

Take a look at the following three possible prior distributions over our simple language:

Screen Shot 2018-10-21 at 1.58.45 PM.png

According to P_1 , your new credence in Ga_2 after observing Ga_1 is P_1(Ga_2 | Ga_1) = 0.80 , while your prior credence in Ga_2 was 0.50. Thus P_1 is an inductive prior; you get more confident in future objects being green when you observe past objects being green.

For P_2 , we have that P_2(Ga_2 | Ga_1) = 0.50 , and P_2(Ga_2) = 0.50 as well. Thus P_2 is a non-inductive prior: observing instances of green things doesn’t make future instances of green things more likely.

And finally, P_3(Ga_2 | Ga_1) = 0.20 , while P_3(Ga_2) = 0.5 . Thus P_3 is an anti-inductive prior. Observing that one object is green makes you more than two times less confident confident that the next object will be green.

The anti-inductive prior can be made even more stark by just increasing the gap between the prior probability of Ga_1 \wedge Ga_2 and Ga_1 \wedge -Ga_2 . It is perfectly consistent with the axioms of probability theory for observing a green object to make you almost entirely certain that the next object you observe will not be green.

Our universe of discourse here was very simple (one predicate and two objects). But the point generalizes. Regardless of how many objects and predicates there are in your language, you can have non-inductive or anti-inductive priors. And it isn’t even the case that there are fewer anti-inductive priors than inductive priors!

The deeper point here is that the prior is doing all the epistemic work. Your prior isn’t just an initial credence distribution over possible hypotheses, it also dictates how you will respond to any possible evidence you might receive. That’s why it’s a mistake to think of prior-setting and updating-by-conditionalization as two distinct processes. The results of updating by conditionalization are determined entirely by the form of your prior!

This really emphasizes the importance of having good criterion for setting priors. If we’re trying to formalize scientific inquiry, it’s really important to make sure our formalism rules out the possibility of anti-induction. But this just amounts to requiring rational agents to have constraints on their priors that go above and beyond the probability axioms!

What are these constraints? Do they select one unique best prior? The challenge is that actually finding a uniquely rationally justifiable prior is really hard. Carnap tried a bunch of different techniques for generating such a prior and was unsatisfied with all of them, and there isn’t any real consensus on what exactly this unique prior would be. Even worse, all such suggestions seem to end up being hostage to problems of language dependence – that is, that the “uniquely best prior” changes when you make an arbitrary translation from your language into a different language.

It looks to me like our best option is to abandon the idea of a single best prior (and with it, the notion that rational agents with the same total evidence can’t disagree). This doesn’t have to lead to total epistemic anarchy, where all beliefs are just as rational as all others. Instead, we can place constraints on the set of rationally permissible priors that prohibit things like anti-induction. While identifying a set of constraints seems like a tough task, it seems much more feasible than the task of justifying objective Bayesianism.

Making sense of improbability

Imagine that you take a coin that you believe to be fair and flip it 20 times. Each time it lands heads. You say to your friend: “Wow, what a crazy coincidence! There was a 1 in 220 chance of this outcome. That’s less than one in a million! Super surprising.”

Your friend replies: “I don’t understand. What’s so crazy about the result you got? Any other possible outcome (say, HHTHTTTHTHHHTHTTHHHH) had an equal probability as getting all heads. So what’s so surprising?”

Responding to this is a little tricky. After all, it is the case that for a fair coin, the probability of 20 heads = the probability of HHTHTTTHTHHHTHTTHHHH = roughly one in a million.

Simpler Example_ Five Tosses.png

So in some sense your friend is right that there’s something unusual about saying that one of these outcomes is more surprising than another.

You might answer by saying “Well, let’s parse up the possible outcomes by the number of heads and tails. The outcome I got had 20 heads and 0 tails. Your example outcome had 12 heads and 8 tails. There are many many ways of getting 12 heads and 8 tails than of getting 20 heads and 0 tails, right? And there’s only one way of getting all 20 heads. So that’s why it’s so surprising.”

Probability vs. Number of heads (1).png

Your friend replies: “But hold on, now you’re just throwing out information. Sure my example outcome had 12 heads and 8 tails. But while there’s many ways of getting that number of heads and tails, there’s only exactly one way of getting the result I named! You’re only saying that your outcome is less likely because you’ve glossed over the details of my outcome that make it equally unlikely: the order of heads and tails!”

I think this is a pretty powerful response. What we want is a way to say that HHHHHHHHHHHHHHHHHHHH is surprising while HHTHTTTHTHHHTHTTHHHH is not, not that 20 heads is surprising while 12 heads and 8 tails is unsurprising. But it’s not immediately clear how we can say this.

Consider the information theoretic formalization of surprise, in which the surprisingness of an event E is proportional to the negative log of the probability of that event: Sur(E) = -log(P(E)). There are some nice reasons for this being a good definition of surprise, and it tells us that two equiprobable events should be equally surprising. If E is the event of observing all heads and E’ is the event of observing the sequence HHTHTTTHTHHHTHTTHHHH, then P(E) = P(E’) = 1/220. Correspondingly, Sur(E) = Sur(E’). So according to one reasonable formalization of what we mean by surprisingness, the two sequences of coin tosses are equally surprising. And yet, we want to say that there is something more epistemically significant about the first than the second.

(By the way, observing 20 heads is roughly 6.7 times more surprising than observing 12 heads and 8 tails, according to the above definition. We can plot the surprise curve to see how maximum surprise occurs at the two ends of the distribution, at which point it is 20 bits.)

Surprise vs. number of heads (1).png

So there is our puzzle: in what sense does it make sense to say that observing 20 heads in a row is more surprising than observing the sequence HHTHTTTHTHHHTHTTHHHH? We certainly have strong intuitions that this is true, but do these intuitions make sense? How can we ground the intuitive implausibility of getting 20 heads? In this post I’ll try to point towards a solution to this puzzle.

Okay, so I want to start out by categorizing three different perspectives on the observed sequence of coin tosses. These correspond to (1) looking at just the outcome, (2) looking at the way in which the observation affects the rest of your beliefs, and (3) looking at how the observation affects your expectation of future observations. In probability terms, these correspond to the P(E), P(T| T) and P(E’ | E).

Looking at things through the first perspective, all outcomes are equiprobable, so there is nothing more epistemically significant about one than the other.

But considering the second way of thinking about things, there can be big differences in the significance of two equally probable observations. For instance, suppose that our set of theories under consideration are just the set of all possible biases of the coin, and our credences are initially peaked at .5 (an unbiased coin). Observing HHTHTTTHTHHHTHTTHHHH does little to change our prior. It shifts a little bit in the direction of a bias towards heads, but not significantly. On the other hand, observing all heads should have a massive effect on your beliefs, skewing them exponentially in the direction of extreme heads biases.

Importantly, since we’re looking at beliefs about coin bias, our distributions are now insensitive to any details about the coin flip beyond the number of heads and tails! As far as our beliefs about the coin bias go, finding only the first 8 to be tails looks identical to finding the last 8 to be tails. We’re not throwing out the information about the particular pattern of heads and tails, it’s just become irrelevant for the purposes of consideration of the possible biases of the coin.

Visualizing change in beliefs about coin bias.png

If we want to give a single value to quantify the difference in epistemic states resulting from the two observations, we can try looking at features of these distributions. For instance, we could look at the change in entropy of our distribution if we see E and compare it to the change in entropy upon seeing E’. This gives us a measure of how different observations might affect our uncertainty levels. (In our example, observing HHTHTTTHTHHHTHTTHHHH decreases uncertainty by about 0.8 bits, while observing all heads decreases uncertainty by 1.4 bits.) We could also compare the means of the posterior distributions after each observation, and see which is shifted most from the mean of the prior distribution. (In this case, our two means are 0.57 and 0.91).

Now, this was all looking at things through what I called perspective #2 above: how observations affect beliefs. Sometimes a more concrete way to understand the effect of intuitively implausible events is to look at how they affect specific predictions about future events. This is the approach of perspective #3. Sticking with our coin, we ask not about the bias of the coin, but about how we expect it to land on the next flip. To assess this, we look at the posterior predictive distributions for each posterior:

Posterior Predictive Distributions.png

It shouldn’t be too surprising that observing all heads makes you more confident that the next coin will land heads than observing HHTHTTTHTHHHTHTTHHHH. But looking at this graph gives a precise answer to how much more confident you should be. And it’s somewhat easier to think about than the entire distribution over coin biases.

I’ll leave you with an example puzzle that relates to anthropic reasoning.

Say that one day you win the lottery. Yay! Super surprising! What an improbable event! But now compare this to the event that some stranger Bob Smith wins the lottery. This doesn’t seem so surprising. But supposing that Bob Smith buys lottery tickets at the same rate as you, the probability that you win is identical to the probability that Bob Smith wins. So… why is it any more surprising when you win?

This seems like a weird question. Then again, so did the coin-flipping question we started with. We want to respond with something like “I’m not saying that it’s improbable that some random person wins the lottery. I’m interested in the probability of me winning the lottery. And if we parse up the outcomes as that either I win the lottery or that somebody else wins the lottery, then clearly it’s much more improbable that I win than that somebody else wins.”

But this is exactly parallel to the earlier “I’m not interested in the precise sequence of coin flips, I’m just interested in the number of heads versus tails.” And the response to it is identical in form: If Bob Smith, a particular individual whose existence you are aware of, wins the lottery and you know it, then it’s cheating to throw away those details and just say “Somebody other than me won the lottery.” When you update your beliefs, you should take into account all of your evidence.

Does the framework I presented here help at all with this case?

A simple probability puzzle

In front of you is an urn containing some unknown quantity of balls. These balls are labeled 1, 2, 3, etc. They’ve been jumbled about so as to be in no particular order within the urn. You initially consider it equally likely that the urn contains 1 ball as that it contains 2 balls, 3 balls, and so on, up to 100 balls, which is the maximum capacity of the urn.

Now you reach in to draw out a ball and read the number on it: 34. What is the most likely theory for how many balls the urn contains?

 

 

(…)

 

(Think of an answer before reading on.)

 

(…)

 

 

The answer turns out to be 34!

Hopefully this is a little unintuitive. Specifically, what seems wrong is that you draw out a ball and then conclude that this is the ball with the largest value on it. Shouldn’t extreme results be unlikely? But remember, the balls were randomly jumbled about inside the urn. So whether or not the number on the ball you drew is at the beginning, middle, or end of the set of numbers is pretty much irrelevant.

What is relevant is the likelihood: Pr(There are N balls | I drew a ball numbered 34). And the value of this is simply 1/N.

In general, comparing the theory that there are N balls to the theory that there are M balls, we look at the likelihood ratio: Pr(There are N balls | I drew a ball numbered 34) / Pr(There are M balls | I drew a ball numbered 34). This is simply M/N.

Thus we see that our prior odds get updated by a factor that favors smaller values of N, as long as N ≥ 34. The likelihood is zero up to N = 33, maxes at 34, and then decreases steadily after it as N goes to infinity. Since our prior was evenly spread out between N = 1 and 100 and zero everywhere else, our posterior will be peaked at 34 and decline until 100, after which it will drop to zero.

One way to make this result seem more intuitive is to realize that while strictly speaking the most probable number of balls in the urn is 34, it’s not that much more probable than 35 or 36. The actual probability of 34 is still quite small, it just happens to be a little bit more probable than its larger neighbors. And indeed, for larger values of the maximum capacity of the urn, the relative difference between the posterior probability of 34 and that of 35 decreases.