This is counterintuitive in an interesting way.
You’d think that since P(Q1|~∀xQx) = 1⁄2 and P(Q1|∀xQx) = 1, observing Q1 is evidence in favor of ∀xQx.
And it is, but the hidden catch is that this depends on the implication that ∀xQx->Q1, and that implication is exactly the same amount of evidence against ∀xQx.
It’s also an amusing answer to the end of part 1 exercise.
i once had to go to the doctor so he could fish a lego out of my nose. So, that was worse than eating all the cabbage or spilling all the milk I think. More scary, and probably more expensive, depending on how the insurance worked out.
Truth is really important sometimes, but so far I’ve been bad about identifying when.
I know a fair bit about cognitive biases and ideal probabilistic reasoning, and I’m pretty good at applying it to scientific papers that I read or that people link through Facebook. But these applications are usually not important.
But, when it comes to my schoolwork and personal relationships, I commit the planning fallacy routinely, and make bad predictions against base rates. And I spend no time analyzing these kinds of mistakes or applying what I know about biases and probability theory.
If I really operationalized my belief that only some truths are important, I’d prioritize truths and apply my rationality knowledge to the top priorities. That would be awesome.
The first and third ones, about info sometimes being worthless, just made me think of Vaniver’s article on value of information calculations. So, I mean, it sounded very LessWrongy to me, very much the kind of thing you’d hear here.
The second one made me think of nuclear secrets, which made me think of HPMOR. Again, it seems like the kind of thing that this community would recognize the value of.
I think my reactions to these were biased, though, by being told how I was expected to feel about them. I always like to subvert that, and feel a little proud of myself when what I’m reading fails to describe me.
I’m pretty sure that the Cauchy likelihood, like the other members of the t family, is a weighted mixture of normal distributions. (Gamma distribution over the inverse of the variance)
EDIT: There’s a paper on this, “Scale mixtures of normal distributions” by Andrews and Mallows, if you want the details
Hmm. Considering that I was trying to come up with an example to illustrate how explicit the assumptions are, the assumptions aren’t that explicit in my example are they?
Prior knowledge about the world --> mathematical constraints --> prior probability distribution
The assumptions I used to get the constraints are that the best estimate of your next measurement is the average of your previous ones, and that the best estimate of its squared deviation from that average is some number s^2, maybe the variance of your previous observations. But those aren’t states of the world, those are assumptions about your inference behavior.
Then I added later that the real assumptions are that you’re making unbiased measurements of some unchanging quantity mu, and that the mechanism of your instrument is unchanging. These are facts about the world. But these are not the assumptions that I used to derive the constraints, and I don’t show how they lead to the former assumptions. In fact, I don’t think they do.
Well. Let me assure you that the assumptions that lead to the constraints are supposed to be facts about the world. But I don’t see how that’s supposed to work.
it seems like a weird response to say “oh, well who cares about explicit assumptions anyways?”
Yeah, sorry. I was getting a little off topic there. It’s just that in your post, you were able to connect the explicit assumptions being true to some kind of performance guarantee. Here I was musing on the fact that I couldn’t. It was meant to undermine my point, not to support it.
What does it mean to “assume that the prior satisfies these constraints”?
?? The answer to this is so obvious that I think I’ve misunderstood you. In my example, the constraints are on moments of the prior density. In many other cases, the constraints are symmetry constraints, which are also easy to express mathematically.
But then you bring up concrete statements about the world? Are you asking how you get from your prior knowledge about the world to constraints on the prior distribution?
EDIT: you don’t “assume a constraint”, a constraint follows from an assumption. Can you re-ask the question?
On the other hand, an argument I hear is that Bayesian methods make their assumptions explicit because they have an explicit prior. If I were to write this as an assumption and guarantee, I would write:
Assumption: The data were generated from the prior.
Guarantee: I will perform at least as well as any other method.
While I agree that this is an assumption and guarantee of Bayesian methods, there are two problems that I have with drawing the conclusion that “Bayesian methods make their assumptions explicit”. The first is that it can often be very difficult to understand how a prior behaves; so while we could say “The data were generated from the prior” is an explicit assumption, it may be unclear what exactly that assumption entails. However, a bigger issue is that “The data were generated from the prior” is an assumption that very rarely holds; indeed, in many cases the underlying process is deterministic (if you’re a subjective Bayesian then this isn’t necessarily a problem, but it does certainly mean that the assumption given above doesn’t hold).
In a post addressing a crowd where a lot of people read Jaynes, you’re not addressing the Jaynesian perspective on where priors come from.
When E. T. Jaynes does statistics, the assumptions are made very clear.
The Jaynesian approach is this:
your prior knowledge puts constraints on the prior distribution,
and the prior distribution is the distribution of maximum entropy that meets all those constraints.
The assumptions are explicit because the constraints are explicit.
As an example, imagine that you’re measuring something you’ve measured before. In a lot of situations, you’ll end up with constraints on the mean and variance of your prior distribution. This is because you reason in the following way:
You think that your best estimate for the next measurement is the average of your previous measurements.
Your previous estimates have given you a sense of the general scale of the errors you get. You express this as an estimate of the squared difference between your estimated next measurement and your real next measurement.
If we take “best estimate” to mean “minimum expected least squared error”, then these are constraints on the mean and variance of the prior distribution. If you maximize entropy subject to these constraints, you get a normal distribution. And that’s your prior.
The assumptions are quite explicit here. You’ve assumed that these measurements are measuring some unchanging quantity in an unbiased way, and that the magnitude of the error tends to stay the same. And you haven’t assumed any other relationship between the next measurement and the previous ones. You definitely didn’t assume any autocorrelation, for example, because you didn’t use the previous measurement in any of your estimates/constraints.
But since we’re not assuming that the data are generated from the prior, we don’t have the corresponding guarantee. Which brings up the question, what do we have? What’s the point of this?
A paper I’m reading right now (Portilla & Simoncelli, 2000, “A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients”) puts it this way:
Guarantee: “The maximum entropy density is optimal in the sense that it does not introduce any constraints on the [prior] besides those [we assumed].”
It’s not a performance guarantee. Just a guarantee against hidden assumptions, I guess?
Despite apparently having nothing to do with performance, it makes a lot of sense from within Jaynes’s perspective. To him, probability theory is logic: figuring out which conclusions are supported by your premises. It just extends deductive logic, allowing you to see to what degree each conclusion is supported by your premises.
And, I think your criticism of the Dutch Book argument applies here: is your time better spent making sure you don’t have any hidden assumptions, or writing more programs to make more money? But it definitely makes your assumptions explicit, that’s just part of the whole “prob theory as logic” paradigm.
(But I don’t really understand what it means to not introduce any constraints besides those assumed, or why maxent is the procedure that achieves this. That’s why I quoted the guarantee from someone who I assume does understand)
okay, and you were just trying to make sure that Manfred knows that all this probability-of-distributions speech you’re speaking isn’t, as he seems to think, about the degree-of-belief-in-my-current-state-of-ignorance distribution for the first roll. Gotcha.
Okay… but do we agree that the degree-of-belief distribution for the first roll is (1/3, 1⁄3, 1⁄3), whether it’s a fair die or a completely biased in an unknown way die?
Because I’m pretty sure that’s what Manfred’s talking about when he says
There is a single correct distribution for our starting information, which is (1/3,1/3,1/3),
and I think him going on to say
the “distribution across possible distributions” is just a delta function there.
was a mistake, because you were talking about different things.
EDIT:
I thought so too, which is why I asked him what he thought a delta function in the distribution space meant.
Ah. Yes. Okay. I am literally saying only things that you know, aren’t I. My bad.
I like your writing style. For something technical, it feels very personal. And you keep it very concise while also easy to read—is there a lot of trimming down that goes on, or do you just write it that way?
In my experience with Bayesian biostatisticians, they don’t talk much about the information a prior represents. But they’re also not just using common ones. They talk a lot about its “properties”—priors with “really nice properties”. As for as I can tell, they mean two things:
Computational properties
The way the distribution shifts as you get evidence. They think about this in a lot of detail, and they like priors that lead to behavior they think is reasonable.
I think this amounts to the same thing. The way they think and infer about the problem is determiend by their information. So, when they create a robot that thinks and infers in the same way, they are creating one with the same information as they have.
But, as a procedure for creating a prior that represents your information, it’s very different from Jaynes’s procedure. Jaynes’s procedure being, stating your prior information very precisely, and then finding symmetries or constraints for maximum entropy I guess.
I’m very happy you’re writing about logical uncertainty btw, it’s been on my mind a lot lately.
According to Jaynes, it’s actually not—I don’t have the page number on me, unfortunately. But the way he does it is by discretizing the space of possibilities, and taking the limit as the number of discrete possibilities goes to infinity. It’s not the limit of the entropy H, since that goes to infinity, it’s the limit of H—log(n). It turns out to be a little different from integrating p*Log(p).
Whoa, you think the only correct interpretation of “there’s a die that returns 1, 2, or 3” is to be absolutely certain that it’s fair? Or what do you think a delta function in the distribution space means?
I haven’t been able to follow this whole thread of conversation, but I think it’s pretty clear you’re talking about different things here.
Obviously, the long-run frequency distribution of the die can be many different things. One of them, (1/3, 1⁄3, 1⁄3), represents fairness, and is just one among many possibilities.
Equally obviously, the probability distribution that represents rational expectations about the first flip is only one thing. Manfred claims that it’s (1/3, 1⁄3, 1⁄3), which doesn’t represent fairness. It could equally well represent being certain that it’s biased to land on only one side every time, but you have no idea which side.
I was actually surprised to read back over the post and see that it was about intelligence; in my memory I had translated it into an article about a one-dimensional scale of ability, which I also imagined including, as a critical component, ability to have effective collaborations.
I guess it was the example of Einstein that brought that to mind. Part of what impressed me the most about his career was that, when he was working on general relativity, he collaborated heavily with a mathematician. Marcel Grossman, I think.
There are two separate simplifying assumptions you can make, when modeling this situation. One is assuming intelligence is the most important factor. The other is assuming that ability is one-dimensional. I suppose that this post makes both of them.
You’re implicitly modeling difficulty as a single number, and ability as another single number that determines the maximum difficulty of the problems you can solve.
However, according to Feynman, you can sometimes solve problems that smarter people can’t by having “a different box of tools”. The quote:
That book also showed how to differentiate parameters under the integral sign—it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals.
The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.
I don’t think this changes the picture very much, though. You just divide the people in the field according to their box of tools. Pick the group with the box of tools that can solve the problem, and your model applies to them.
You make a few different points, and I’ve tried operationalizing two of them.
Time is expensive:
when considering a job, estimate how much time you’ll be able to take off from it and when you’ll be able to retire, and what you’ll do with that time
do this estimation for a lot of jobs, even those that don’t seem appealing, to find the opportunity cost of the ones that do seem appealing
I don’t mean I’ve actually done these, not yet anyway. I just read the article.
Another point was, comparative advantage:
Observe time usage and look up costs to hire people to do what I’m doing
Look up ways to turn time into money. I’m getting paid a fixed stipend, but there must be TA or tutoring opportunities
With that information at hand, I can recognize the opportunity to, for example, trade my cleaning time for tutoring time and keep some cash
can’t dispute the conclusion but I object to the argument.
Because, the argument doesn’t apply to this model: “Probabilities are calculated from a random walk, except for large unlikely changes, or changes when the price is near zero.”
Approximate models always have a range of applicability. Showing that the model gives absurd results outside the range of applicability for which it’s intended doesn’t mean much.
a magically murdered and revived centaur is a big political problem, though, between Hogwarts and the centaurs of the forest.
Unless this kind of thing is routine, why would he expect to get away with this?
EDIT: There’s also another explanation for why he took so long, which is he was in the Firenze’s mind, learning exactly what the centaurs had divined.
I don’t understand this yet, which isn’t too surprising since I haven’t read the background posts yet. However, all the “roughly speaking” summaries of the more exact stuff are enough to show me that this article is talking about something I’m curious about, so I’ll be reading in more detail later probably.