Perhaps this formulation is nice:

0 = (P(H|E)-P(H))P(E) + (P(H|~E)-P(H))P(~E)

The expected change in probability is zero (for if you expected change you would have already changed).

Since P(E) and P(~E) are both positive, to maintain balance if P(H|E)-P(H) < 0 then P(H|~E)-P(H) > 0. If P(E) is large then P(~E) is small, so (P(H|~E)-P(H)) must be large to counteract (P(H|E)-P(H)) and maintain balance.

...

Barkley, you don't realize that Bayes's Theorem is precisely what describes the normative update in beliefs over time? That this is the whole point of Bayes's Theorem?

Before black swans were observed, no one expected to encounter a black swan, and everyone expected to encounter another white swan on occasion. A black swan is huge evidence against, a white swan is tiny additional evidence for. Had they been normative, the two quantities would have balanced exactly.

I'm not sure what to say here. Maybe point to Probability Theory: The Logic of Science or A Technical Explanation of Technical Explanation? I don't know where this misunderstanding is coming from, but I'm learning a valuable lesson in how much Bayesian algebra someone can know without realizing which material phenomena it describes.

Eliezer Yudkowsky, The word "normative" has stood in the way of my understanding what you mean, at least the first few times I saw you use it, before I pegged you as getting it from the heuristics and biases people. It greatly confused me many times when I first encountered them. It's jargon, so it shouldn't be surprising that different fields use it to mean rather different things.

The heuristics and biases people use it to mean "correct," because social scientists aren't allowed to use that word. I think there's a valuable lesson about academics, institutions, or taboos in there, but I'm not sure what it is. As far as I can tell, they are the only people that use it this way.

My dictionary defines normative as "of, relating to, or prescribing a norm or standard." It's confusing enough that it carries those two or three meanings, but to make it mean "correct" as well is asking for trouble or in-groups.

a more general law, which I would name Conservation of Expected Evidence

I thought it was pretty clear that I was coining the phrase. I'm certainly not the first person to point out the law. E.g. Robin notes that our best estimate of anything should have no predictable trend. In any case, I posted the mathematical derivation and you certainly don't have to take my word about anything.

One minor correction, Eliezer: the link to your essay uses the text "An Intuitive Expectation of Bayesian Reasoning." I think you titled that essay "An Intuitive EXPLANATION of Bayesian Reasoning." (I am 99.9999% sure of this, and would therefore pay especial attention to any evidence inconsistent with this proposition.)

But you can't have it both ways - as a matter of probability theory, not mere fairness.

You've proved your case - but there's still enough wriggle room that it won't make much practical difference. One example from global warming, which predicts higher temperature on average in Europe - unless it diverts the gulf stream, in which case it predicts lower average temperatures. Consider the two statements: 1) If average temperatures go up in Europe, or down, this is evidence for global warming. 2) If average temperatures go up in Europe, and the gulf stream isn't diverted, or average temperatures go down, while the gulf stream is diverted, this is evidence of global warming.

1) is nonsense, 2) is true. Lots of people say statements that sound like 1), when they mean something like 2). Add an extra detail, and the symmetry is broken.

This weakens the practical power of your point; if an accused witch is afraid, that shows she's guilty; if she's not afraid, in a way which causes the inquisitor to be suspicious, she's also guilty. That argument is flawed, but it isn't a logical flaw (since the similar statement 2) is true).

Then we're back to arguing the legitimacy of these "extra details".

"Of course you are assuming a strong form of Bayesianism here. Why do we have to accept that strong form?"

Because it's mathematically proven. You might as well ask "Why do we have to accept the strong form of arithmetic?"

"So, if some evidence slightly moves the expectation in a particular direction, but does not push it across the 50% line from wherever it started, what is the big whoop?"

Because (in this case especially!) small probabilities can have large consequences. If we invent a marvelous new cure for acne, with a 1% chance of death to the patient, it's well below 50% and no specific person using the "medication" would expect to die, but no sane doctor would ever sanction such a "medication".

"Why is 50% special here?"

People seem to have a little arrow in their heads saying whether they "believe in" or "don't believe in" a proposition. If there are two possibilities, 50% is the point at which the little arrow goes from "not believe" to "believe".

Eliezer - what if the presence of the gene was decided by an omnipotent being called Omega? Then you'd break out the Spearmint, right?

Fantastic heuristic! It's like x=y·(z/y)+(1-y)·(x-z)/(1-y) for the rationalist's soul :)

It's worth noting, though, that you can rationally expect your credence in a certain belief "to increase", in the following sense: If I roll a die, and I'm about to show you the result, your credence that it didn't land 6 is now 5/6, and you're 5/6 sure that this credence it about to increase to 1.

I think this is what makes people feel like they can have a non-trivial expected value for their new beliefs: you can expect an increase or expect a decrease, but quantitatively the two possibilities exactly cancel each out in the expected value of your belief.

you can't possibly expect the resulting game plan to shift your beliefs (on average) in a particular direction.

But you can act to change the probability distribution of your future beliefs (just not its mean). That's the entire point of testing a belief. If you have a 50% belief that a ball is under a certain cup, then by lifting the cup, you can be certain than your future belief will be in the set {0%,100%} (with equal probability for 0 and 100, hence the same mean as now).

Getting the right shape of the probability distribution of future belief is the whole skill in testing a hypothesis.

What Peter said. Barkley, do you question that P(H) = P(H,E) + P(H, ~E) or do you question that P(H,E) = P(H|E)*P(E)?

Barkley, it looks to me like Eli derived it using the sum and product rules of probability theory.

What I mean, Barkley, is that the expression P(H|E), as held at time t=0, should - normatively - describe the belief about H you will hold at time t=2 if you see evidence E at time t=1. Thus, statements true in probability theory about the decomposition of P(H) imply the normative law of Conservation of Expected Evidence, if you accept that probability theory is normative for real-world problems where no one has ever seen an infinite set.

If you don't think probability theory is valid in the real world, I have some Dutch Book trades I'd like to make with you. But that's a separate topic, and in any case, most readers of this blog will at least understand what I intend to convey when I speak from within the view that probability theory is normative.

I guess I was a Bayesian before I knew what it meant....

The hyperlink "An Intuitive Explanation of Bayesian Reasoning" is broken. The current location of that essay is here: http://yudkowsky.net/rational/bayes

I'll modify my advice. If the probability that "I do action A in order to increase my subjective probability of position P" is greater given P than given not P, then doing A in order to increase my subjective probability of position P will be evidence in favor of P.

So in many cases, there will such a plan that I can devise. Let's see Eliezer find a way out of this one.

There is more discussion of this post here as part of the Rerunning the Sequences series.

Actually, the Omega situation is a perfect example. Someone facing the two boxes would like to increase his subjective probability that there is a million in the second box, and he is able to do this by deciding to take only the second box. If he decides to take both, on the other hand, he should decrease his credence in the presence of the million, even before opening the box.

per the Black Swan:

The set of potential multicolored variations of Swans is infinite (purple, brown, grey, blue, green, etc). We can not prove any one of them do not exist. But every day that proceeds where we don't see these swans gives us a higher probability they do not exist. It never equals 1, but it's darn close.

The problem with the Black Swan parable is not that it's untrue, but rather unimportant. The set of things we have no evidence of is infinite. To then pounce across an unexpected observation (eg, a Black Swan, that Kevin Federline is a relatively good parent, last week's liquidity run on mortgage lenders), and say, "aha! You were all wrong!" merely sets up a staw man, that everything we reasonably don't anticipate and plan for is assumed to have had a probability of zero.

In reality, when you want to pay money for extreme events you overpay, that is, the implied probability is overweighted because sellers can't insure against these events. London bookmakers offer only 250-1 odds against a perpetual motion machine being discovered, 100-1 that aliens won't be proven. In option markets you have a volatility smile so that extreme events get higher and higher implied volatilities as you move away from the mean, meaning their probability is not assumed Gaussian.

The bottom line is that "absence of evidence is not evidence of absence" merely uses hindsight to attack a caricature of beliefs, and seems to suggests something practically important. In practice, people lose money on lottery tickets (or hurricane insurance, or buing a 3-delta put), so exploiting this is a fool's game.

"no one expected to encounter a white swan, and everyone expected to encounter another black swan on occasion. A white swan is huge evidence against, a black swan is tiny additional evidence for." I presume you meant the reverse of this?

Stuart, if the extra details are observable and specified in advance, the legitimacy is clear-cut.

Barkley, I'm an infinite set atheist, all real-world problems are finite; and you seem to be assuming that priors are arbitrary but likelihood ratios are fixed eternal and known, which is a strange position; and in any case what does that have to do with something as simple as Conservation of Expected Evidence? If anyone attempts to make an infinite-set scenario that violates CEE, it disproves their setup by reductio ad absurdum, and reinforces the ancient wisdom of E. T. Jaynes that no infinity may be assumed except as the proven limit of a finite problem.

One reason is Cox's theorem, which shows any quantitative measure of plausibility must obey the axioms of probability theory. Then this result, conservation of expected evidence, is a theorem.

What is the "confidence level"? Why is 50% special here?

Closely related is the law of total expectation: https://en.wikipedia.org/wiki/Law_of_total_expectation

It states that E[E[X|Y]]=E[X].

Hi, I'm new here but I've been following the sequences in the suggested order up to this point.

I have no problem with the main idea of this article. I say this only so that everyone knows that I'm nitpicking. If you're not interested in nitpicking then just ignore this post.

I don't think that the example given bellow is a very good one to demonstrate the concept of Conservation of Expected Evidence:

If you argue that God, to test humanity's faith, refuses to reveal His existence, then the miracles described in the Bible >must argue against the existence of God.

Assuming I'm reading this correctly:

Our Prior is P(G) = The probability that God Exists (let's assume this is the Judeo-Christain God since that seems to be the intended target)

P(T) = the probability that God is Testing Humanity by not revealing his existence

P(M) = the probability that the Miracles of the bible are true.

The issue that I find with this is that P(G|T) = 1

If God is testing Humanity by hiding his existence there is a 100% chance that God Exists. I was going to write out the whole Bayesian equation to explain why this is true, but I think it's pretty intuitive. P(T) cannot be evidence for P(G) since it assumes that P(G) is true.

Another issue is that the way this is written you're implying that P(M) = P(~T). But this is not true, since the Miracles of the bible existing is not the direct opposite of God testing humanity by not reveling his existence. Unless you intend to completely twist the argument that most people are making when they say assert P(T) as truth. They aren't saying or even implying that God wants there to be no evidence at all of his existence. Most theist would instead argue that the existence of miracles are a part of God's test for humanity. They say that God sent us miraculous signs and prophets instead of just coming down and saying "Hey humanity, I'm God" because he wanted to test our faith. Had they the mathematical language, they would say that P(T|M) > P(T), meaning M serves as evidence of T. Not P(M) = P(~T)

Though this whole concept of God "testing humanity by not revealing himself" does seem more like an example of Belief in belief, where P(T) was devised as a means to justify the existence of an invisible God, I still feel like the example you've given is a bit of a stretch.

Here's an example which doesn't bear on Conservation of Expected Evidence as math, but does bear on the statement,

"There is no possible plan you can devise, no clever strategy, no cunning device, by which you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before."

taken at face value.

It's called the Cable Guy Paradox; it was created by Alan Hájek, a philosopher the Australian National University. (I personally think the term Paradox is a little strong for this scenario.)

Here it is: the cable guy is coming tomorrow, but cannot say exactly when. He may arrive any time between 8 am and 4 pm. You and a friend agree that the probability density for his arrival should be uniform over that interval. Your friend challenges you to a bet: even money for the event that the cable guy arrives before noon. You get to pick which side of the bet you want to take -- by expected utility, you should be indifferent. Here's the curious thing: if you pick the morning bet, then almost surely there will be times in the morning when you would prefer to switch to the afternoon bet.

This would seem to be a situation in which "you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before," even though the equation P(H) = P(H|E)P(E) + P(H|~E)P(~E) is not violated. I'm not sure, but I think it's due to multiple possible interpretations of the word "before".

Um, no, if a study shows that people who chew gum also have a gene GXTP27 or whatever, which also protects against cancer, I cannot plan to increase my subjective probability that I have gene GXTP27 by starting to chew gum.

See also: "evidential decision theory", why nearly all decision theorists do not believe in.

This post was one of the most helpful for me personally, but I recently realized this isn't true in an absolute sense: "There is no possible plan you can devise, no clever strategy, no cunning device, by which you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before."

Suppose the statement "I perform action A" is more probable given position P than given not-P. Then if I start planning to perform action A, this will be evidence that I will perform A. Therefore it will also be evidence for position P. So there is some plan that I can devise such that I can expect my confidence in P to be higher than before I devised the plan.

In general, of course, unless P is a position relating to my actions or habits, this effect will not be very large.

Hi, new here.

I was wondering if I've interpreted this correctly:

'For a true Bayesian, it is impossible to seek evidence that confirms a theory. There is no possible plan you can devise, no clever strategy, no cunning device, by which you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before. You can only ever seek evidence to test a theory, not to confirm it.'

Does this mean that it is impossible to prove the truth of a theory? Because the only evidence that can exist is evidence that falsifies the theory, or supports it?

For example, something people know about gravity and objects under it's influence, is that on Earth objects will accelerate at something like 9.81ms^-2. If we dropped a thousand different objects and observed their acceleration, and found it to be 9.81ms^-2, we would have a thousand pieces of evidence supporting the theory, and zero pieces to falsify the theory. We all believe that 9.81 is correct, and we teach that it is the truth, but we can never really know, because new evidence could someday appear that challenges the theory, correct?

Thanks

Can someone tell me if I understand this correctly : He is saying that we must be clear before hand what constitutes evidence for and what constitutes evidence against and what doesn't constitute evidence either way?

Because in his examples it seems that what is being changed is what counts as evidence. It seems that no matter what transpires (in the witch trials for example) it is counted as evidence for. This is not the same as changing the hypothesis to fit the facts. The hypothesis was always 'she's a witch'. Then the evidence is interpreted as supportive of the hypothesis no matter what.

Mantel cox log rank tests compare observations and expectations too...