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Clever as usual Nick.

Re: most genetic mutations being deleterious (i.e. fitness-reducing), yes, that is the case, both empirically and theoretically, although the precise probability that a mutation is beneficial depends on its effect size. Mutations of small effect are more likely to be beneficial. The theoretical argument, due to R. A. Fisher (The Genetical Theory of Natural Selection, 1930), is a geometric argument, and is very clever and elegant. Imagine that the phenotype z of an individual can be described as a point in a 2-D space (the argument generalizes to any number of dimensions). Now imagine that there's some optimal (fitness-maximizing) phenotype theta, which is some distance d away from the individual's phenotype. The smaller d is, the closer your phenotype is to the optimum, and so the higher your fitness. A mutation changes the phenotype, moving it some distance r in the 2-D phenotype space, in a random direction (a random direction because whether or not a mutation occurs has nothing to do with its fitness effect). You can think of r as the effect size of the mutation; a mutation of large effect is one producing a big phenotypic change (large r). A beneficial mutation is one that moves you from z to some point closer to the optimum phenotype, a deleterious mutation is one that moves you further away. It can easily be shown that, the smaller r is (i.e. the smaller the mutation's effect on phenotype, and thus on fitness), the more likely it is to be beneficial. In the limit of very small r, the probability that a mutation is beneficial approaches a maximum of 50% for a 2-D phenotype. Conversely, mutations of large effect are very unlikely to be beneficial, because even if they move the phenotype in the direction of the optimum, they can overshoot the optimum and so can leave the phenotype even further from the optimum than it was before. Small mutations eliminate the possibility of overshoots, which is why they're more likely to be beneficial. So yes, the majority of mutations will be deleterious, but that's especially so for mutations of large effect (and especially if your phenotype is already near-optimal). Rees Kassen at Ottawa is among those who've done some very clever experiments to confirm this and other more detailed predictions that can be derived from Fisher's model. It really works!

Here's a picture to go with the words (sorry, long URL):

Thanks for (unintentionally) giving me the excuse to bore your readers with one of my favorite results from evolutionary theory. Never ceases to amaze me that Fisher, writing in 1930 and so knowing hardly anything about the physical basis of genetics and development, could derive some quite exact and extremely general results about the genetic basis of evolution. Indeed, I wonder if knowing everything we know about molecular and developmental biology just would've gotten in the way of seeing the essence of the problem.

I seem to remember reading or hearing (probably on Quirks and Quarks or Discovery Channel) that our brains are wired with the assumption that events in the world are correlated. In general it makes us more likely to pay attention to the bad news because we think it could happen to us. And it sometimes leads us to be too willing to believe stuff that just isn't true - like the gambler's fallacy.

Yet most news about non-local weather is bad news: floods, drought, wildfires, hurricanes, tornadoes, mudslides, snowstorms, you name it.

The reason is that unexpected behavior in a complex, human designed system, be it a car, the economy or society itself, is almost certainly bad news, because of entropy which returns us to a less organised state of nature - which to the human mind is destruction.

In short, our notion of "good" is skewed and improbable to begin with, and there's a "hostile" universe out there that operates under different rules, intent on killing us. Guess why most forms of religion are so detached from reality?

While distribution might be part of it, I think a much larger driver is the media's need to make the news personal in order to resonate with viewers.

Malaria stories are a pretty good example of this. There's good news and bad news about malaria. Death rates have dropped 26%. But that's just a statistic, 2 seconds of airtime for a anchor to read off at the top of the Medical News segment. If this were a treatment, we'd have a survivor to interview at least, someone who benefitted from a fancy new drug. But most of this drop is from prevention in the form of mosquito control and net distribution. Try to get a story about one of the 230,000 people who didn't get malaria last year because of these programs, and what do you have? A story about the ordinary life of an ordinary person. There may be a story to tell, but it's nothing to do with malaria. A much more engaging story is framed around one of the 655,000 families who've lost someone to malaria, highlighting that this is still a large problem that is far from being solved.

The same goes for crime. There's good news and bad news. Crime rates have been falling since the mid-'90s...over 15 years. But an interview with someone who wasn't mugged? That's not a story. Someone who was? Now there's something worth broadcasting.

Not to disagree particularly, but I think that there are other aspects, a couple of which I want to bring up. You know the saying that no news is good news. OC, that is ambiguous, but one meaning is that when things go pretty much as expected, that is good, but it is not news. This means that the mean of news is not zero. Given that news is a deviation from expectation, all news has a negative aspect, even good news. This is more clearly seen in medieval and ancient cultures. In our modern culture we embrace change, but in premodern times all change was considered bad, even if it had good aspects. While modern culture is different, psychologists are aware that even good events in a person's life can cause stress. Nearly all news has a negative aspect.

Also, I do think that there is a bias in reporting news, because people want to hear bad news. The misfortune of others gives us comfort. This is not schadenfreude, but more akin to relief, like when we say that we are lucky because we almost got run over by a car. ;) Like the cajun said about why they throw a party after a funeral: "We're sorry he's dead, but we're glad we're not."

Nick - one thing we were talking about was the nature of complex systems. It would take a fundamental re-design of the road, engine, tires, social norms about stopping for traffic lights etc for that car to get you to Maniwaki at an average speed of 200 km/hr. To make a complex system work significantly better, usually quite a number of different elements of that system have to be upgraded. To make a complex system work significantly worse, that is, for you not to make it to Maniwaki at all, all it takes is for one element to go wrong - road wash-out, tranny failure, some idiot driver.

In your post you've formulated this idea mathematically, but I don't find that mathematical formulation as intuitive.

And, yes, it's also, as others have noted, that our brains have evolved to be more keenly attuned to "leopard" than "nice sunny day".

b.t.w., any suggestions on good and bad news stories (other than the Euro) would be very welcome.

Another example is your personal health. Almost every day is the same, you're fine. And then you get cancer - that's news.

Cancer is unfortunately much more likely than growing wings or a second brain.

I would note too that there are large minorities of ideologically brittle individuals who are committed to the idea that the world is a horrible place. If there is any news, however unreliable, that the world is awful, they are triumphant about it, and if there is good news they are indignant and claim a cover-up.

I hear this reaction all the time if there is a news item about a decrease in crime, domestic violence, poverty, or good economic news. I witnessed it in spectacular fashion last month at a conference where someone presented a paper saying the prevalence of child sexual abuse was overestimated. Unbelievably, some people are outraged at the notion that things can get better, or that our past gloom may have been in error.

Min and Frances:

Suppose (OK, no need to suppose?) I were a small c conservative, who didn't like any change in any direction. Then a day when there was no change would be good news. If change were a mean zero normal distribution, then changes to my utility would have a skewed distribution. Lots of days when all changes are small, so I am slightly happier than I expect to be, but it's less than 1 SD, so it's not worth reporting. Ad a small number of days when there's a big change, and my utility is more than 1 SD lower than i expected, so it gets reported.

Now, any designed system, like a new car, is very much like a small c conservative. Since the designers and builders (presumably) optimised all the settings originally, any change is a bad thing. (If a change made the car go faster, or use less gas, or corner better, the designers would have built it that way to begin with.) So, if all changes to the car are normally distributed, the probability distribution of my utility will be skewed.

My math model was trying to formalise some complex designed system, like a car. H* is the original manufacturer's setting of the car's mechanicals. V(t) is all the things that wear and tear and nature does to the car's mechanicals. Even if V(t) is normally distributed, W(H*+V(t)) will have a skewed distribution.

Interesting post. This reminds me a bit of the asymmetry in business cycles. Crashes in output tend to be shorter and steeper than booms in output. That reminded me if the fact that although you sometimes see houses accidentally burn down in 30 minutes, you never see houses accidentally get built in 30 minutes. (Although the laws of quantum mechanics don't preclude such a possibility.)

That made me wonder whether the asymmetry in business cycles has anything to do with the asymmetry in houses burning down. Thus if GM decides to hire another 1000 workers for a plant in Ontario, it might take a year to build the plant. But if they decide to lay off 1000 workers, they can do so almost immediately.

In the US we don't have mini-recessions. It's all or nothing. Either unemployment rises by no more than 0.6%, or it soars by at least 1.9%. (The only exception was the steel strike of 1959, which did produce a mini-recession of 0.8% higher unemployment.)

So as soon as people recognize a "recession" is underway, they dramatically shift their plans to account for an environment where unemployment will eventually rise by 1.9% or more.

Max: yep. A human body is just like a car. The designer (either God or Darwin, take your pick) has already roughly optimised the settings of the mechanicals. So all changes are for the worse. Good news is when there's no change. But if the changes are normally distributed, and so not skewed, the Utility of those changes will have a skewed distribution.

Scott: "This reminds me a bit of the asymmetry in business cycles."

Wow! Yep! How did I miss that?! Is this Milton Friedman's "plucking model"? (which I've never really understood)?

NIck - "If W(R) is a strictly concave function, that is symmetric around its maximum value at W(H*)"

O.k., so in your model, good news (H>H*) is bad news, because the system can't handle it. Just like being too intelligent might be a curse, because society is designed for average people, and you'd spend your entire time being frustrated and thinking "this is so stupid!" Similarly being too tall is a problem (even though height and health are generally positively correlated).

I think it would be simpler just to model it as a weakest link technology. E.g. a holiday dinner is as happy and peaceful as the least happy and peaceful person at the table. To get one standard deviation above the average, everyone has to be one standard deviation above the average. To get one standard deviation below the average, only one person has to be a standard deviation below. (Actually, this seems to be to be a strong argument for having very small holiday gatherings).

All good news is bad news seems overly pessimistic.

Frances: OK, a "weakest link" technology Q=min{X1,X2,X3} would be skewed left, I think. But wouldn't a multiplicative technology Q=X1.X2.X3 be skewed right? (I may have my left and rights muddled).

I'm unhappy about the metaphysics of any statement about reality being fundamentally weakest link as opposed to multiplicative.

Being taller or more intelligent than average must be a curse, otherwise the Darwinian process would have made us all taller and more intelligent? Brains and height must be costly things. There was a big argument about this on Razib Khan's Gene Expression blog a few weeks back, but I couldn't quite follow it all.

Nick, I think a comment of mine from last night may have been eaten by the spam filter...

Jeremy: I retrieved your comment. Sorry about that. I was hoping you might show up on this post.

Your explanation was very clear. I could "see" it even without looking at the picture. Presumably it works the same in 1 D space as well? But I think there must be some sort of assumption about the symmetry of the genetic fitness function with respect to the phenotype?

Your original math seems OK: it is a corollary of Jensen's inequality (if the expectation of a convex function is >= the function of the expectation, then the expectation of a concave function is <= the function of the expectation.) But it doesn't seem to have had the intended consequence of illuminating the discussion, which is confounded on two not very compatible ideas:

1. Reality delivers news which is in expectation neutral and in variation symmetrical, but our evaluation of this news is skewed [Nick version 1.]

2. Reality delivers news which is in expectation negative because there are more ways that things can go wrong than go right [Frances, Tolstoy, Nick version2?] A great many functions of expectation will be negative when the expectation itself is negative.

Phil: "1. Reality delivers news which is in expectation neutral and in variation symmetrical, but our evaluation of this news is skewed [Nick version 1.]"

That's my story. News can't be in expectation negative, by definition (unless you have biased expectations).

I remember Jensen's Inequality. But can it be used to prove that the *skewness* of a concave function of a variable, around the optimum, is greater than the skewness of the variable itself? So that a non-skewed variable still gives a skewed function of that variable?

There is also something peculiar about the numerical car breakdown example. The expected distance figure 99.5 implies an assumption that 1. the car can only break down once per trip, and 2. the car is equally likely to break down at any point on the route. This in turn implies a declining hazard rate (if breakdown is an absorbing state, and the unconditioned probability of breakdown is uniform, then the probability of breakdown conditioned on prior survival must be declining.) But the example goes on to assume (quite plausibly) that an old car has a higher breakdown probability than a new one; i.e., the hazard rate is increasing.

In other words, Jensen's Inequality (as I understood it) is about what happens to the first moment of the distribution when you run it through a concave function. I need to know what happens to the third moment, when you run it through a concave function, around the maximum of that function.


"Your explanation was very clear."


"Presumably it works the same in 1-D space as well?"

I think so. Fisher actually derived a formula for the probability that a mutation will be beneficial, as a function of d, r, and the dimensionality of the space, but I'm too lazy to look it up.

"But I think there must be some sort of assumption about the symmetry of the genetic fitness function with respect to the phenotype?"

Yes, Fisher was assuming that fitness depends only on the distance d between the current phenotype and the optimum, not on the direction. That's why the diagram shows perfect circles, not ovals or irregular blobs or whatever. There are various reasons why you wouldn't necessarily expect perfect circles, or their higher- or lower-dimensional equivalent, but I'm not sure how important that is in practice (it's not really my field). As I said, in practice many predictions from Fisher's original model hold up empirically. More broadly, you can start talking about "rugged" fitness "landscapes" with multiple peaks (local optima) at different heights. Such landscapes create new possibilities that you can't really think about using Fisher's single-peak model. Sean Rice is one evolutionary theoretician who's worked on these more general cases, and relevant empirical data are starting to come in (again, your neighbor Rees Kassen is one person doing experimental work on this). And then of course there's density and frequency dependence (fitness of a given phenotype depends on the absolute and relative abundances of individuals with different phenotypes), which is a whole 'nother kettle of fish that you can't even really think about properly with the "fitness landscape" metaphor...

Jeremy: given Mendelian genetics, plus a large number of genes, wouldn't the distribution of r be normal? And if so, and given Fisher's model, wouldn't the distribution of genetic fitness across the population be skewed? (A long tail of people who die without grandkids?)

And I'm trying to fit that together with Gengis Khan and that Irish guy (Brian of the seven veils?) who are supposed to have great^n grandsired half of Eurasia.

Nick - knowing you and old cars I'm still debating whether that makes you car trip analogy more accurate or biased in some way.

Frances - On news stories, obviously the Japanese earthquake/tsunami/meltdown would be a major bad news story. The whole Arab spring thing would be a huge story as well, though good or bad? Maybe good so far on balance, though not unambiguous.

Then of course there was the whole Justin Bieber paternity thing....

Jim - "Explain the economic significance of the Justin Bieber paternity thing" - you know, that's a challenge, I like it.

Nick - an interesting suggestion from across the room - positive externalities tend to be internalized, negative ones aren't, so this is why bad stuff is in the public domain?

I'm starting to see the advantages of your way of modelling the problem.

"given Mendelian genetics, plus a large number of genes, wouldn't the distribution of r be normal? And if so, and given Fisher's model, wouldn't the distribution of genetic fitness across the population be skewed? (A long tail of people who die without grandkids?) And I'm trying to fit that together with Gengis Khan and that Irish guy (Brian of the seven veils?) who are supposed to have great grandsired half of Eurasia."

Now you're getting into a number of other topics, including but not limited to quantitative genetics, the form of the genotype-phenotype and phenotype-fitness maps, and the coalescent process. Responding would require me to provide an overview of a largish fraction of the entire field of population genetics in one blog comment, which would be a big ask for someone like me, even if I didn't have to mark exams. ;-)

Jeremy: no worries. Thanks.

Here's one exception that acts to confirm Nick's explanation: lottery news. You don't see headlines saying "hundreds of thousands of people lost money on the lottery yesterday"; you do see stories about the winners. Lotteries skew positive.

I'm with Jeremy on this one. It's really not about convexity but rather about optimization. The number of states of the world that we would consider to be good is minuscule (insanely minuscule) compared to the number of states that we would consider to be bad. So random changes are bad and overwhelmingly so because of the high dimensionality of the space of possible changes. We actually had this debate before in this post by Mike Moffatt. The debate then was over Frances' proposal that "unknown unknowns" were overwhelmingly likely to be bad, which I defended (vehemently). Didn't seem to get much traction though, and it seems to be the case that the precautionary principle is held in low regard among some economists. To statistical physicists (and probably also evolutionary biologists) it seems fairly self-evident though.

Actually I guess it's about convexity (and Jensen's inequality) in the sense that utility is convex in the vicinity of the optimum (or it wouldn't be an optimum). So, yes, good point Nick.

K - " We actually had this debate before" - yup, pretty much exactly one year ago, when we were trying to think of the best and worst of 2010.

But even if utility is linear in the distance from the optimum then a random change is still on average going to take you further away from the optimum even if there is no chance of going through to the other side of the optimum. Think of two points in two dimensional space one unit apart (the first is the optimum, the second is where we are at). Now take a step of 1 unit in a random direction. You will find yourself on a unit circle centered centered on the second point, most of which is outside the unit circle centered on the optimum. If you keep taking random steps you will keep moving further away on average. It's a brownian motion so on average you will be about sqrt(n) away from the optimum after n steps.

I think your point about r being normal makes sense. But so long as it's *small* (compared to the distance to the optimum) then the iso-fitness surface that you start on looks locally like a plane (instead of an n-dimensional surface) and then the odds of increasing fitness (for one step) is 50-50. Bit take a big step and you are right. We'll all die off.

K: "...it seems to be the case that the precautionary principle is held in low regard among some economists. To statistical physicists (and probably also evolutionary biologists) it seems fairly self-evident though."

Speaking as an evolutionary biologist (well, an evolutionarily-oriented ecologist), I personally wouldn't consider it self-evident. Neither would many other organisms, I suspect. There's certainly something to be said for it. But in the right circumstances, it's a good idea--essential, even--to change things up, and take risks, even risks with the potential to lead to big losses. There are conditions that select for high mutation rates, for instance. Heck, the fact that most species reproduce sexually, at least occasionally, rather than just cloning themselves is probably the single most dramatic illustration offered by nature that there's such a thing as too conservative.

I meant "*but* take a big step". Stupid iPhone!

And I meant "n-dimensional sphere" not "n-dimensional surface". Can't really blame the iPhone for that one.

Jeremy: "But in the right circumstances, it's a good idea--essential, even--to change things up, and take risks"

Well heck then, bring on the global warming. It's almost all unknown unknowns. :-)

K: "But even if utility is linear in the distance from the optimum then a random change is still on average going to take you further away from the optimum even if there is no chance of going through to the other side of the optimum."

Yes. I glossed over this in previous comments.

But seriously, sexual reproduction looks (to me) like a great way to take big steps around inside a large but known to be fairly optimal region of phase space and to ensure that useful genes are combined with a large variety of different other genes to ensure that they won't be easily eliminated, eg by being associated universally with some gene that suddenly becomes detrimental. So it looks like a good risk mitigation strategy for an organism that is faced with a quickly varying, highly complex environment. Mutation, on the other hand has a high risk of taking you radically outside the known safe region.

The precautionary principle, though, doesn't say you shouldn't try stuff that might be bad. Sometimes we have to do experiments and evolution, of course, is all about (mostly failed) experiments. It says that the unknown, unmodeled effects of your experiment are more likely to be bad than good so think about how much net benefit you are expecting and try to establish strict bounds on the possible impacts rather than relying the moments of the distributions of models that you know to capture an absurdly minuscule portion of the future phase space. And try to avoid doing things that have significant unknown impacts on all the organisms at the same time. Evolution doesn't try specific mutations on everyone all at once either.

It strikes me that evolution, in fact, practices the ultimate conservative form of the precautionary principle, attempting the necessary experiments on the minimal conceivable subject, the single organism. It would simply be impossible to design a mechanism that explores the available phase space in a manner that is more conservative than that.

Nick: The skewness coefficient is just an expectation .....E[{(X-Mean)/Sigma}^3]. Sigma (std. deviaiton) is always positive. Take Mean = 0. Then, for positive X, (X^3) is strictly convex, and Jensen's inequality can be applied to the skewness coeeficient (under the conditions stated above).

Dave: thanks. I'm really bad at math and stats. I sort of see what you are saying, I think. Which means I'm right, right? Jensen's inequality means that a concave welfare function converts an unskewed distribution of R(t) into a skewed distribution of W(H*+R(t))?

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