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Bayesian Epistemology: The Short Version
Feb 28th
There may be worlds in which it is possible to know a thing for certain, but in ours there is always a very tiny chance that you are being deceived by an alien reptile conspiracy. Some people throw up their hands in response to this and start buying lottery tickets, because “there’s always a chance, right?” The more general version of this naïve epistemological theory is “use wishful thinking to choose among all beliefs that are even remotely plausible.”
There are better ways of choosing beliefs than this, and in fact there is one particular epistemology that is mathematically guaranteed to give you the best chance of being right: Bayesian induction with a Solomonoff prior.¹
Bayesian Induction
In math:
So we’ve got this hypothesis that my roommate is a reptiloid alien, which we’ll call A, and we want to know how likely it is now that we’ve just seen that he has two sets of eyelids, an observation which we’ll call B. This is the posterior probability of A, which is P(A|B) in the formula above. (You read that as “the probability of A given B.” No butts are involved.)
To figure out that probability, we need to know three different things:
- The prior probability of A, which is how likely A was before we’d even heard of B. If we didn’t already have an idea of likely A was, then we’d want to use the Solomonoff prior for A, which we’ll get to in a minute. (Note that the more likely A was before we saw B, the more likely it should be afterwards.)
- The conditional probability of B, given A, which is the P(B|A) in the formula. If my roommate is a reptiloid, then it is very likely that he’ll have two sets of eyelids, but it’s not guaranteed. (Nothing is.) Maybe reptiloids don’t have any eyelids at all. (The more likely it is for reptiloids to have two eyelids, the more likely it is for my roommate to be a reptiloid if he does have two sets.)
- The prior probability of B, which is how likely B is. If there’s a common birth defect that causes people to have two pairs of eyelids, then it should be less likely that my roommate is an alien, because he probably just has the birth defect. But if there is no such birth defect, well, we should be more suspicious. (The more likely B is, the less likely our hypothesis A is.)
Once we have those numbers worked out, we can do some multiplication and some division and figure out exactly how paranoid I should be.
In English, with no math:
Represent all of your beliefs as probabilities or degrees of belief. When you encounter new evidence, update your beliefs to be more or less likely.
The Solomonoff Prior
To make any of this Bayesian induction stuff work, you need to be able to have some kind of prior probability for hypotheses before you see any evidence at all. The simplest way to do this would be to use an ignorance prior, which would mean treating all hypotheses as equally likely before you see evidence for or against them. But starting out with the assumption that my roommate is as likely to be a reptiloid as not seems… over-generous. We can do better.
Occam’s razor is a pretty handy principle, and we can adopt a version of it to serve as our universal prior: the simpler a hypothesis is, the more likely it is to be true. All we need now is a mathematical definition of simple and we’re set. How about:
The prior probability of a hypothesis is 1/x, where x is the number of bits it would take to write a computer program that prints a complete description of the hypothesis.
I’ll pry that definition apart tomorrow and explain how it ticks.
1: This is actually a slight lie. If the Church-Turing thesis is false, Solomonoff induction will perform poorly in cases where hypercomputation is relevant. But this possibility is exotic enough that it can be ignored for our purposes.
Stop Being Wrong: A Moral Imperative
Feb 27th
I have met people who exaggerate the differences [between the morality of different cultures], because they have not distinguished between differences of morality and differences of belief about facts. For example, one man said to me, “Three hundred years ago people in England were putting witches to death. Was that what you call the Rule of Human Nature or Right Conduct?” But surely the reason we do not execute witches is that we do not believe there are such things. If we did-if we really thought that there were people going about who had sold themselves to the devil and received supernatural powers from him in return and were using these powers to kill their neighbours or drive them mad or bring bad weather, surely we would all agree that if anyone deserved the death penalty, then these filthy quislings did. There is no difference of moral principle here: the difference is simply about matter of fact. It may be a great advance in knowledge not to believe in witches: there is no moral advance in not executing them when you do not think they are there. You would not call a man humane for ceasing to set mousetraps if he did so because he believed there were no mice in the house.
- C. S. Lewis
To behave in the most ethical manner possible, it is not enough to have the right theory of normative ethics. When I choose an action, what I am really doing is choosing a particular possible world that I expect my action to bring about. If I am hungry, I prefer possible worlds in which I am eating a donut to worlds in which I am not. By choosing to drive to a nearby Dunkin Donuts I am choosing a possible world of the former sort, an entirely rational thing to do. If I were a classical utilitarian, I would want to choose actions that would lead to possible worlds with more happiness in them than the other options available to me. Donuts for everyone, as it were. Any moral theory can be interpreted in this way.¹
And already there are several ways in which my knowledge (or lack thereof) will have a tremendous influence on my ability to make morally good decisions. If there are actions available to me that I’m not aware of, I won’t be able to choose them no matter how good they are. If I have forgotten that I have an entire box of Krispy Kreme donuts in my fridge, then choosing to drive to Dunkin Donuts is a poor decision, even though it is the best one I can make.
I can also make poor moral choices because I possess incorrect knowledge. If the Dunkin Donuts closed two weeks ago, then driving out to it would be an awful choice–I wouldn’t get any donuts at all that way.
These concerns also apply, of course, to decisions about matters weightier than donut acquisition.
Let’s say I’m a parent concerned with maximizing the welfare of my child. He’s just reached the age at which he should be vaccinated against a variety of infectious diseases, ostensibly because doing so is good for him. But I watched Jenny McCarthy appear on Larry King Live a couple years ago, and now believe that vaccination will raise my son’s risk of developing autism. I choose not to have him vaccinated, because I prefer worlds in which my son does not have autism to worlds in which he does.
Two years later, he catches Whooping Cough and dies. This is not an outcome I am particularly pleased with.
As a popular webcomic has pointed out, my decision was causally equivalent to murder.
Homework: how could I have done better? (The title of this post may be relevant.)
1: But not always specifically in terms of donuts. It would be quite strange to talk about a duty to eat donuts, for example.
Unhappy With Utility
Feb 26th
Philosophy is a multifaceted, sprawling academic discipline with an astonishing amount of accumulated historical baggage. While this is occasionally a good thing, it more often produces obnoxious and unnecessary misunderstandings. One such misunderstanding that has been particularly aggravating to me in recent weeks concerns the terms “utility” and “utility maximization”, which each have two meanings in philosophy which are substantially different but easily confused.¹
The first kind of utility is experience-utility, a measure of the amount of happiness associated with an outcome; it is the subject of the moral theory of utilitarianism, originally developed by Jeremy Bentham and John Stuart Mill over the course of the late-18th and mid-19th centuries. Bentham and Mill argued that happiness is the greatest good, and so good actions are those which increase the amount of happiness in the world. Maximizing experience-utility is a Level 1 moral imperative; it is a theory of what people ought to do.²
The second kind of utility is decision-utility, a measure of preference or desire for an outcome; it is a key concept in economics, game theory, ethics, and artificial intelligence. This kind of utility can be formalized and discussed mathematically, and it has been proven (Math warning!) that you can construct a mathematical function (the utility function) for every “rational” agent which incorporates all of their preferences. Maximizing decision-utility is a description of the behavior of rational agents.³
Consider the case of a woman, Anne, who is walking through a park on the way to an important business meeting. As she passes a pond, she sees that there is a young boy, Billy, about 20 feet from shore, and it is immediately clear from the way he is flailing that he will soon drown. Anne is faced with a choice: she can dive into the pond and rescue the drowning boy, but then she will show up to her meeting very late and soaking wet. (She was a lifeguard in college, so it is all but certain that she will succeed with her rescue.) Or she can hurry on to her meeting. Anne is perfectly selfish and concerned only with her own happiness, so her experience-utility and decision-utility are always equal for a particular outcome; she prefers outcomes in proportion to how happy they will make her. Unfortunately for Billy, she knows that helping him won’t make her happy and that being on time for her meeting will. Billy drowns.
Now imagine that instead of Anne being presented with the problem of the drowning boy, Carl is the one walking through the park that day. Like Anne, Carl knows that he’ll be much happier if he has a successful meeting and doesn’t have to get drenched; his experience-utility will be higher if he walks away. But quite unlike Anne, Carl likes to read John Stuart Mill and considers himself a classical utilitarian. He realizes that the happiness that Billy and his family will feel if he survives would drastically outweigh the unhappiness that he himself will experience if he misses his meeting. He acts rationally and saves Billy, which maximizes his decision-utility and the world’s total amount of experience-utility, even though it means sacrificing some of his own experience-utility.
If I wasn’t distinguishing between the two varieties of utility in these hypotheticals, I could criticize Anne for failing to maximize utility because she has made the world an unhappier place, and then I could criticize Carl for failing to maximize utility because he has made himself unhappy. If I had longwindedly described scenarios for Danielle and Eric, the irrational but otherwise identical twins of Anne and Carl who chose the options they didn’t prefer, I could criticize them both for failing to maximize utility as well. But while they would have the advantage of being brief, none of those criticisms would be very helpful or informative. Please keep your utilities straight. It’s always ugly when someone mixes up Water Works and the Electric Company.
1: Even my beloved Wikipedia fails to make this distinction. I’m not brave enough to wade in and fix that article, but more courageous readers are encouraged to do what they can.
2: Or so it is argued. There are a number of serious problems with purely hedonistic moral systems, which I may explore in greater depth in the future.
3: Humans are not rational agents by this definition, but we resemble them enough that economic models can pretend that we are and still prove useful for many purposes; economics tends to go horribly wrong when it forgets that people are not actually perfectly rational utility maximizers.
I Believe…
Feb 25th
(Loosely inspired by Common Sense Atheism and a passage from Neil Gaiman’s American Gods.)
…that everything is made out of math. Our universe (and any multiverses of which it may be a part) is a mathematical object which happens to describe us; nothing more and nothing less. (Physicist Max Tegmark explains the theory here, in a manner much clearer than I could hope to achieve.)
…that reductionism is the only sensible way to account for phenomena not traditionally classified as physics. Phrased another way, everything is physics and we’ve just mucked up our system of classification. Nothing exists that cannot be fully understood by understanding its component parts; “emergence” may be a convenient shorthand for certain kinds of complex behavior, but no new properties or information are ever actually introduced.
…that there is no real morality as it is conventionally understood. No system of ethics is written into the fabric of our universe; the most that can be said is that people are real, that they really do have preferences, and that those preferences can be described by some kind of mathematical function (though probably not one that satisfies the Von Neumann-Morgenstern axioms).
…that brains are not computers in any useful sense. Granted, they’re almost certainly Turing-equivalent and I expect it will not be many more decades before they can be simulated by computers, but it is not helpful to think of them as digital machines. They are made out of meat, and so they do meat-things. Biology has more to contribute to our understanding of the human mind than computer science does, and computer scientists would often do well to remember this.
…that there are no such things as faeries.
…that computer programs can be persons.
…that at some point in the next century or so, a superintelligent machine will be built and the human era will end. I am increasingly convinced that it will end badly.
…that there are a few parts of humanity worth attempting to save, though not many.
…that whether or not they are art, video games can be as valuable as movies or literature or music, and for many of the same reasons.
…that being human is an offense great enough to make most everyday prejudices unnecessary.
…that there’s nothing wrong with sex. Or having two boyfriends. Or two girlfriends. Or some more complex combination thereof.
…that politics isn’t worth worrying about in the vast majority of cases.
…
How To Obliterate Solipsism Before Breakfast
Dec 19th
Wikipedia tells us that:
Solipsism is the philosophical idea that only one’s own mind is sure to exist.
This is obviously dumb and wrong, but philosophers tend to be quite bad at backing up that intuition. Typically, they claim that solipsism is ridiculous or meaningless or has bad consequences (though usually with bigger words), and then dismiss it. Some agree that it is true, say what a great tragedy it is, and then carry on as if it were false for lack of anything better to do. I find this very annoying, because refuting solipsism isn’t actually very difficult if you approach it from the right angle.¹
To better argue against it, I’ll clarify Wikipedia’s definition of solipsism further and break it down into two distinct claims:
- One’s mind is sure to exist.
- Nothing else is sure to exist.
The first claim dates back to Rene Descartes’s declaration that “cogito, ergo sum;” I think, therefore I am. The cogito argument has been (justly) criticized as relying on unjustified assumptions about individual identity, but we can pare it down to a weaker (and less elegant) argument along the lines of “thinking is occurring, therefore there is a thinking thing.” This isn’t precisely equivalent to that first claim above, but we can reword it as “A mind is sure to exist” without doing too much damage to the philosophy of solipsism; it’s certainly not a clear enough victory to qualify as obliteration, so I’ll turn to the second claim.
I love strong negative claims because they make it very clear what it’ll take to refute them. In this case, if someone can demonstrate that anything other than a single mind exists, solipsism has no choice but to vanish in a puff of logic.² But before I can do that, I should probably nail down the definitions I’m using and get rid of as much ambiguity as I can manage.
So what does it mean for something to exist, then? Most definitions of existence share a common form: something exists if it is present in a specific system. For example, I could say that something exists in the everyday sense if it is present in the system of the physical universe. By this definition, dinosaur fossils exist, but the invisible dragon in my garage does not. More abstractly, I could say that the integer 5 exists in the formal system of arithmetic, but bleem, the integer between 3 and 4, does not. These kinds of existence are not interchangeable, or else I could say that the integer 5 physically exists, and that dinosaur fossils are mathematical symbols.
Unfortunately, any kind of existence that relies on such a system can’t be the existence that solipsism is concerned with, as systems describe the relationships between things. If I have a relationship between two things, then at least one of those things must be something other than the mind I already know about, and my definition would be assuming what I’m trying to prove. Circular logic is a no-no, so the definition of existence I’m looking for must be an absolute one, that allows me to say that a thing either does or does not exist regardless of its relationship to anything else.
But now I’m trapped. If I can’t use relationships to determine existence, then I can no longer say that “thinking is occurring, therefore there is a thinking thing,” because logical implication is a relationship in the system of logic! It is no longer certain that a mind exists, and so solipsism’s first claim is false.
There is no reasonable definition of existence for which solipsism is true; it is a philosophical position which is logically impossible. On the bright side, it provides a thought experiment that’s helpful for clarifying the nature of existence, which is a bad thing to be confused about when trying to answer other philosophical questions.
1: There are philosophers who have made this argument before, but for some unfathomable reason they don’t get very much attention.
2: Credit for this phrase is due to the tragically dead Douglas Adams.