Re: Hume, induction, and probability

[Kathryn Blackmond Laskey]

Ronald,

>Whenever you condition, you are assuming that evidence was drawn from
>the distribution about which you are making a probability statement.

Well, yes. I build a mathematical model of the joint distribution of hypothesis and evidence, usually expressed as P(H)P(E|H). Then I condition on E to derive P(H|E). As Clark pointed out, once I have constructed this model, the manipulations I do with the model to obtain P(H|E) are deduction.

But I don't want just to do deduction. I want to use my model to draw conclusions about the world.

I'm looking around me. When I look out the window, I see the Potomac river, cars driving by, trees, office buildings... I also feel the keys touching my fingers, feel a twinge of pain in the toe I caught in a door last week, hear voices down the hall... In addition, I have memories of my childhood, the things I have learned in school, from my reading, from conferences... All these things are data I observe, that I assume are produced by the universe of which I am a conscious subsystem. Different "types of universe" would be expected to produce different kinds of data.

The data I see seem highly characteristic of the type of universe I've been calling a "learnable universe" (I hope to make this concept mathematically precise some day). They seem very uncharacteristic of a universe being typed out by monkeys. Ergo, my posterior probability on "learnable universe" is much, much higher than my posterior probability that I'm being typed out by monkeys.

I formalize this argument as a model P(H)P(E|H), perform deductive inference to compute P(H|E), and then apply the result back to the world. I claim that my result P(H|E) models my belief about H, updated by the evidence E. That step, applying my model to make a claim about the world (or at least what I believe about the world), is not deduction. I never said it was. I can't prove to anyone, including myself, that this belief is "right."

The applicability of this conclusion to the world assumes that the connection between E and H postulated in my model actually obtains in the world, and that my observations on E are accurate, and that my prior beliefs about H are not too dogmatic. Any time we apply a mathematical model to the world, the conclusions we draw are subject to the assumption that the model is a sufficiently good representation for the purpose to which we are putting it.

If the joint distribution P(H,E) is a bad model, then the answer I get will be bad. If it's an accurate model of my beliefs about the world, but a bad model of the world, then the answer will be a correct reflection of my beliefs, which will be wrong. You're right that I can't prove the applicability of my model to the world. All I can say about this analysis is that in my opinion, our living in a "learnable" universe is justified by the evidence. I could, of course, be wrong.

>If you are simply choosing between models in a world without time, then
>there is no point in talking about induction and no point in talking
>about the future.

That's not what I meant.

Conditional on a "typed out by monkeys" universe, tomorrow has nothing to do with today. Ergo, what I am observing now has nothing to do with what will happen tomorrow, and consequently I have very vague beliefs about what will happen tomorrow, which don't change when I get evidence today.

Conditional on my "learnable universe" model there is time and causality and lots of conditional independence. I use observations to learn about the structure and parameters of my temporal, causal model, and then confidently make predictions about tomorrow.

The latter model has much higher probability to me than the former, for reasons I've discussed. Therefore, I make confident predictions about the future. Of course, if I'm wrong (and I can't prove I'm not) then all my predictions will go haywire tomorrow. However, I happen to believe (of course I could be wrong) that I have strong grounds for being confident in my predictions.

>If you believe that time exists and are attempting to make a scientific
>statement about how rules in the world hold over time, then you are
>interested in the question of induction. Looking at a mass of prior
>data can give you a hypothesis about how rules held in the world in the
>past. If you try to update your posterior on induction being true in
>the present or future, you are assuming stationarity, i.e. that evidence
>you have about induction in the past is germane to the present. This
>means that you are assuming an underlying distribution that does not
>change across time, which is equivalent to the assumption that induction
>is valid. This is circular.

No.

In one of my models ("learnable universe") I assume at least some form of stationarity, and that the past is germane to the present and future. Conditional on that model, what I see *right now* (including my memories of the past -- which are encoded as memory traces in the present state of the world) is the kind of thing I'd expect to see. This is not circular. I am postulating a model and looking at what kind of data I would expect under the model.

In another model ("monkeys typing") I make no such assumptions. There's no connection between past, present and future. What I think are memories of my past are not real memories -- they've been conjured up and planted in my mind by the typing monkeys. My plans for tomorrow will never happen because in another moment I'm going to de-materialize as the monkeys lapse back into gibberish. Under that model, I have only extremely vague predictions about tomorrow. Under that model, my existence is monstrously improbable.

Assuming a non-dogmatic prior, my posterior odds ratio for "learnable universe" against "monkeys typing" is enormous. Of course, there are lesser degrees of "unlearnability" that are not so easily refuted, and those have non-negligible posterior probability.

This is not circular reasoning.

>>In the very most unlearnable of the
>>unlearnable hypotheses, the past, present and future have absolutely
>>nothing to do with each other. The present materialized exactly as it is
>>right now, including my memories of my childhood and this quaint idea I
>>seem to have that I'm learning, and is going to dissolve again in an
>>instant. For me (maybe not for you) this hypothesis has very low posterior
>>probability, because the likelihood of my seeing something I can understand
>>so well as a temporally evolving universe is quite high under the "edge of
>>chaos" hypothesis and miniscule under this "very, VERY unlearnable"
>>hypotheiss.
>
>This is a classic example in which, by *construction*, evidence cannot
>influence your posterior. Any attempt to construct a posterior based on
>evidence violates the premise.

I don't understand what you mean. Within the "monkeys typing" universe model, evidence about today does not influence my prediction for tomorrow, because tomorrow has nothing to do with today. However, evidence about today is quite germane to the question of whether I'm in a "learnable" or "monkeys typing" universe. Hence, it's germane to the question of whether what happens today has anything to do with what's going to happen tomorrow.

>The template for the particular form of circular reasoning in question here
>requires a premise about an inference rule (or precondition thereof) and
>then the use of that rule to establish the premise.

That is NOT what I am doing.

I'm not using Bayes Rule to establish Bayes Rule. Bayes Rule is a valid form of logical deduction, and stands on that merit.

I do aim to formalize the concepts "simple universe," "complex but learnable universe," and "unlearnable universe" (or some suitable modifications thereof), and then derive characteristics we would expect to see in each kind of universe. Note that these universe-types will of necessity be relational concepts: they will concern not just what "goes on" in the universe, but also the mapping between what "goes on" in the universe and what it "represents to itself" about what is going on. What will consciousnesses "see" (assuming there are consciousnesses to see anything) in each kind of universe? Then I can compare these predictions with what we see in our universe.

Once I've done this, I will have articulated hypotheses (universe types), evidence, and probabilistic predictions on evidence given hypotheses. I can apply the mathematics of Bayes Rule and come up with posterior degrees of belief on the hypotheses. That part will be deduction. Whether the deduction results in "reasonable" degrees of belief about the world depends on the validity of applying the model to the world. As a subjectivist, I leave the decision of how much credence to put in my conclusions up to you.

To say this another way -- I do not attempt to establish by induction that induction is valid in general. I aim to argue by induction that ours is a universe in which induction "works." I also argue (and I hope to formalize this some day) that it is probable that I would be able to do this in a universe in which induction works, and improbable in any other kind of universe. This provides strong evidence that ours is a universe in which induction works. This is a perfectly reasonable and non-circular argument.

>The classic example
>of how to fill this template is the "counter-induction" rule. We assume
>that things that have worked in the past will *not* work in the future
>(and vice versa). We apply this rule in a universe where induction
>holds and observe that the rule has failed. From this, we conclude, due
>to the counter-inductive principle, that the rule must work in the
>future. Each new failure only strengthens our belief in
>counter-induction.

Sure. This reinforces Clark's point that if you have a bad model you'll make bad inferences. No one can prove any of us doesn't have a horribly bad model. If the model you just cited is your model, then you'll go on making abysmal predictions quite confidently. But it's not circular reasoning, it's just a bad model.

But note that you are missing the second part of my argument above. You argue by counter-induction that ours is a universe in which counter-induction works. But you cannot establish that it is probable that you should be able to do this in only those universes in which counter-induction works. In fact, you've shown that you can argue by counter-induction that counter-induction works, in a universe in which it does *not* work. Therefore, the likelihood ratio does not favor counter-induction as a valid general principle.

It seems that in our universe, very bad theories eventually die out, although it may take a while. I conjecture this happens because we're a learnable universe. I have attempted to argue NOT that this conjecture can be proven (it can't!), but that it enjoys considerable evidential support. I am arguing that many of the properties we have inferred about our world seem characteristic of what we are coming to understand as general properties of learnable universes.

Kathy