At the end of my post about other ways of looking at probability, I showed you a graph of evidence against probability. This is the relevant graph:
Looking at this graph was one of the most useful things I’ve ever done as a Bayesian. It shows, as I explained, exactly where most of the difficulty is in proving things, in coming up with hypotheses, etc. Another interesting aspect is the symmetry, showing that, to a Bayesian, confirmation and disconfirmation are pretty much the same thing. Probability doesn’t have a prejudice for or against any hypothesis, you just gather evidence and slide along the graph. Naïvely, the concept of Popper falsification doesn’t look terribly useful or relevant or particularly true.
So whence comes the success of the idea?
We can never confirm a hypothesis, only falsify it. Thus goes the saying. And even the staunchest Bayesian may look at the symmetry of Bayes, at the success of Popper, and awkwardly mumble some excuse. Or maybe I’m being unkind to staunch Bayesians and they’ve already thought about this. Whichever the case, if we have already rescued a hypothesis from the dark pits of hugely negative evidence once and almost confirmed it, it doesn’t immediately seem that far-fetched that we may once again bring it back from the dead, after it’s been falsified.
Whence comes the asymmetry? I phrased it all in a fairly suggestive way. Let’s think about a real-life example of a falsified theory that we all can agree will never be dug back up from its grave: Newtonian mechanics. What prevents Newtonian mechanics from coming back?
And then the answer becomes obvious: quantum mechanics does. Special relativity and general relativity and quantum mechanics are dancing on top of Newtonian mechanics’ grave and will stay there until shot dead. And that’s the progress of science, in a nutshell. That’s the meaning of falsification, to a Bayesian. That’s why we can only falsify hypotheses, but never confirm them.
The only way to kill a hypothesis is by bringing a new one to life.
Let’s suppose we’re at a point where Newtonian mechanics seems to be pretty much confirmed, is quite high. Then we observe the “anomalous” precession of the perihelion of Mercury. That’s strange. Einstein looks at the character of physical law, he looks at the Maxwell equations. Newtonian mechanics has action-at-a-distance, doesn’t it? is fairly low, so observing that anomalous precession is quite vexing to the theory.
Then Einstein comes up with his general theory of relativity, one that seems to take on the Machian spirit of previous laws. An object travelling at constant velocity should not be able to distinguish the world from one where it’s stationary and the universe is travelling at constant and opposite velocity: that’s Newton. An object under constant acceleration should not be able to distinguish the world from one where it’s stationary and the universe is under constant and opposite acceleration, which means that accelerated matter ought to generate gravitational waves: that’s Einstein.
Then we started collecting evidence that was increasingly vexing to Newtonian mechanics: the aforementioned precession of the perihelion, the deflection of light by the Sun, the gravitational redshift of light. That’s evidence that’s not covered at all by Newton, while it’s in fact predicted and entailed by General Relativity: is close to zero, is close to unity. The likelihood ratio there is pretty lopsided: GR captures all of Newton’s observed predictions, and makes new predictions that are not made by Newton but are observed nonetheless.
How, dear reader, how could Newton ever recover from that blow? Yes, at one point Newtonian mechanics hadn’t even been imagined, its prior probability ridiculously tiny. The same is true of General Relativity. Once GR came, Newtonian mechanics was once again driven down to the gutter. Is there any evidence that could bring it back up?
Of course not. Popper falsification comes, not from any asymmetry in Bayes’ Theorem itself, but from the fact that the negation of any given specific hypothesis consists of the disjunction of a huge number of alternatives. If you have observed evidence that’s not covered by your preferred hypothesis, you can be damn sure there’s an alternative that captures all the predictions of the old one, and also makes other predictions that are forbidden by it, in one sweeping blow.
The strength of a hypothesis is in what it prohibits, not in what it allows. Why is that true?
A given hypothesis only has so much probability it can allocate to different observations. A hypothesis that prohibits a lot of possible outcomes concentrates its probability mass more sharply on the outcomes it does allow.
But hypotheses don’t exist in a vacuum, so to speak. There’s always other hypotheses. Suppose a given hypothesis predicts a given datum with probability . That’s quite a bit, isn’t it? But Bayes’ Theorem goes:
I don’t know what the posterior probability for is if I don’t know how well other hypotheses predict the data. If it turns out that then the likelihood ratio between those hypotheses is and is favoured by the data as opposed to . And if there’s some hypothesis such that then my likelihood ratio for that data will be and it will no longer back up either.
Hypotheses are in eternal competition amongst themselves. The hypothesis that concentrates the most probability mass in the actual observed outcomes will invariably be the winner, and that’s why the real strength of a hypothesis is in what it prohibits, not in what it allows. The bland, allows-everything hypothesis will always lose bits against the sure-thing hypothesis (the hypothesis that gives probability to the observed outcomes and to all others).
But if that’s the case, why isn’t the sure-thing hypothesis always the winner? Well, because of priors, of course. For every possible imaginable mutually exclusive outcome, there is a sure-thing hypothesis that predicts it with probability 1, which means that before they’re observed, all those hypotheses are on equal footing, and they all have very tiny priors. A perfectly rational agent will have an Occamian prior over these hypotheses, so that simpler hypotheses that predict the data with fairly high probability, even if not probability one, can be favoured. Especially when new observations come that cut down on the sure-thing hypotheses even further.
The strength of a hypothesis is in what it prohibits, not in what it allows, because if it doesn’t concentrate enough probability mass on the actual observed outcomes (and therefore takes a lot of probability mass from the forbidden ones), the hypotheses that do will beat it to death.
Who are they competing with, though? That’s a relatively silly question, one might feel. The hypotheses are always competing amongst themselves, that’s what I just said, isn’t it? And Bayes’ Theorem is symmetrical there. Let’s take a look at the three-hypothesis case:
The probability of the conjunction of the hypothesis and the data is what’s actually competing, there. Now, suppose in that case, is winning, and its probability is much larger than the other two. Then, dominates and we can say that
That is, asymptotically, each hypothesis competes only with the one that has the highest posterior. But even this might not be enough to visualise it. Let’s use a toy example.
Suppose there is some experiment that has two possible outcomes: Success and Failure. Furthermore, suppose we have some hypotheses such that and . Then, if I run that experiment times, of which came up Success:
Suppose, then, that I have three hypotheses such that , , and . Further, suppose the priors for those three hypotheses are ,, and . Finally, suppose . What happens if I graph the evidence for each of these hypotheses as a function of ?
Remember, evidence is given by the formula where are the odds for . So:
Since the probabilities are exponential in , the graph of the evidence is best approximated by a line. But the interesting thing there is the behaviour of those lines.
The green line is , the blue one is , and the red one is . As you can see, isn’t even in the graph when is small. At that point, and are competing amongst themselves, and is nowhere to be seen. As we observe more successes, however, gets more and more confirmation. Eventually, becomes more likely than . At that point, suddenly the blue line, which describes , changes slope and starts losing probability.
Why does that happen? Because, as I suggested earlier, the hypotheses don’t really compete equally and fairly amongst each other. Rather, they’re mostly competing with the most likely hypothesis other than themselves. When becomes more likely than , starts competing with that hypothesis instead, and starts losing. However, is still only competing with because that’s still the most likely hypothesis. When becomes more likely than , however, then even starts competing with it, because became the most likely hypothesis that’s not itself.
And this agrees nicely with our intuition, too! When you’re collecting evidence and trying to prove or falsify a hypothesis, you don’t shoot for the underdog; you try to take down the leading one! People are trying to replace Quantum Mechanics and General Relativity, not Newtonian mechanics. Newton has lost, and sure, maybe by prospect Theory of Everything does better than Newton did, but who cares, that’s no longer the relevant theory.
As a final example, let’s examine the four hypothesis case, where , , and . The yellow line is , the red one is , the light blue is , and the dark blue is the new .
As you can see, at any given point, hypotheses only compete with their strongest opponent, and as soon as some other opponent takes the lead, everyone feels it. The strength of a hypothesis is in what it prohibits, not in what it allows, and the winning hypothesis is the one that will beat the other ones to death.