Tag Archives: Bayesian

Ignoring Small Chances

On September 9, 1713, so the story goes, Nicholas Bernoulli proposed the following problem in the theory of games of chance, after 1768 known as the St Petersburg paradox …:

Peter tosses a coin and continues to do so until it should land heads when it comes to the ground. He agrees to give Paul one ducat if he gets heads on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled.

Nicholas Bernoulli … suggested that more than five tosses of heads are [seen as] morally impossible [and so ignored]. This proposition is experimentally tested through the elicitation of subjects‘ willingness-to-pay for various truncated versions of the Petersburg gamble that differ in the maximum payoff. … All gambles that involved probability levels smaller than 1/16 and maximum payoffs greater than 16 Euro elicited the same distribution of valuations. … The payoffs were as described …. but in Euros rather than in ducats. … The more senior students seemed to have a higher willingness-to-pay. … Offers increase significantly with income. (more)

This isn’t plausibly explained by risk aversion, nor by a general neglect of possibilities with a <5% chance. I suspect this is more about analysis complexity, i.e., about limiting the number of possibilities we’ll consider at any one time. I also suspect this bodes ill for existential risk mitigation.

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My Surprises

I am surprised that:

  1. I exist at all; the vast majority of possible things do not exist.
  2. I am alive; the vast majority of real things are dead.
  3. I have a brain; the vast majority of living things have none.
  4. I am a mammal; the vast majority of brains aren’t.
  5. I am a human; the vast majority of mammals aren’t.
  6. I am richer than the vast majority who have ever lived.
  7. I am alive earlier than the vast majority of human-like folks who will ever live.
  8. I am richer and smarter than most humans alive today.
  9. I write a popular blog, and unusually interesting articles.

Now how bothered should I be by these surprises? The bigger is some particular surprise, the more eager I should be to seek alternative theories, under which that surprise would be smaller.  But what alternative accounts could weaken these surprises?

One hypothesis that does the trick is the simulation argument – the idea that I’m really part of a simulation created in the distant future to explore their past world.  It lessens the surprise of #5-9, and maybe also #2-4 as well. Does this mean I should take the simulation argument a bit more seriously than I otherwise would?

Added 9a:  I find anything unusually interesting to be “surprising.” Sometimes that is of course just an accident, but the more surprising something is, the more one should seek alternate explanations.  If you can’t find them, you’ll just have to go back to considering them an accident.

Yes the fact that I am cognitively able to actually be surprised predicts other things, and given that fact those other things are no longer surprising.  But the fact that I am able to be surprised is itself surprising!

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What Is “Belief”?

Richard Chappell has a couple of recent posts on the rationality of disagreement. As this fave topic of mine appears rarely in the blogsphere, let me not miss this opportunity to discuss it.

In response to the essential question “why exactly should I believe I am right and you are wrong,” Richard at least sometimes endorses the answer “I’m just lucky.” This puzzled me; on what basis could you conclude it is you and not the other person who has made a key mistake? But talking privately with Richard, I now understand that he focuses on what he calls “fundamental” disagreement, where all parties are confident they share the same info and have made no analysis mistakes.

In contrast, my focus is on cases where parties assume they would agree if they shared the same info and analysis steps.  These are just very different issues, I think.  Unfortunately, they appear to be more related than they are, because of a key ambiguity in what we mean by “belief.”  Many common versions of this concept do not “carve nature at the relevant joints.”  Let me explain.

Every decision we make is influenced by a mess of tangled influences that can defy easy classification. But one important distinction, I think, is between (A) influences that come most directly from inside of us, i.e., from who we are, and (B) influences that come most directly from outside of us. (Yes, of course, indirectly each influence can come from everywhere.) Among outside influences, we can also usefully distinguish between (B1) influences which we intend to track the particular outside things that we are reasoning about, from (B2) influences that come from rather unrelated sources.

For example, our attitude toward rain soon might be influenced by (A) our dark personality, that makes us expect dark things, and from (B1) seeing dark clouds, which is closely connected to the processes that make rain.  Our attitude toward rain might also be influenced by (B2) broad social pressures to make weather forecasts match the emotional mood of our associates, even when this has little relation to if there will be rain.

Differing attitudes between people on rain soon is mainly problematic regarding (B1) aspects of our mental attitudes which we intend to have track that rain. Yes of course if we are different inside, and are ok with remaining different in such ways, then it is ok for our decisions to be influenced by such differences. But such divergence is not so ok regarding the aspects of our minds that we intend to track things outside our minds.

Imagine that two minds intend for certain aspects of their mental states to track the same outside object, but then they find consistent or predictable differences between their designated mental aspects. In this case these two minds may suspect that their intentions have failed. That is, their disagreement may be evidence suggesting that for at least one of them other influences have contaminated mental aspects that person had intended would just track that outside object.

This is to me the interesting question in rationality of disagreement; how do we best help our minds to track the world outside us in the face of apparent disagreements? This is just a very different question from what sort of internal mental differences we are comfortable with having and acknowledging.

Unfortunately most discussion about “beliefs” and “opinions” are ambiguous regarding whether those who hold such things intend for them to just be mental aspects that track outside objects, or whether such things are intended to also reflect and express key internal differences. Do you want your “belief” in rain to just track the chance it will rain, or do you also want it to reflect your optimism toward life, your social independence, etc.?  Until one makes more clear what mental aspects exactly are referred to by the word “belief”, it seem very hard to answer such questions.

This ambiguity also clouds our standard formal theories. Let me explain.  In standard expected-utility decision theory, the two big influences on actions are probabilities and utilities, with probabilities coming from a min-info “prior” plus context-dependent info. Most econ models of decision making assume that all decision makers use expected utility and have the same prior. For example, agents might start with the same prior, get differing info about rain, take actions based on their differing info and values, and then change their beliefs about rain after seeing the actions of others. In such models, info and thus probability is (B1) what comes from outside agents to influence their decisions, while utility (A) comes from inside. Each probability is designed to be influenced only by the thing it is “about,” minimizing influence from (A) internal mental features or (B2) unrelated outside sources.

In philosophy, however, it is common to talk about the possibility that different people have differing priors. Also, for every set of consistent decisions one could make, there are an infinite number of different pairs of probabilities and utilities that produce those decisions. So one can actually model any situation with several expected-utility folks making decisions as either one with common priors or with uncommon priors.

Thus in contrast to the practice of most economists, philosophers’ use of “belief” (and “probability” and “prior”) confuses or mixes (A) internal and (B) external sources of our mental states. Because of this, it seems pointless for me to argue with philosophers about whether rational priors are common, or whether one can reasonably have differing “beliefs” given the same info and no analysis mistakes. We would do better to negotiate clearer language to talk about the parts of our mental states that we intend to track what our decisions are about.

Since I’m an economist, I’m comfortable with the usual econ habit of using “probability” to denote such outside influences intended to track the objects of our reasoning.  (Such usage basically defines priors to common.) But I’m willing to cede words like “probability”, “belief” or “opinion” to other purposes, if other important connotations need to be considered.

However, somewhere in our lexicon for discussing mental states we need words to refer to something like what econ models usually mean by “probabilities”, i.e., aspects of our mental states that we intend to track the objects of our reasoning, and to be minimally influenced by other aspects of our mental states.

(Of course all this can be applied to “beliefs” about our own minds, if we consider influences coming from our minds as if it were something outside.)

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How Hopeless A PhD?

Imagine that you have some estimate in your mind of the odds of becoming a professor, given that you start a Ph.D. program. Now imagine you see an article titled “The disposable academic: Why doing a PhD is often a waste of time.” How much do you expect that to change your estimate? Yeah, it should lower your estimate a bit.

Now consider actually reading the article. How much on average do you expect your estimate to change then? If your belief changes are rational, you should never expect your estimates to change – they might go up, might go down, but on average stay the same. OK, but do you so expect in this case?

Here is the article; test yourself. Quotes below the fold. Continue reading "How Hopeless A PhD?" »

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New Hard Steps Results

If planets like ours are common but intelligent life like ours is rare, then it should be rare that life on a planet evolves to our level of development before life is no longer possible on that planet.  If Earth was “lucky” in this way, and if life had to go through a series of stages of varying difficulty to reach our level, how long should each stage have taken?

Now these stages could be of quite different difficulties, taking quite different unconditional expected times to complete.  But back in ’98 I noticed (and posted) an interesting non-intuitive result: if each stage is “exponential,” with a constant per time chance c to jump to the next level, then all “hard step” durations are similarly distributed, no matter what their relative difficulty.  (Joint step times are drawn from a uniform distribution.)  So we should see a history of roughly equally spaced hard step transition events in Earth’s history.

Prof. David J. Aldous, of U.C. Berkeley Dept. of Statistics, has just posted some generalizations of this result. While my result generalizes trivially to any per time success chance function C(t) that is nearly a constant C(t) = c near t=0, Aldous also generalized my similarly-distributed result to any function that is nearly linear C(t) = c*t near t=0.  He also generalized my result to any arbitrary tree of possible paths.  Each link in the tree can have arbitrarily varying difficulty, at each node in the tree many processes compete to be the first to succeed, and the one that wins this contest determines the system’s direction in the tree.

While Aldous warns us against over-reliance on simple models, this does I think gives a bit more reason to expect our history to consist of a sequence of roughly equally spaced hard step transitions.

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Real Rationality

Bayesian probability is a great model of rationality that gets lots of important things right, but there are two ways in which its simple version, the one that comes most easily to mind, is extremely misleading.

One way is that it is too easy to assume that all our thoughts are conscious – in fact we are aware of only a tiny fraction of what goes on in our minds, perhaps only one part in a thousand. We have to deal with not only “running on error-prone hardware”, but worse, relying on purposely misleading inputs. Our subconscious often makes coordinated efforts to mislead us on particular topics.

But at least many folks are aware of and try to deal with this; for example, I’ve seen a lot of good related posts on this at Less Wrong lately. There is, however an even bigger way in which the simple Bayesian model is extremely misleading, and I’ve seen no discussion of it at Less Wrong. We may see one part in a thousand of our minds, but that fraction pales by comparison to the fact that we are each only one part in seven billion of living humanity.

Taking this fact seriously requires even bigger changes to how we think about rationality. OK, we don’t need to consider it for topics that only we can influence. But for most interesting important topics, it matters far more what the entire world does than what we personally do. For such topics, rationality consists mainly in the world having and using good systems (academia, news media, wikipedia, prediction markets, etc.) for generating and distributing reliable beliefs on which everyone can act.

When seven billion minds are involved, the overwhelming consideration must be managing a division of labor, so that we don’t each have to redo the same work. Together we must manage systems for deciding who should be heard on what. Given such systems, each of us will make our strongest contributions, by far, by fitting into these systems.

So to promote rationality on interesting important topics, your overwhelming consideration simply must be: on what topics will the world’s systems for deciding who to hear on what listen substantially to you?   Your efforts to ponder and make progress will be largely wasted if you focus on topics where none of the world’s “who to hear on what” systems rate you as someone worth hearing.  You must not only find something worth saying, but also something that will be heard.

Yes, existing who-to-hear systems are far from perfect, but that fact simply does not make it rational for you to work on topics where a better system would approve you, if only such systems existed. Wishes are not horses. It might make sense for you to work on reforming our systems, but even then your best efforts will work through channels where current systems can rate you as a person to hear on that meta topic.

When what matters is how the world acts, not how you act, rationality on your part consists mainly in improving the rationality of the world’s beliefs, as determined by its main systems for deciding who to believe about what.  Just wishing we had other systems, or acting as if we had them, is delusion, not rationality.

From a conversation with Steve Rayhawk.

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Why Neglect Big Topics

From an ’04 review of Robert Triver’s ’02 book Natural Selection and Social Theory:

[Trivers’] directions on writing a classic paper …:

1. Pick an important topic.
2. Try to do a little sustained thinking on the topic, always keeping close to the task at hand.
3. Generalize outward from your chosen topic.
4. Write in the language of your discipline but, of course, try to do so simply and clearly.
5. If at all possible, reorganize existing evidence around your theory.

Those hoping this advice would get them on the fast lane to their own version of parent-offspring conflict theory or a new and groundbreaking take on reciprocal altruism may find themselves disappointed.  Most of these instructions fall into the category of easier said than done, but as Trivers also notes, “it still seems remarkable to me how often people bypass what are more important subjects to work on less important ones.”

Neglect of important subjects is remarkable if we assume academics mainly seek intellectual progress.  But it makes a lot more sense if we realize academics are not Bayesian:  Academics change their beliefs only when a sufficiently impressive work appears to earn that respect, even if that work provides little info.  And the more apparently important is the topic, the more impressive a work must be to change beliefs.

So a paper suggesting academics change their opinion on a very important subject will be held to a higher standard of impressiveness.  It must use more impressive math, data, analysis, or have a more prestigious author.  “Extraordinary claims require extraordinary evidence” is true, but not so much for evidential reasons.  Academics whose contributions might be informative but cannot rise to this higher impressiveness standard are well advised to stay away from important topics.

These academic blindsides in principle offer an opportunity for bloggers to contribute to intellectual progress via thoughtful posts that add info but are not impressive enough by academic standards, and via drawing reasonable conclusions from these and other unimpressive sources to which academics refuse to listen.  But if blogger customers will not actually pay much for such progress, it is not clear bloggers will bother.

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Contrarian Excuses

On average, contrarian views are less accurate than standard views.  Honest contrarians should admit this, that neutral outsiders should assign most contrarian views a lower probability than standard views, though perhaps a high enough probability to warrant further investigation.  Honest contrarians who expect reasonable outsiders to give their contrarian view more than normal credence should point to strong outside indicators that correlate enough with contrarians tending more to be right.

Most contrarians, however, prefer less honest positions, like:

  1. “They Laughed At Galileo Too”  Many contrarians seem content merely to point out that contrarian views have sometimes turned out to be right.  Have they no higher aspirations?
  2. “Standard Experts Are Biased” Yes of course we can identify many biases that plausibly afflict standard experts.  But we can also see at least as many biases that plausibly afflict contrarians.  No fair assuming you are less biased just because you feel that way.
  3. “We’ve More Detail Than Critics” Some contrarians say that only explicitly offered detailed arguments and analysis should count; it shouldn’t matter who agrees or disagrees.  And since advocates usually offer more detail in support of their specific arguments than critics offer in response, they automatically win.  They may not have written up their arguments in a standard or accessible style, or published them in standard places, or even submitted them for publication.  But by their “how much stuff we’ve written/done” standard, they win.
  4. “Few Who Study Us Disagree” Some contrarians accept that who agrees or disagrees matters, but say only those who have reviewed most available detail should count.  Since critics have less patience than advocates for studying advocate detail, advocates win.  If many critics do read and reject them, advocates can just add more detail and then complain critics haven’t read that.  If critics do read more advocates can complain critics aren’t of the right sort, e.g., not enough math, sociology, or whatever.  There is usually some way to define “valid” critics so they are outnumbered by advocates.

If you want outsiders to believe you, then you don’t get to choose their rationality standard.  The question is what should rational outsiders believe, given the evidence available to them, and their limited attention.  Ask yourself carefully:  if most contrarians are wrong, why should they believe your cause is different?

(Inspired by this recent argument with Eliezer Yudkowsky.)

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Why Academics Aren’t Bayesian

Bryan Caplan asks Why Aren’t Academic Economists Bayesians?:

Almost all economic models assume that human beings are Bayesians, … [but] academic economists are not Bayesians.  And they’re proud of it!

This is clearest for theorists.  Their epistemology is simple: Either something has been (a) proven with certainty, or (b) no one knows – and no intellectually respectable person will say more.  If no one has proven that Comparative Advantage still holds with imperfect competition, transportation costs, and indivisibilities, only an ignoramus would jump the gun and recommend free trade in a world with these characteristics. …

Empirical economists’ deviation from Bayesianism is more subtle.  Their epistemology is rooted in classical statistics.  The respectable researcher comes to the data an agnostic, and leaves believing “whatever the data say.”  When there’s no data that meets their standards, they mimic the theorists’ snobby agnosticism.  If you mention “common sense,” they’ll scoff.

I’ve argued that the main social function of academia is to let students, patrons, readers, etc. affiliate with credentialed-as-impressive minds.  If so, academic beliefs are secondary – the important thing is to clearly show respect to those who make impressive displays like theorems or difficult data analysis. And the obvious way for academics to use their beliefs to show respect for impressive folks is to have academic beliefs track the most impressive recent academic work.

So it won’t do to have beliefs bounce around with every little common sense thing anyone says, however informative those may be.  That would give too much respect to not-very-impressive sources of common sense. Instead, beliefs must stay fixed until an impressive enough theorem or data analysis comes along where beliefs should change out of respect for it.  Academics also avoid keeping beliefs pretty much the same when each new study hardly adds much evidence – that wouldn’t offer enough respect to the new display.

Relative to the Bayesians that academic economic theorists typically assume populate the world, real academics over-react or under-react to evidence, as needed to show respect for impressive academic displays. This helps assure the customers of academia that by affiliating with the most respected academics, they are affiliating with very impressive minds.

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Simple Forecasts Best

A fascinating tale, from the book Dance With Chance:

During the 1970s … It bugged the professor greatly that [business] practitioners were making these predictions without recourse to the latest, most theoretically sophisticated methods developed by statisticians like himself. Instead, they preferred simpler techniques which – they said – allowed them to explain their forecasts more easily to senior management. The outraged author … embarked on a research project that would demonstrate the superiority of the latest statistical techniques. …

The professor and his research assistant collected [111] sets of economic and business data over time from a wide range of economic and business sources. … Each series was split into two parts: earlier data and later data. The researchers pretended that the later part hadn’t happened yet and proceeded to fit various statistical techniques, both simple and statistically sophisticated, to the earlier data. Treating this earlier data as “the past,” they then used each of the techniques to predict “the future,” whereupon they sat back and started to compare their “predictions” with what had actually happened.  Horror of horrors, the practitioners’ simple, boss-pleasing techniques turned out to be more accurate than the statisticians’ clever, statistically sophisticated methods. … One of the simplest methods, known as “single exponential smoothing,” in fact appeared to be one of the most accurate. … Continue reading "Simple Forecasts Best" »

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