Category: Epistemology

The Anthropic Dice Killer

~ squarishbracket ~ 38 Comments

Today we discuss anthropic reasoning.

The Problem

Imagine the following scenario:

A mad killer has locked you in a room. You are trapped and alone, with only your knowledge of your situation to help you out.

One piece of information that you have is that you are aware of the maniacal schemes of your captor. His plans began by capturing one random person. He then rolled a pair of dice to determine their fate. If the dice landed snake eyes (both 1), then the captive would be killed. If not, then they would be let free.

But if they are let free, the killer will search for new victims, and this time bring back ten new people and lock them alone in rooms. He will then determine their fate just as before, with a pair of dice. Snake eyes means they die, otherwise they will be let free and he will search for new victims.

His murder spree will continue until the first time he rolls snake eyes. Then he will kill the group that he currently has imprisoned and retire from the serial-killer life.

Now. You become aware of a risky way out of the room you are locked in and to freedom. The chances of surviving this escape route are only 50%. Your choices are thus either (1) to traverse the escape route with a 50% chance of survival or (2) to just wait for the killer to roll his dice, and hope that it doesn’t land snake eyes.

What should you do?

 

 

 

(Think about it before reading on)

 

 

 

A plausible-sounding answer

Your chance of dying if you stay and wait is just the chance that the dice lands snake eyes. The probability of snake eyes is just 1/36 (1/6 for each dice landing 1).

So your chance of death is only 1/36 (≈ 3%) if you wait, and it’s 50% if you try to run for it. Clearly, you are better off waiting!

But…

You guessed it, things aren’t that easy. You have extra information about your situation besides just how the dice works, and you should use it. In particular, the killing pattern of your captor turns out to be very useful information.

Ask the following question: Out of all of the people that have been captured or will be captured at some point by this madman, how many of them will end up dying? This is just the very last group, which, incidentally, is the largest group.

Consider: if the dice land snake eyes the first time they are rolled, then only one person is ever captured, and this person dies. So the fraction of those captured that die is 100%.

If they lands snake eyes the second time they are rolled, then 11 people total are captured, 10 of whom die. So the fraction of those captured that die is 10/11, or ≈ 91%.

If it’s the third time, then 111 people total are captured, 100 of whom die. Now the fraction is just over 90%.

In general, no matter how many times the dice rolls before landing snake eyes, it always ends up that over 90% of those captured end up being in the last round, and thus end up dying.

So! This looks like bad news for you… you’ve been captured, and over 90% of those that are captured always die. Thus, your chance of death is guaranteed to be greater than 90%.

The escape route with a 50% survival chance is looking nicer now, right?

Wtf is this kind of reasoning??

What we just did is called anthropic reasoning. Anthropic reasoning really just means updating on all of the information available to you, including indexical information (information about your existence, age, location, and so on). In this case, the initial argument neglected the very crucial information that you are one of the people that were captured by the killer. When updating on this information, we get an answer that is very very different from what we started with. And in this life-or-death scenario, this is an important difference!

You might still feel hesitant about the answer we got. After all, if you expect a 90% chance of death, this means that you expect a 90% chance for the dice to land snake eyes. But it’s not that you think the dice are biased or anything… Isn’t this just blatantly contradictory?

This is a convincing-sounding rebuttal, but it’s subtly wrong. The key point is that even though the dice are fair, there is a selection bias in the results you are seeing. This selection bias amounts to the fact that when the dice inevitably lands snake-eyes, there are more people around to see it. The fact that you are more likely than 1/36 to see snake-eyes is kind of like the fact that if you are given the ticket of a random concert-goer, you have a higher chance of ending seeing a really popular band than if you just looked at the current proportion of shows performed by really popular bands.

It’s kind of like the fact that in your life you will spend more time waiting in long lines than short lines, and that on average your friends have more friends than you. This all seems counterintuitive and wrong until you think closely about the selection biases involved.

Anyway, I want to impress upon you that 90% really is the right answer, so I’ll throw some math at you. Let’s calculate in full detail what fraction of the group ends up surviving on average.

Screen Shot 2018-08-02 at 1.16.15 AM

By the way, the discrepancy between the baseline chance of death (1/36) and the anthropic chance of death (90%) can be made as large as you like by manipulating the starting problem. Suppose that instead of 1/36, the chance of the group dying was 1/100, and instead of the group multiplying by 10 in size each round, it grew by a factor of 100. Then the baseline chance of death would be 1%, and the anthropic probability would be 99%.

We can find the general formula for any such scenario:

Screen Shot 2018-08-02 at 4.54.30 AM.png

IF ANYBODY CAN SOLVE THIS, PLEASE TELL ME! I’ve been trying for too long now and would really like an analytic general solution. 🙂

There is a lot more to be said about this thought experiment, but I’ll leave it there for now. In the next post, I’ll present a slight variant on this thought experiment that appears to give us a way to get direct Bayesian evidence for different theories of consciousness! Stay tuned.

What do I find conceptually puzzling?

~ squarishbracket ~ 6 Comments

There are lots of things that I don’t know, like, say, what the birth rate in Sweden is or what the effect of poverty on IQ is. There are also lots of things that I find really confusing and hard to understand, like quantum field theory and monetary policy. There’s also a special category of things that I find conceptually puzzling. These things aren’t difficult to grasp because the facts about them are difficult to understand or require learning complicated jargon. Instead, they’re difficult to grasp because I suspect that I’m confused about the concepts in use.

This is a much deeper level of confusion. It can’t be adjudicated by just reading lots of facts about the subject matter. It requires philosophical reflection on the nature of these concepts, which can sometimes leave me totally confused about everything and grasping for the solid ground of mere factual ignorance.

As such, it feels like a big deal when something I’ve been conceptually puzzled about becomes clear. I want to compile a list for future reference of things that I’m currently conceptually puzzled about and things that I’ve become un-puzzled about. (This is not a complete list, but I believe it touches on the major themes.)

Things I’m conceptually puzzled about

What is the relationship between consciousness and physics?

I’ve written about this here.

Essentially, at this point every available viewpoint on consciousness seems wrong to me.

Eliminativism amounts to a denial of pretty much the only thing that we can be sure can’t be denied – that we are having conscious experiences. Physicalism entails the claim that facts about conscious experience can be derived from laws of physics, which is wrong as a matter of logic.

Dualism entails that the laws of physics by themselves cannot account for the behavior of the matter in our brains, which is wrong. And epiphenomenalism entails that our beliefs about our own conscious experience are almost certainly wrong, and are no better representations of our actual conscious experiences than random chance.

How do we make sense of decision theory if we deny libertarian free will?

Written about this here and here.

Decision theory is ultimately about finding the decision D that maximizes expected utility EU(D). But to do this calculation, we have to decide what the set of possible decisions we are searching is.

EU confusion

Make this set too large, and you end up getting fantastical and impossible results (like that the optimal decision is to snap your fingers and make the world into a utopia). Make it too small, and you end up getting underwhelming results (in the extreme case, you just get that the optimal decision is to do exactly what you are going to do, since this is the only thing you can do in a strictly deterministic world).

We want to find a nice middle ground between these two – a boundary where we can say “inside here the things that are actually possible for us to do, and outside are those that are not.” But any principled distinction between what’s in the set and what’s not must be based on some conception of some actions being “truly possible” to us, and others being truly impossible. I don’t know how to make this distinction in the absence of a robust conception of libertarian free will.

Are there objectively right choices of priors?

I’ve written about this here.

If you say no, then there are no objectively right answers to questions like “What should I believe given the evidence I have?” And if you say yes, then you have to deal with thought experiments like the cube problem, where any choice of priors looks arbitrary and unjustifiable.

(If you are going to be handed a cube, and all you know is that it has a volume less than 1 cm3, then setting maximum entropy priors over volumes gives different answers than setting maximum entropy priors over side areas or side lengths. This means that what qualifies as “maximally uncertain” depends on whether we frame our reasoning in terms of side length, areas, or cube volume. Other approaches besides MaxEnt have similar problems of concept dependence.)

How should we deal with infinities in decision theory?

I wrote about this here, here, here, and here.

The basic problem is that expected utility theory does great at delivering reasonable answers when the rewards are finite, but becomes wacky when the rewards become infinite. There are a huge amount of examples of this. For instance, in the St. Petersburg paradox, you are given the option to play a game with an infinite expected payout, suggesting that you should buy in to the game no matter how high the cost. You end up making obviously irrational choices, such as spending $1,000,000 on the hope that a fair coin will land heads 20 times in a row. Variants of this involve the inability of EU theory to distinguish between obviously better and worse bets that have infinite expected value.

And Pascal’s mugging is an even worse case. Roughly speaking, a person comes up to you and threatens you with infinite torture if you don’t submit to them and give them 20 dollars. Now, the probability that this threat is credible is surely tiny. But it is non-zero! (as long as you don’t think it is literally logically impossible for this threat to come true)

An infinite penalty times a finite probability is still an infinite expected penalty. So we stand to gain an infinite expected utility by just handing over the 20 dollars. This seems ridiculous, but I don’t know any reasonable formalization of decision theory that allows me to refute it.

Is causality fundamental?

Causality has been nicely formalized by Pearl’s probabilistic graphical models. This is a simple extension of probability theory, out of which naturally falls causality and counterfactuals.

One can use this framework to represent the states of fundamental particles and how they change over time and interact with one another. What I’m confused about is that in some ways of looking at it, the causal relations appear to be useful but un-fundamental constructs for the sake of easing calculations. In other ways of looking at it, causal relations are necessarily built into the structure of the world, and we can go out and empirically discover them. I don’t know which is right. (Sorry for the vagueness in this one – it’s confusing enough to me that I have trouble even precisely phrasing the dilemma).

How should we deal with the apparent dependence of inductive reasoning upon our choices of concepts?

I’ve written about this here. Beyond just the problem of concept-dependence in our choices of priors, there’s also the problem presented by the grue/bleen thought experiment.

This thought experiment proposes two new concepts: grue (= the set of things that are either green before 2100 or blue after 2100) and bleen (the inverse of grue). It then shows that if we reasoned in terms of grue and bleen, standard induction would have us concluding that all emeralds will suddenly turn blue after 2100. (We repeatedly observed them being grue before 2100, so we should conclude that they will be grue after 2100.)

In other words, choose the wrong concepts and induction breaks down. This is really disturbing – choices of concepts should be merely pragmatic matters! They shouldn’t function as fatal epistemic handicaps. And given that they appear to, we need to develop some criterion we can use to determine what concepts are good and what concepts are bad.

The trouble with this is that the only proposals I’ve seen for such a criterion reference the idea of concepts that “carve reality at its joints”; in other words, the world is composed of green and blue things, not grue and bleen things, so we should use the former rather than the latter. But this relies on the outcome of our inductive process to draw conclusions about the starting step on which this outcome depends!

I don’t know how to cash out “good choices of concepts” without ultimately reasoning circularly. I also don’t even know how to make sense of the idea of concepts being better or worse for more than merely pragmatic reasons.

How should we reason about self defeating beliefs?

The classic self-defeating belief is “This statement is a lie.” If you believe it, then you are compelled to disbelieve it, eliminating the need to believe it in the first place. Broadly speaking, self-defeating beliefs are those that undermine the justifications for belief in them.

Here’s an example that might actually apply in the real world: Black holes glow. The process of emission is known as Hawking radiation. In principle, any configuration of particles with a mass less than the black hole can be emitted from it. Larger configurations are less likely to be emitted, but even configurations such as a human brain have a non-zero probability of being emitted. Henceforth, we will call such configurations black hole brains.

Now, imagine discovering some cosmological evidence that the era in which life can naturally arise on planets circling stars is finite, and that after this era there will be an infinite stretch of time during which all that exists are black holes and their radiation. In such a universe, the expected number of black hole brains produced is infinite (a tiny finite probability multiplied by an infinite stretch of time), while the expected number of “ordinary” brains produced is finite (assuming a finite spatial extent as well).

What this means is that discovering this cosmological evidence should give you an extremely strong boost in credence that you are a black hole brain. (Simply because most brains in your exact situation are black hole brains.) But most black hole brains have completely unreliable beliefs about their environment! They are produced by a stochastic process which cares nothing for producing brains with reliable beliefs. So if you believe that you are a black hole brain, then you should suddenly doubt all of your experiences and beliefs. In particular, you have no reason to think that the cosmological evidence you received was veridical at all!

I don’t know how to deal with this. It seems perfectly possible to find evidence for a scenario that suggests that we are black hole brains (I’d say that we have already found such evidence, multiple times). But then it seems we have no way to rationally respond to this evidence! In fact, if we do a naive application of Bayes’ theorem here, we find that the probability of receiving any evidence in support of black hole brains to be 0!

So we have a few options. First, we could rule out any possible skeptical scenarios like black hole brains, as well as anything that could provide any amount of evidence for them (no matter how tiny). Or we could accept the possibility of such scenarios but face paralysis upon actually encountering evidence for them! Both of these seem clearly wrong, but I don’t know what else to do.

How should we reason about our own existence and indexical statements in general?

This is called anthropic reasoning. I haven’t written about it on this blog, but expect future posts on it.

A thought experiment: imagine a murderous psychopath who has decided to go on an unusual rampage. He will start by abducting one random person. He rolls a pair of dice, and kills the person if they land snake eyes (1, 1). If not, he lets them free and hunts down ten new people. Once again, he rolls his pair of die. If he gets snake eyes he kills all ten. Otherwise he frees them and kidnaps 100 new people. On and on until he eventually gets snake eyes, at which point his murder spree ends.

Now, you wake up and find that you have been abducted. You don’t know how many others have been abducted alongside you. The murderer is about to roll the dice. What is your chance of survival?

Your first thought might be that your chance of death is just the chance of both dice landing 1: 1/36. But think instead about the proportion of all people that are ever abducted by him that end up dying. This value ends up being roughly 90%! So once you condition upon the information that you have been captured, you end up being much more worried about your survival chance.

But at the same time, it seems really wrong to be watching the two dice tumble and internally thinking that there is a 90% chance that they land snake eyes. It’s as if you’re imagining that there’s some weird anthropic “force” pushing the dice towards snake eyes. There’s way more to say about this, but I’ll leave it for future posts.

Things I’ve become un-puzzled about

Newcomb’s problem – one box or two box?

To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly.

– Nozick, 1969

I’ve spent months and months being hopelessly puzzled about Newcomb’s problem. I now am convinced that there’s an unambiguous right answer, which is to take the one box. I wrote up a dialogue here explaining the justification for this choice.

In a few words, you should one-box because one-boxing makes it nearly certain that the simulation of you run by the predictor also one-boxed, thus making it nearly certain that you will get 1 million dollars. The dependence between your action and the simulation is not an ordinary causal dependence, nor even a spurious correlation – it is a logical dependence arising from the shared input-output structure. It is the same type of dependence that exists in the clone prisoner dilemma, where you can defect or cooperate with an individual you are assured is identical to you in every single way. When you take into account this logical dependence (also called subjunctive dependence), the answer is unambiguous: one-boxing is the way to go.

Summing up:

Things I remain conceptually confused about:

  • Consciousness
  • Decision theory & free will
  • Objective priors
  • Infinities in decision theory
  • Fundamentality of causality
  • Dependence of induction on concept choice
  • Self-defeating beliefs
  • Anthropic reasoning

Some simple probability puzzles

~ squarishbracket ~ 3 Comments

(Most of these are taken from Ian Hacking’s Introduction to Probability and Inductive Logic.)

  1. About as many boys as girls are born in hospitals. Many babies are born every week at City General. In Cornwall, a country town, there is a small hospital where only a few babies are born every week.

    Define a normal week as one where between 45% and 55% of babies are female. An unusual week is one where more than 55% or less than 45% are girls.

    Which of the following is true:
    (a) Unusual weeks occur equally often at City General and at Cornwall.
    (b) Unusual weeks are more common at City General than at Cornwall.
    (c) Unusual weeks are more common at Cornwall than at City General.

  2. Pia is 31 years old, single, outspoken, and smart. She was a philosophy major. When a student, she was an ardent supporter of Native American rights, and she picketed a department store that had no facilities for nursing mothers.

    Which of the following statements are most probable? Which are least probable?

    (a) Pia is an active feminist.
    (b) Pia is a bank teller.
    (c) Pia works in a small bookstore.
    (d) Pia is a bank teller and an active feminist.
    (e) Pia is a bank teller and an active feminist who takes yoga classes.
    (f) Pia works in a small bookstore and is an active feminist who takes yoga classes.

  3. You have been called to jury duty in a town with only green and blue taxis. Green taxis dominate the market, with 85% of the taxis on the road.

    On a misty winter night a taxi sideswiped another car and drove off. A witness said it was a blue cab. This witness is tested under similar conditions, and gets the color right 80% of the time.

    You conclude about the sideswiping taxi:
    (a) The probability that it is blue is 80%.
    (b) It is probably blue, but with a lower probability than 80%.
    (c) It is equally likely to be blue or green.
    (d) It is more likely than not to be green.

  4. You are a physician. You think that it’s quite likely that a patient of yours has strep throat. You take five swabs from the throat of this patient and send them to a lab for testing.

    If the patient has strep throat, the lab results are right 70% of the time. If not, then the lab is right 90% of the time.

    The test results come back: YES, NO, NO, YES, NO

    You conclude:
    (a) The results are worthless.
    (b) It is likely that the patient does not have strep throat.
    (c) It is slightly more likely than not that the patient does have strep throat.
    (d) It is very much more likely than not that the patient does have strep throat.

  5. In a country, all families wants a boy. They keep having babies till a boy is born. What is the expected ratio of boys and girls in the country?
  6.  Answer the following series of questions:

    If you flip a fair coin twice, do you have the same chance of getting HH as you have of getting HT?

    If you flip the coin repeatedly until you get HH, does this result in the same average number of flips as if you repeat until you get HT?

    If you flip it repeatedly until either HH emerges or HT emerges, is either outcome equally likely?

    You play a game with a friend in which you each choose a sequence of three possible flips (e.g HHT and TTH). You then flip the coin repeatedly until one of the two patterns emerges, and whosever pattern it is wins the game. You get to see your friend’s choice of pattern before deciding yours. Are you ever able to bias the game in your favor?

    Are you always able to bias the game in your favor?

 

Solutions (and lessons)

  1. The correct answer is (a): Unusual weeks occur more often at Cornwall than at City General. Even though the chance of a boy is the same at Cornwall as it is at City General, the percentage of boys from week to week is larger in the smaller city (for N patients a week, the percentage boys goes like 1/sqrt(N)). Indeed, if you think about an extreme case where Cornwall has only one birth a week, then every week will be an unusual week (100% boys or 0% boys).
  2. There is room to debate the exact answer but whatever it is, it has to obey some constraints. Namely, the most probable statement cannot be (d), (e), or (f), and the least probable statement cannot be (a), (b), or (c). Why? Because of the conjunction rule of probability: each of (d), (e), and (f) are conjunctions of (a), (b), and (c), so they cannot be more likely. P(A & B) ≤ P(A).

    It turns out that most people violate this constraint. Many people answer that (f) is the most probable description, and (b) is the least probable. This result is commonly interpreted to reveal a cognitive bias known as the representativeness heuristic – essentially, that our judgements of likelihood are done by considering which descriptions most closely resemble the known facts. In this case,

    Another factor to consider is that prior to considering the evidence, your odds on a given person being a bank teller as opposed to working in a small bookstore should be heavily weighted towards her being a bank teller. There are just far more bank tellers than small bookstore workers (maybe a factor of around 20:1). This does not necessarily mean that (b) is more likely than (c), but it does mean that the evidence must discriminate strongly enough against her being a bank teller so as to overcome the prior odds.

    This leads us to another lesson, which is to not neglect the base rate. It is easy to ignore the prior odds when it feels like we have strong evidence (Pia’s age, her personality, her major, etc.). But the base rate on small bookstore workers and bank tellers are very relevant to the final judgement.

  3. The correct answer is (d) – it is more likely than not that the sideswiper was green. This is a basic case of base rate neglect – many people would see that the witness is right 80% of the time and conclude that the witness’s testimony has an 80% chance of being correct. But this is ignoring the prior odds on the content of the witness’s testimony.

    In this case, there were prior odds of 17:3 (85%:15%) in favor of the taxi being green. The evidence had a strength of 1:4 (20%:80%), resulting in the final odds being 17:12 in favor of the taxi being green. Translating from odds to probabilities, we get a roughly 59% chance of the taxi having been green.

    We could have concluded (d) very simply by just comparing the prior probability (85% for green) with the evidence (80% for blue), and noticing that the evidence would not be strong enough to make blue more likely than green (since 85% > 80%). Being able to very quickly translate between statistics and conclusions is a valuable skill to foster.

  4. The right answer is (d). We calculate this just like we did the last time:

    The results were YES, NO, NO, YES, NO.

    Each YES provides evidence with strength 7:1 (70%/10%) in favor of strep, and each NO provides evidence with strength 1:3 (30%/90%).

    So our strength of evidence is 7:1 ⋅ 1:3 ⋅ 1:3 ⋅ 7:1 ⋅ 1:3 = 49:27, or roughly 1.81:1 in favor of strep. This might be a little surprising… we got more NOs than YESs and the NO was correct 90% of the time for people without strep, compared to the YES being correct only 70% of the time in people with strep.

    Since the evidence is in favor of strep, and we started out already thinking that strep was quite likely, in the end we should be very convinced that they have strep. If our prior on the patient having strep was 75% (3:1 odds), then our probability after getting evidence will be 84% (49:9 odds).

    Again, surprising! The patient who sees these results and hears the doctor declaring that the test strengthens their belief that the patient has strep might feel that this is irrational and object to the conclusion. But the doctor would be right!

  5. Supposing as before that the chance of any given birth being a boy is equal to the chance of it being a girl, we end up concluding…

    The expected ratio of boys and girls in the country is 1! That is, this strategy doesn’t allow you to “cheat” – it has no impact at all on the ratio. Why? I’ll leave this one for you to figure out. Here’s a diagram for a hint:

    36666658_10216831977421805_8359037287605993472_n

    This is important because it applies to the problem of p-hacking. Imagine that all researchers just repeatedly do studies until they get the results they like, and only publish these results. Now suppose that all the researchers in the world are required to publish every study that they do. Now, can they still get a bias in favor of results they like? No! Even though they always stop when getting the result they like, the aggregate of their studies is unbiased evidence. They can’t game the system!

  6.  Answers, in order:

    If you flip a fair coin twice, do you have the same chance of getting HH as you have of getting HT? (Yes)

    If you flip it repeatedly until you get HH, does this result in the same average number of flips as if you repeat until you get HT? (No)

    If you flip it repeatedly until either HH emerges or HT emerges, is either outcome equally likely? (Yes)

    You play a game with a friend in which you each choose a sequence of three coin flips (e.g HHT and TTH). You then flip a coin repeatedly until one of the two patterns emerges, and whosever pattern it is wins the game. You get to see your friend’s choice of pattern before deciding yours. Are you ever able to bias the game in your favor? (Yes)

    Are you always able to bias the game in your favor? (Yes!)

    Here’s a wiki page with a good explanation of this: LINK. A table from that page illustrating a winning strategy for any choice your friend makes:

    1st player’s choice 2nd player’s choice Odds in favour of 2nd player
    HHH THH 7 to 1
    HHT THH 3 to 1
    HTH HHT 2 to 1
    HTT HHT 2 to 1
    THH TTH 2 to 1
    THT TTH 2 to 1
    TTH HTT 3 to 1
    TTT HTT 7 to 1

Against moral realism

~ squarishbracket ~ 3 Comments

Here’s my primary problem with moral realism: I can’t think of any acceptable epistemic framework that would give us a way to justifiably update our beliefs in the objective truth of moral claims. I.e. I can’t think of any reasonable account of how we could have justified beliefs in objectively true moral principles.

Here’s a sketch of a plausible-seeming account of epistemology. Broad-strokes, there are two sources of justified belief: deduction and induction.

Deduction refers to the process by which we define some axioms and then see what logically follows from them. So, for instance, the axioms of Peano Arithmetic entail the theorem that 1+1=2 – or, in Peano’s language, S(0) + S(0) = S(S(0)). The central reason why reasoning by deduction is reliable is that the truths established are true by definition – they are made true by the way we have constructed our terms, and are thus true in every possible world.

Induction is scientific reasoning – it is the process of taking prior beliefs, observing evidence, and then updating these beliefs (via Bayes’ rule, for instance). The central reason why induction is reliable comes from the notion of causal entanglement. When we make an observation and update our beliefs based upon this observation, the brain state “believes X” has become causally entangled with the truth of the the statement X. So, for instance, if I observe a positive result on a pregnancy test, then my belief in the statement “I am pregnant” has become causally entangled with the truth of the statement “I am pregnant.” It is exactly this that justifies our use of induction in reasoning about the world.

Now, where do moral claims fall? They are not derived from deductive reasoning… that is, we cannot just arbitrarily define right and wrong however we like, and then derive morality from these definitions.

And they are also not truths that can be established through inductive reasoning; after all, objective moral truths are not the types of things that have any causal effects on the world.

In other words, even if there are objective moral truths, we would have no way of forming justified beliefs about this. To my mind, this is a pretty devastating situation for a moral realist. Think about it like this: a moral realist who doesn’t think that moral truths have causal power over the world must accept that all of their beliefs about morality are completely causally independent of their truth. If we imagine keeping all the descriptive truths about the world fixed, and only altering the normative truths, then none of the moral realist’s moral beliefs would change.

So how do they know that they’re in the world where their moral beliefs actually do align with the moral reality? Can they point to any reason why their moral beliefs are more likely to be true than any other moral statements? As far as I can tell, no, they can’t!

Now, you might just object to the particular epistemology I’ve offered up, and suggest some new principle by which we can become acquainted with moral truth. This is the path of many professional philosophers I have talked to.

But every attempt that I’ve heard of for doing this begs the question or resorts to just gesturing at really deeply held intuitions of objectivity. If you talk to philosophers, you’ll hear appeals to a mysterious cognitive ability to reflect on concepts and “detect their intrinsic properties”, even if these properties have no way of interacting with the world, or elaborate descriptions of the nature of “self-evident truths.”

(Which reminds me of this meme)

self-evident-truth-5153703.png

None of this deals with the central issue in moral epistemology, as I see it. This central issue is: How can a moral realist think that their beliefs about morality are any more likely to be true than any random choice of a moral framework?

Explanation is asymmetric

~ squarishbracket ~ Leave a comment

We all regularly reason in terms of the concept of explanation, but rarely think hard about what exactly we mean by this explanation. What constitutes a scientific explanation? In this post, I’ll point out some features of explanation that may not be immediately obvious.

Let’s start with one account of explanation that should seem intuitively plausible. This is the idea that to explain X to a person is to give that person some information I that would have allowed them to predict X.

For instance, suppose that Janae wants an explanation of why Ari is not pregnant. Once we tell Janae that Ari is a biological male, she is satisfied and feels that the lack of pregnancy has been explained. Why? Well, because had Janae known that Ari was a male, she would have been able to predict that Ari would not get pregnant.

Let’s call this the “predictive theory of explanation.” On this view, explanation and prediction go hand-in-hand. When somebody learns a fact that explains a phenomenon, they have also learned a fact that allows them to predict that phenomenon.

 To spell this out very explicitly, suppose that Janae’s state of knowledge at some initial time is expressed by

K1 = “Males cannot get pregnant.”

At this point, Janae clearly cannot conclude anything about whether Ari is pregnant. But now Janae learns a new piece of information, and her state of knowledge is updated to

K2 = “Ari is a male & males cannot get pregnant.”

Now Janae is warranted in adding the deduction

K’ = “Ari cannot get pregnant”

This suggests that added information explains Ari’s non-pregnancy for the same reason that it allows the deduction of Ari’s non-pregnancy.

Now, let’s consider a problem with this view: the problem of relevance.

Suppose a man named John is not pregnant, and somebody explains this with the following two premises:

  1. People who take birth control pills almost certainly don’t get pregnant.
  2. John takes birth control pills regularly.

Now, these two premises do successfully predict that John will not get pregnant. But the fact that John takes birth control pills regularly gives no explanation at all of his lack of pregnancy. Naively applying the predictive theory of explanation gives the wrong answer here.

You might have also been suspicious of the predictive theory of explanation on the grounds that it relied on purely logical deduction and a binary conception of knowledge, not allowing us to accommodate the uncertainty inherent in scientific reasoning. We can fix this by saying something like the following:

What it is to explain X to somebody that knows K is to give them information I such that

(1) P(X | K) is small, and
(2) P(X | K, I) is large.

“Small” and “large’ here are intentionally vague; it wouldn’t make sense to draw a precise line in the probabilities.

The idea here is that explanations are good insofar as they (1) make their explanandum sufficiently likely, where (2) it would be insufficiently likely without them.

We can think of this as a correlational account of explanation. It attempts to root explanations in sufficiently strong correlations.

First of all, we can notice that this doesn’t suffer from a problem with irrelevant information. We can find relevance relationships by looking for independencies between variables. So maybe this is a good definition of scientific explanation?

Unfortunately, this “correlational account of explanation” has its own problems.

Take the following example.

uploaded image

This flagpole casts a shadow of length L because of the angle of elevation of the sun and the height of the flagpole (H). In other words, we can explain the length of the shadow with the following pieces of information:

I1 =  “The angle of elevation of the sun is θ”
I2 = “The height of the lamp post is H”
I3 = Details involving the rectilinear propagation of light and the formation of shadows

Both the predictive and correlational theory of explanation work fine here. If somebody wanted an explanation for why the shadow’s length is L, then telling them I1, I2, and I3 would suffice. Why? Because I1, I2, and I3 jointly allow us to predict the shadow’s length! Easy.

X = “The length of the shadow is L.”
(I1 & I2 & I3) ⇒ X
So I1 & I2 & I3 explain X.

And similarly, P(X | I1 & I2 & I3) is large, and P(X) is small. So on the correlational account, the information given explains X.

But now, consider the following argument:

(I1 & I3 & X) ⇒ I2
So I1 & I3 & X explain I2.

The predictive theory of explanation applies here. If we know the length of the shadow and the angle of elevation of the sun, we can deduce the height of the flagpole. And the correlational account tells us the same thing.

But it’s clearly wrong to say that the explanation for the height of the flagpole is the length of the shadow!

What this reveals is an asymmetry in our notion of explanation. If somebody already knows how light propagates and also knows θ, then telling them H explains L. But telling them L does not explain H!

In other words, the correlational theory of explanation fails, because correlation possesses symmetry properties that explanation does not.

This thought experiment also points the way to a more complete account of explanation. Namely, the relevant asymmetry between the length of the shadow and the height of the flagpole is one of causality. The reason why the height of the flagpole explains the shadow length but not vice versa, is that the flagpole is the cause of the shadow and not the reverse.

In other words, what this reveals to us is that scientific explanation is fundamentally about finding causes, not merely prediction or statistical correlation. This causal theory of explanation can be summarized in the following:

An explanation of A is a description of its causes that renders it intelligible.

More explicitly, an explanation of A (relative to background knowledge K) is a set of causes of A that render X intelligible to a rational agent that knows K.

Query sensitivity of evidential updates

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Plausible reasoning, unlike logical deduction, is sensitive not only to the information at hand but also to the query process by which the information was obtained.

Judea Pearl, Probabilistic Reasoning in Intelligent Systems

This quote references an interesting feature of inductive reasoning that’s worth unpacking. It is indicative of the different level of complexity involved in formalizing induction than that involved in formalizing deduction.

A very simple example of this:

You spread a rumor to your neighbor N. A few days later you hear the same rumor from another neighbor N’. Should you increase your belief in the rumor now that N acknowledges it, or should you determine first whether N heard it from N’?

Clearly, if the only source of information for N’ was N, then your belief should not change. But if N’ independently confirmed the validity of the rumor, you have good reason to increase your belief in it.

In general, when you have both top-down (predictive) and bottom-up (explanatory/diagnostic) inferences in evidential reasoning, it is important to be able to trace back queries. If not, one runs the risk of engaging in circular reasoning.

So far this is all fairly obvious. Now here’s an example that’s more subtle.

Three prisoners problem

Three prisoners have been tried for murder, and their verdicts will be read tomorrow morning. Only one will be declared guilty, and the other two will be declared innocent.

Before sentencing, Prisoner A asks the guard (who knows which prisoner will be declared guilty) to do him a favor and give a letter to one of the other two prisoners who will be released (since only one person will be declared guilty, Prisoner A knows that at least one of the other two prisoners will be released). The guard does so, and later, Prisoner A asks him which of the two prisoners (B or C) he gave the letter two. The guard responds “I gave the letter to Prisoner B.”

Now Prisoner A reasons as follows:

“Previously, my chances of being executed were one in three. Now that I know that B will be released, only C and I remain as candidates for being declared guilty. So now my chances are one in two.”

Is this wrong?

Denote “A is guilty” as GA, and “B is innocent” as IB. Now, since GA → IB, we have that P(IB | GA) = 1. This tells us that we can write

P(GA | IB) = P(IB | GA) P(GA) / P(IB)
= P(GA) / P(IB) = ⅓ / ⅔ = ½

The problem with this argument is that we have excluded some of the context of the guard’s response, namely, that the guard could only have answered “I gave the letter to Prisoner B” or “I gave the letter to Prisoner C.” In other words, the fact “Prisoner B will be declared innocent” leads to the wrong conclusion about the credibility of A’s guilt.

Let’s instead condition on IB’ = “Guard says that B will be declared innocent.” Now we get

P(GA | IB’) = P(IB’ | GA) P(GA) / P(IB’) = ½ ⋅ ⅓ / ½ = ⅓.

It’s not sufficient to just condition on what the guard said. We must consider the range of possible statements that the guard could have made.

In general, we cannot only assess the impact of propositions implied by information. We must also consider what information we could have received.

Things get clearer if we consider a similar thought experiment.

1000 prisoners problem

You are one of 1000 prisoners awaiting sentencing, with the knowledge that only one of you will be declared guilty. You come across a slip of paper from the court listing 998 prisoners; each name marked ‘innocent’. You look through all 998 names and find that your name is missing.

This should worry you greatly – your chances of being declared guilty have gone from 1 in 1000 to 1 in 2.

But imagine that you now see the query that produced the list.

Query: “Print the names of any 998 innocent right-handed prisoners.”

If you are the only left-handed prisoner, then you should thank your lucky stars. Why? Because now that you know that the query couldn’t have produced your name, the fact that it didn’t gives you no information. In other words, your chances have gone back from 1:2 to 1:1000.

In this example you can see very clearly why information about the possible outputs of a query is relevant to how we should update on the actual output of the query. We must know the process by which we attain information in order to be able to accommodate that information into our beliefs.

But now what if we don’t have this information? Suppose that you only run into the list of prisoners, and have no additional knowledge about how it was produced. Well, then we must consider all possible queries that might have produced this output!

This is no small matter.

For simplicity, let’s reconsider the simpler example with just three prisoners: Prisoners A, B, and C. Imagine that you are Prisoner A.

You come across a slip of paper from the court containing the statement I, where

I = “Prisoner B will be declared innocent”.

Now, we must assess the impact of I on the proposition GA = “Prisoner A is guilty.”

P(GA | I) = P(I | GA) P(GA) / P(I) = ⅓ P(I | GA) / P(I)

The result of this calculation depends upon P(I | GA), or in other words, how likely it is that the slip would declare Prisoner B innocent, given that you are guilty. This depends on the query process, and can vary anywhere from 0 to 1. Let’s just give this variable a name: P(I | GA) = p.

We’ll also need to know two other probabilities: (1) that the slip declares B innocent given that B is guilty, and (2) that it declares B innocent given that C is guilty. We’ll assume that the slip cannot be lying (i.e. that the first of these is zero), and name the second probability q = P(I | GC).

P(I | GA) = p (slip could declare either B or C innocent)
P(I | GB) = 0 (slip could declare either A or C innocent)
P(I | GC) = q (slip could declare either A or B innocent)

Now we have

P(GA | I) = ⅓ p / P(I)
=⅓ p / [P(I | GA) P(GA) + P(I | GB) P(GB) + P(I | GC) P(GC)]
= ⅓ p / (⅓ p + 0 + ⅓ q)
= p/(p + q)

How do we assess this value, given that p and q are unknown? The Bayesian solution is to treat the probabilities p and q as random variables, and specify probability distributions over their possible values: f(p) and g(q). This distribution should contain all of your prior knowledge about the plausible queries that might have produced I.

The final answer is obtained by integrating over all possible values of p and q.

P(GA | I) = E[p/(p + q)]
= ∫ p/(p + q) f(p) g(q) dp dq

Supposing that our distributions over p and q are maximally uncertain, the final distribution we obtain is

P(GA | I) = ∫ p/(p + q) dp dq
= 0.5

Now suppose that we know that the slip could not declare A (yourself) innocent (as we do in the original three prisoners problem). Then we know that q = 1 (since if C is guilty and A couldn’t be on the slip, B is the only possible choice). This gives us

P(GA | I) = ∫ p/(p + 1) f(p) dp

If we are maximally uncertain about the value of p, we obtain

P(GA | I) = ∫ p/(p + 1) dp
= 1 – ln(2)
≈ 0.30685

If, on the other hand, we are sure that the value of p is 50% (i.e., we know that in the case that A is guilty, the guard chooses randomly between B and C), we obtain

P(GA | I) = .5/(.5 + 1) = ⅓

We’ve re-obtained our initial result! Interestingly, we can see that being maximally uncertainty about the guard’s procedure for choosing between B and C gives a different answer than knowing that the guard chooses totally randomly between B and C.

Notice that this is true even though these reflect the same expectation of what choice the guard will make!

I.e., in both cases (total uncertainty about p, and knowledge that p is exactly .5), we should have 50% credence in the guard choosing B. This gives us some insight into the importance of considering different types of uncertainty when doing induction, which is a topic for another post.

Summarizing conscious experience

~ squarishbracket ~ 1 Comment

There’s a puzzle for implementation of probabilistic reasoning in human beings. This is that the start of the reasoning process in humans is conscious experience, and it’s not totally clear how we should update on conscious experiences.

Jeffreys defined a summary of an experience E as a set B of propositions {B1, B2, … Bn} such that for all other propositions in your belief system A, P(A | B) = P(A | B, E).

In other words, B is a minimal set of propositions that fully screens off your experience.

This is a useful concept because summary sentences allow you to isolate everything that is epistemically relevant about conscious experience. if you have a summary B of an experience E, then you only need to know P(A | B) and P(B | E) in order to calculate P(A | E).

Notice that the summary set is subjective; it is defined only in terms of properties of your personal belief network. The set of facts that screens off E for you might be different from the set of facts that screens it off for somebody else.

Quick example.

Consider a brief impression by candlelight of a cloth held some distance away from you. Call this experience E.

Suppose that all you could decipher from E is that the cloth was around 2 meters away from you, and that it was either blue (with probability 60%) or green (with probability 40%). Then the summary set for E might be {“The cloth is blue”, “The cloth is green”, “The cloth is 2 meters away from you”, “The cloth is 3 meters away from you”, etc.}.

If this is the right summary set, then the probabilities P(“The cloth is blue”), P(“The cloth is green”) and P(“The cloth is x meters away from you”) should screen off E from the rest of your beliefs.

One trouble is that it’s not exactly obvious how to go about converting a given experience into a set of summary propositions. We could always be leaving something out. For instance, one more thing we learned upon observing E was the proposition “I can see light.” This is certainly not screened off by the other propositions so far, so we need to add it in as well.

But how do we know that we’ve gotten everything now? If we think a little more, we realize that we have also learned something about the nature of the light given off by the candle flame. We learn that it is capable of reflecting the color of light that we saw!

But now this additional consideration is related to how we interpret the color of the cloth. In other words, not only might we be missing something from our summary set, but that missing piece might be relevant to how we interpret the others.

I’d like to think more about this question: In general, how do we determine the set of propositions that screens off a given experience from the rest of your beliefs? Ultimately, to be able to coherently assess the impact of experiences on your web of beliefs, your model of reality must contain a model of yourself as an experiencer.

The nature of this model is pretty interesting from a philosophical perspective. Does it arise organically out of factual beliefs about the physical world? Well, this is what a physicalist would say. To me, it seems quite plausible that modeling yourself as a conscious experiencer would require a separate set of rules relating physical happenings to conscious experiences. How we should model this set of rules as a set of a priori hypotheses to be updated on seems very unclear to me.

Simple induction

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In front of you is a coin. You don’t know the bias of this coin, but you have some prior probability distribution over possible biases (between 0: always tails, and 1: always heads). This distribution has some statistical properties that characterize it, such as a standard deviation and a mean. And from this prior distribution, you can predict the outcome of the next coin toss.

Now the coin is flipped and lands heads. What is your prediction for the outcome of the next toss?

This is a dead simple example of a case where there is a correct answer to how to reason inductively. It is as correct as any deductive proof, and derives a precise and unambiguous result:

Fixed

This is a law of rational thought, just as rules of logic are laws of rational thought. It’s interesting to me how the understanding of the structure of inductive reasoning begins to erode the apparent boundary between purely logical a priori reasoning and supposedly a posteriori inductive reasoning.

Anyway, here’s one simple conclusion that we can draw from the above image: After the coin lands heads, it should be more likely that the coin will land heads next time. After all, the initial credence was µ, and the final credence is µ multiplied by a value that is necessarily greater than 1.

You probably didn’t need to see an equation to guess that for each toss that lands H, future tosses landing H become more likely. But it’s nice to see the fundamental justification behind this intuition.

We can also examine some special cases. For instance, consider a uniform prior distribution (corresponding to maximum initial uncertainty about the coin bias). For this distribution (π = 1), µ = 1/2 and σ = 1/3. Thus, we arrive at the conclusion that after getting one heads, your credence in the next toss landing heads should be 13/18 (72%, up from 50%).

We can get a sense of the insufficiency of point estimates using this example. Two prior distributions with the same average value will respond very differently to evidence, and thus the final point estimate of the chance of H will differ. But what is interesting is that while the mean is insufficient, just the mean and standard deviation suffice for inferring the value of the next point estimate.

In general, the dynamics are controlled by the term σ/µ. As σ/µ goes to zero (which corresponds to a tiny standard deviation, or a very confident prior), our update goes to zero as well. And as σ/µ gets large (either by a weak prior or a low initial credence in the coin being H-biased), the observation of H causes a greater update.

How large can this term possibly get? Obviously, the updated point estimate should asymptote towards 1, but this is not obvious from the form of the equation we have (it looks like σ/µ can get arbitrarily large, forcing our final point estimate to infinity). What we need to do is optimize the updated point estimate, while taking into account the constraints implied by the relationship between σ and µ.

Priors in the supernatural

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A friend of mine recently told me the following anecdote.

Years back, she had visited an astrologer in India with her boyfriend, who told her the following things: (1) she would end up marrying her boyfriend at the time, (2) down the line they would have two kids, the first a girl and the second a boy, and (3) he predicted the exact dates of birth of both children.

Many years down the line, all of these predictions turned out to be true.

I trust this friend a great deal, and don’t have any reason to think that she misremembered the details or lied to me about them. But at the same time, I recognize that astrology is completely crazy.

Since that conversation, I’ve been thinking about the ways in which we can evaluate our de facto priors in supernatural events by consulting either real-world anecdotes or thought experiments. For instance, if we think that each of these two predictions gave us a likelihood ratio of 100:1 in favor of astrology being true, and if I ended up thinking that astrology was about as likely to be true as false, then I must have started with roughly 1:10,000 odds against astrology being true.

That’s not crazily low for a belief that contradicts much of our understanding of physics. I would have thought that my prior odds would be something much lower, like 1:1010 or something. But really put yourself in that situation.

Imagine that you go to an astrologer, who is able to predict an essentially unpredictable sequence of events years down the line, with incredible accuracy. Suppose that the astrologer tells you who you will marry, how many kids you’ll have, and the dates of birth of each. Would you really be totally unshaken by this experience? Would you really believe that it was more likely to have happened by coincidence?

Yes, yes, I know the official Bayesian response – I read it in Jaynes long ago. For beliefs like astrology that contradict our basic understanding of science and causality, we should always have reserved some amount of credence for alternate explanations, even if we can’t think of any on the spot. This reserve of credence will insure us against jumping in credence to 99% upon seeing a psychic continuously predict the number in your heads, ensuring sanity and a nice simple secular worldview.

But that response is not sufficient to rule out all strong evidence for the supernatural.

Here’s one such category of strong evidence: evidence for which all alternative explanations are ruled out by the laws of physics as strongly as the supernatural hypothesis is ruled out by the laws of physics.

I think that my anecdote is one such case. If it was true, then there is no good natural alternative explanation for it. The reason? Because the information about the dates of birth of my friend’s children did not exist in the world at the time of the prediction, in any way that could be naturally attainable by any human being.

By contrast, imagine you go to a psychic who tells you to put up some fingers behind your back and then predicts over and over again how many fingers you have up. There’s hundreds of alternative explanations for this besides “Psychics are real science has failed us.” The reason that there are these alternative explanations is that the information predicted by the psychic existed in the world at the time of the prediction.

But in the case of my friend’s anecdote, the information predicted by the astrologer was lost far in the chaotic dynamics of the future.

What this rules out is the possibility that the astrologer somehow obtained the information surreptitiously by any natural means. It doesn’t rule out a host of other explanations, such as that my friend’s perception at the time was mistaken, that her memory of the event is skewed, or that she is lying. I could even, as a last resort, consider that possibility that I hallucinated the entire conversation with her. (I’d like to give the formal title “unbelievable propositions” to the set of propositions that are so unlikely that we should sooner believe that we are hallucinating than accept evidence for them.)

But each of these sources of alternative explanations, with the possible exception of the last, can be made significantly less plausible.

Let me use a thought experiment to illustrate this.

Imagine that you are a nuclear physicist who, with a group of fellow colleagues, have decided to test the predictive powers of a fortune teller. You carefully design an experiment in which a source of true quantum randomness will produce a number between 1 and N. Before the number has been produced, when it still exists only as an unrealized possibility in the wave function, you ask the fortune teller to predict its value.

Suppose that they get it correct. For what value of N would you begin to take their fortune telling abilities seriously?

Here’s how I would react to the success, for different values of N.

N = 10: “Haha, that’s a funny coincidence.”

N = 100: “Hm, that’s pretty weird.”

N = 1000: “What…”

N = 10,000: “Wait, WHAT!?”

N = 100,000: “How on Earth?? This is crazy.”

N = 1,000,000: “Ok, I’m completely baffled.”

I think I’d start taking them seriously as early as N = 10,000. This indicates prior odds of roughly 1:10,000 against fortune-telling abilities (roughly the same as my prior odds against astrology, interestingly!). Once again, this seems disconcertingly low.

But let’s try to imagine some alternative explanations.

As far as I can tell, there are only three potential failure points: (1) our understanding of physics, (2) our engineering of the experiment, (3) our perception of the fortune teller’s prediction.

First of all, if our understanding of quantum mechanics is correct, there is no possible way that any agent could do better than random at predicting the number.

Secondly, we stipulated that the experiment was designed meticulously so as to ensure that the information was truly random, and unavailable to the fortune-teller. I don’t think that such an experiment would actually be that hard to design. But let’s go even further and imagine that we’ve designed the experiment so that the fortune teller is not in causal contact with the quantum number-generator until after she has made her prediction.

And thirdly, we can suppose that the prediction is viewed by multiple different people, all of whom affirm that it was correct. We can even go further and imagine that video was taken, and broadcast to millions of viewers, all of whom agreed. Not all of them could just be getting it wrong over and over again. The only possibility is that we’re hallucinating not just the experimental result, but indeed also the public reaction and consensus on the experimental result.

But the hypothesis of a hallucination now becomes inconsistent with our understanding of how the brain works! A hallucination wouldn’t have the effect of creating a perception of a completely coherent reality in which everybody behaves exactly as normal except that they saw the fortune teller make a correct prediction. We’d expect that if this were a hallucination, it would not be so self-consistent.

Pretty much all that’s left, as far as I can tell, is some sort of Cartesian evil demon that’s cleverly messing with our brains to create this bizarre false reality. If this is right, then we’re left weighing the credibility of the laws of physics against the credibility of radical skepticism. And in that battle, I think, the laws of physics lose out. (Consider that the invalidity of radical skepticism is a precondition for the development of laws of physics in the first place.)

The point of all of this is just to sketch an example where I think we’d have a good justification for ruling out all alternative explanations, at least with an equivalent degree of confidence that we have for affirming any of our scientific knowledge.

Let’s bring this all the way back to where we started, with astrology. The conclusion of this blog post is not that I’m now a believer in astrology. I think that there’s enough credence in the buckets of “my friend misremembered details”, “my friend misreported details”, and “I misunderstood details” so that the likelihood ratio I’m faced with is not actually 10,000 to 1. I’d guess it’s something more like 10 to 1.

But I am now that much less confident that astrology is wrong. And I can imagine circumstances under which my confidence would be drastically decreased. While I don’t expect such circumstances to occur, I do find it instructive (and fun!) to think about them. It’s a good test of your epistemology to wonder what it would take for your most deeply-held beliefs to be overturned.