Evolution: Technically Random?

[Vorticity]

OK. My head is about to explode, mostly due to the lack of mathematical understanding in this thread. And I don't mean by Tai Chi, but by almost everyone else here.

First of all, let me mention that I make my living as a mathematical physicist. Probablity and the study of stochastic processes are my bread and butter. (I don't mean by this that you should necessarily take everything I say as gospel, of course, I'm just trying to make it clear that I'm not making this up as I go along. I should also make it clear that I am not a biologist.)

Everytime I hear someone say "evolution is non-random", I wince. This has been a pet peeve of mine for some time.

So, just to get things straight from a pure math point of view: Given the standard defiitions of "random", "stochastic process", "function", it is certainly correct to say that a non-trivial function of a random variable is itself a random variable. That's basic probability theory. Let X be a random variable. Then Y = f(X) is a random variable, unless you define your function in some goofy trivial way like f(X) = 7.

Examples:

1) Let X = -1 or 1 with equal probability. Then Y = 10 + X = 9 or 11 with equal probability. Just because Y does not average to zero does not magically mean that it is not random.

2) For the mathematically inclined: Consider the stochastic differential equation (SDE)

dv/dt = v + W(t), v(0) = 10,

where W(t) is a standard white noise process. The solution will look roughly like a jagged, increasing exponential, i.e. v(t) = 10*e^t + noise. Just because v(t) has an overall trend that can be predicted does not suddenly make it a non-stochastic process.

"Random" does not mean "averages to zero" or "you can't say anything whatsoever about how it'll turn out". Yes, I know, there may be some confusion regarding the vernacular vs. the technical definition of "random", but if we are truly speaking in the mathematical realm, then T'ai is definitely correct.

And regarding BillHoyt's question about a bomb going off or not: Just because we can be virtually certain about the bomb going off, does not make the process non-random. Small random variations in the initial behavior of the explosion will induce various random variations in the subsequent shape of the mushroom cloud, the angular distribution of the bomb's energy, etc. That's what T'ai means when he says "But there certainly are various bomb outputs that one cannot predict with certaintly..."

Also, regarding evolution, someone suggested that the randomness is inherent only at the "micro" level, i.e. the small time or individual level, whereas it gets washed out by the time we get to the "macro" level, i.e. the species level. This does not necessarily follow. It is easy to think of many situations where micro randomness does indeed wash out by the time we get to the macro level, leaving us with what is, for all intents and purposes, a non-random system. Somebody mentioned bridge design: You don't have to take into account quark randomness in designing bridges. That's a good example. However, there are many, many situations where the macroscopic behavior is a very strong function of micro randomness. For example (and this is at least somewhat apropos to the current discussion): Among other things, I've been studying the propogation of diseases in human poulations by computer simulation. For these kinds of systems, the way the disease spreads, and the final numbers of the infected, depend very strongly on the (largely random) actions of individual humans ("agents", we call them), e.g. whether one of the first people to be infected decides to take that vacation to Europe or not. Small variations can make the difference between a global pandemic and a mild local health threat. For a nice reference on some aspects of this (with numerical results), see: "Forecast and control of epidemics in a globalized world", PNAS, vol 101 no. 42. (Not my paper, but a good one.)

Evolution may or may not behave qualitatively as a spreading disease. However, one cannot blithely assert that micro-randomness MUST wash out. In fact, given what I know about biology and population genetics, this seems intuitively unlikely. (Though I'm certainly no expert.)

So to sum up: I think that Tai Chi is perfectly right to say that evolution is random. It is a stochastic process, and thus inherently random. This is not a "technicality" or a "nitpick", but follows unambiguously from the mathematical definition of these terms.

 

[CFLarsen]

So to sum up: I think that Tai Chi is perfectly right to say that evolution is random. It is a stochastic process, and thus inherently random. This is not a "technicality" or a "nitpick", but follows unambiguously from the mathematical definition of these terms.

Where does natural selection come in, then?

 

[sphenisc]

OK. My head is about to explode, mostly due to the lack of mathematical understanding in this thread. And I don't mean by Tai Chi, but by almost everyone else here.
---snip---

So to sum up: I think that Tai Chi is perfectly right to say that evolution is random. It is a stochastic process, and thus inherently random. This is not a "technicality" or a "nitpick", but follows unambiguously from the mathematical definition of these terms.

Thanks, and for a final tour de force could you pick this apart for the hard of thinking among us (me).

http://mathworld.wolfram.com/StochasticProcess.html

Doob (1996) defines a stochastic process is(sic) a family of random variables {x(t,-),t in J} from some probability space (S,S,P) into a state space (S^',S^'). Here, J is the index set of the process.

[The set notation hasn't copied across properly, but you get the gist.]

Cheers

 

[Vorticity]

Where does natural selection come in, then?

Exactly where we all know it comes in: By "weeding out" those gene mutations that cause an organsim to be less fit with regard to its environment.

Keep in mind that arguments about whether the environment, or NS itself, contain inherent random factors are entirely moot. We could stipulate that the environment and NS are completely deterministic, and that the only randomness in the system comes from random gene mutations, and evolution would still be random.

As someone pointed, you'd have the randomness inherent in WHEN a beneficial mutation occurs. In addition, you'd have the randomness inherent in WHICH beneficial mutation occurs. Remember, it is conceivable that there could be more than one mutation that solves a biological problem posed by the environment. Which solution "takes off" could be a highly random things.

Again: "Directed" (e.g. by NS) does not mean non-random. Google "Brownian Ratchet".

 

[CFLarsen]

Exactly where we all know it comes in: By "weeding out" those gene mutations that cause an organsim to be less fit with regard to its environment.

Keep in mind that arguments about whether the environment, or NS itself, contain inherent random factors are entirely moot. We could stipulate that the environment and NS are completely deterministic, and that the only randomness in the system comes from random gene mutations, and evolution would still be random.

As someone pointed, you'd have the randomness inherent in WHEN a beneficial mutation occurs. In addition, you'd have the randomness inherent in WHICH beneficial mutation occurs. Remember, it is conceivable that there could be more than one mutation that solves a biological problem posed by the environment. Which solution "takes off" could be a highly random things.

But the "weeding out" isn't random.

Again: "Directed" (e.g. by NS) does not mean non-random. Google "Brownian Ratchet".

Nope. "Go Google" is not a satisfactory answer here.

 

[sphenisc]

Nope. "Go Google" is not a satisfactory answer here.

Which is why Vorticity included the rest of the post!

 

[Vorticity]

Thanks, and for a final tour de force could you pick this apart for the hard of thinking among us (me).

http://mathworld.wolfram.com/StochasticProcess.html


[The set notation hasn't copied across properly, but you get the gist.]

Cheers

Complicated.

OK. x(t) is the stochastic process, so x(t) is a random variable for every allowable time t. The set of allowable times is J. t could be discrete, e.g. J={0,1,2,..}, or t could be continuous, e.g. J = set of all non-negative real numbers.

Now it gets a bit more difficult. When Doob says "probability space (S,s,P)", that means that S is a set of elements (maybe continous, maybe discrete), s is a set of subsets of S, and P is a probability measure function that maps from s to the real interval [0,1]. In other words, given some subset A of S (which will satisfy A in s), you can calculate the probability that you'll "land" in A as P(A).

Example: Roll a dice. S = {1,2,3,4,5,6}. s = set of all subsets of S. P(1) = 1/6, P(EMPTY SET) = 0, P(1 OR 3) = 1/3, P(EVEN) = P(2 OR 4 OR 6) = 1/2. etc.

The "state space (S',s')" means that S' is the set of all possible values of x(t) (which in general could depend on t, I suppose, but they left that out), and s' is the set of all possible subsets of S' (although sometimes you'd only have s' be a limited set of subsets of S').

The rigorous form of probability theory is weird and hard. They tried less weird and less hard ways of doing it, but Kolmogorov (and others) showed that those versions can lead to inconsistency. So they had to get hard-core on us and do it the difficult way.

 

[Vorticity]

But the "weeding out" isn't random.

It doesn't have to be. I address that in the second two paragraphs of my post.

Nope. "Go Google" is not a satisfactory answer here.

I know. The "Brownian ratchet" thing is not critical. I just threw it in as a fun example of a "directed" process that is non-random. It doesn't matter.

A brownian ratchet is a "wiggly ratchet". It'd like to wiggle both ways, but only "up" wiggles are allowed or accepted. If memory serves, molecular brownian ratchets are the ultimate means by which we contract our muscles.

 

[BillHoyt]

This bears repeating:

T'ai is going for the equivocation fallacy. He wants to move us to accept it, though, by accepting his major premise: that random processes always lead to random results. This is a way of smearing the reputation of randomness so that it looks more like chaos. He wants to set up the image of genes in some kind of molecular pogrom that can't possibly lead to evolution.

 

[epepke]

Natural selection is a statistical filter; sometimes bad mutations aren't discarded, sometimes beneficial mutations are. The output still as elements of randomness (I'd prefer the term stochastic) in it.

Yes, that's much better. Evolution is a stochastic process.

 

[Vorticity]

This bears repeating:

T'ai is going for the equivocation fallacy. He wants to move us to accept it, though, by accepting his major premise: that random processes always lead to random results. This is a way of smearing the reputation of randomness so that it looks more like chaos. He wants to set up the image of genes in some kind of molecular pogrom that can't possibly lead to evolution.

I couldn't remember what the "equivocation fallacy" is; I had to look it up:

From: http://atheism.about.com/library/FAQs/skepticism/blfaq_fall_equivocation.htm

This fallacy is perhaps the most simple and obvious of the fallacies of ambiguity. Here, a single term is used with two or more meanings in the same argument. The basic form of this fallacy is:

a. Premise: [statement(s) using term X in sense 1]
b. Premise: [statement using term X in sense 2] AND/OR Conclusion: [statement using term X in sense 2]

The term equivocation comes from the Latin terms equi (equal) and vox (voice) - and means "with equal voice". When a term is used univocally in an argument, it always has the same meaning, but when it is used equivocally, more than one meaning is given equal voice.

Do you mean that T'ai is trying to equate the two meanings of random, i.e. the mathematical meaning and the vernacular meaning (i.e. "averages to zero" or "you can't say anything about how it'll turn out", etc.) ?

I've seen no evidence of this. It looks to me like he's been clear all along that he's using the mathematical definition. Furthermore, there seems to be an undercurrent in the criticism of Tai's arguments in this thread that seem to imply that he's some kind of creationist. I've seen no evidence of this either.

But back on track:

You mentioned chaos. Chaos and randomness are distinct in terms of their mathematical definitions. Generally, use of the word "chaos" implies that you are dealing with a deterministic system, whose behavior displays "sensitive dependance on initial conditions", along with various other bits of esoteric math stuff that we needn't get into. Were Tai to claim that evolution is like chaos, that would actually be contradictory to the argument that he is trying to make.

 

[drkitten]

It doesn't have to be. I address that in the second two paragraphs of my post.

Actually, I 'd argue that in a strict definition you use above, even the weeding out is "random." But this ends up not being especially relevant either.

For example, we can imagine playing a variation of blackjack where the players get to select their first card (deliberately), but then get dealt a second (uniformly random) card from the deck. Obviously, my strategy of "pick an ace" is a better strategy overall than a competing strategy of "pick a six" -- but the "overall" superiority is only in a statistical sense. It's still "random" (albeit in the strict sense, not in the colloquial) whether or not I win any individual hand.

Mutations are like that. A mutation that lets you make your own vitamin C (and makes you therefore immune to scurvy) is not going to be a positive mutation if you happen to live in an era of good harvests where vitamin C deficiency never comes up. A mutation that gives you an allergy to bee stings will not affect you if you happen never to be stung by a bee in your life. A mutation that lets you digest lactose as an adult is probably a positive mutation, unless it happens that you don't eat milk as an adult anyway....

The real problem is that the meaning of "random" in the sense that creationists use it is, as Hoyt points out, different than the meaning of "random" as used by probabilists in a technical sense. Using the same word merely reinforces the ambiguity. Sand may technically be "rock" in a technical sense to a crystallographer --- but not to a construction engineer trying to design a foundation for a house, and certainly not for a theologian talking about "on this rock I will build my church." Graphite and diamond are both "carbon crystals," but don't try to tell that to your fiance when you are trying to give her an engagement HB pencil.

 

[CFLarsen]

It doesn't have to be. I address that in the second two paragraphs of my post.

No, no, no.

Natural selection is not random, because that's the process of survival of the fittest.

It's not as if each member of each species flips a coin to see who gets to pass on his genes. The ones who are best suited to the environment "wins".

If the "weeding out" process were random, we would not have survival of the fittest. We would have survival of the luckiest.

But that's not evolution.

 

[BillHoyt]

I couldn't remember what the "equivocation fallacy" is; I had to look it up:

From: http://atheism.about.com/library/FAQs/skepticism/blfaq_fall_equivocation.htm

Do you mean that T'ai is trying to equate the two meanings of random, i.e. the mathematical meaning and the vernacular meaning (i.e. "averages to zero" or "you can't say anything about how it'll turn out", etc.) ?

I've seen no evidence of this. It looks to me like he's been clear all along that he's using the mathematical definition. Furthermore, there seems to be an undercurrent in the criticism of Tai's arguments in this thread that seem to imply that he's some kind of creationist. I've seen no evidence of this either.

But back on track:

You mentioned chaos. Chaos and randomness are distinct in terms of their mathematical definitions. Generally, use of the word "chaos" implies that you are dealing with a deterministic system, whose behavior displays "sensitive dependance on initial conditions", along with various other bits of esoteric math stuff that we needn't get into. Were Tai to claim that evolution is like chaos, that would actually be contradictory to the argument that he is trying to make.

Vorticity,

Here is T'ai's opening post:

(That is, NS is a function acting on things, some of which are random. Or put another way, evolution is the non-random selection of random variation and some other things)

the going by this conception, doesn't that make Evolution random (ie. unable to be predicted with certaintly beforehand), and therefore saying evolution is random is not wrong, but in fact a true and reasonable statement?

For example, say I have fair coin, with Heads or Tails which I code as 0 or 1. Then I calculate some function, I choose 2^outcome+10. The output is still random, either 11 or 12.

Take any non-trivial function f, and calculate f(something random). Won't the output f be random?

I highlighted the key section in red. A stochastic process can be predicted with certainty. In fact, the exact certainty is also predicted by any number of statistical measures: std dev, variance, CI. The processes will shape a given genome by one of three methods:

o directional selecton - introducing skew along the pdf's allele axis

o disruptive selection - creating two distinct peaks (where there'd previously been a standard gaussian) along the pdf's allele axis

o stabilizing selection - reduction of the variance about the mean

These are predictable outcomes of the combination of underlying stochastic and deterministic processes. Just as the bomb will go off.

I know T'ai hasn't yet unveiled his "gotcha", but, trust me, it was being set up. This mouse doessn't have to step up to bite the cheese before he sees the mouse trap.

 

[drkitten]

Natural selection is not random, because that's the process of survival of the fittest.

It's not as if each member of each species flips a coin to see who gets to pass on his genes. The ones who are best suited to the environment "wins".

If the "weeding out" process were random, we would not have survival of the fittest. We would have survival of the luckiest.

Um, wrong. Don't think coin flips, think die rolls. But I get to roll one of those geekly icosahedral dice used in role playing games, and you have to roll a standard cubical die. Higher value wins. Which one of us is "fittest"?

Which one of us will win this next roll?

Those are two separate questions, with two entirely separate answers.

In the long run, I'm obviously fitter. In the short run, you may get luckier than I do. Speaking more formally, I can predict that with probability 1 ("certainty"), I will eventually have an arbitrarily good win/loss record. Over any finite time, I can predict "with high confidence" that I will win more often than I will lose. But can I guarantee that I won't lose the next hand coming up? Of course not.

 

[Vorticity]

No, no, no.

Natural selection is not random, because that's the process of survival of the fittest.

It's not as if each member of each species flips a coin to see who gets to pass on his genes. The ones who are best suited to the environment "wins".

If the "weeding out" process were random, we would not have survival of the fittest. We would have survival of the luckiest.

But that's not evolution.

Claus, did you even read the words I posted in response to you above?

I didn't say that the "weeding out" is random. In fact, I specifically argued that even if the "weeding out" part of the process is not random at all, then evolution would still be "random" in the technical sense.

 

[Vorticity]

Vorticity,

Here is T'ai's opening post:

snipped...

the going by this conception, doesn't that make Evolution random (ie. unable to be predicted with certaintly beforehand), and therefore saying evolution is random is not wrong, but in fact a true and reasonable statement?

snipped...

I highlighted the key section in red. A stochastic process can be predicted with certainty. In fact, the exact certainty is also predicted by any number of statistical measures: std dev, variance, CI. The processes will shape a given genome by one of three methods:

o directional selecton - introducing skew along the pdf's allele axis

o disruptive selection - creating two distinct peaks (where there'd previously been a standard gaussian) along the pdf's allele axis

o stabilizing selection - reduction of the variance about the mean

These are predictable outcomes of the combination of underlying stochastic and deterministic processes. Just as the bomb will go off.

Ok. I see that we are using different definitions of the phrase "predicted with certainty". You seem to mean that, given the statistical traits of a stochastic process x(t), we can predict in advance the probability distribution of x for each t. This is not the standard meaning of the word "predict". For example, you cannot tell me a prior what specific value x will take at t=14.3. That value is clearly random, if x(t) is truly a non-trivial stochastic process. Your enumeration above of the various ways in which the random component of a process can effect the ultimate distribution of the values of a stochastic process in no way means that the values themselves may be predicted. They are just as random as ever.

I know T'ai hasn't yet unveiled his "gotcha", but, trust me, it was being set up. This mouse doessn't have to step up to bite the cheese before he sees the mouse trap.

Do you really think that T'ai is a secret creationist? I've often thought of making a very similar argument as his (though worded differently) on this board, and I'm certainly no creationist.

[self-protection]
At any rate, I only agree with Tai's current point, and not necessarily with any hypothetical future "gotcha".
[/self-protection]

 

[sphenisc]

Complicated.

OK. x(t) is the stochastic process, so x(t) is a random variable for every allowable time t. The set of allowable times is J. t could be discrete, e.g. J={0,1,2,..}, or t could be continuous, e.g. J = set of all non-negative real numbers.

Now it gets a bit more difficult. When Doob says "probability space (S,s,P)", that means that S is a set of elements (maybe continous, maybe discrete), s is a set of subsets of S, and P is a probability measure function that maps from s to the real interval [0,1]. In other words, given some subset A of S (which will satisfy A in s), you can calculate the probability that you'll "land" in A as P(A).

Example: Roll a dice. S = {1,2,3,4,5,6}. s = set of all subsets of S. P(1) = 1/6, P(EMPTY SET) = 0, P(1 OR 3) = 1/3, P(EVEN) = P(2 OR 4 OR 6) = 1/2. etc.

The "state space (S',s')" means that S' is the set of all possible values of x(t) (which in general could depend on t, I suppose, but they left that out), and s' is the set of all possible subsets of S' (although sometimes you'd only have s' be a limited set of subsets of S').

The rigorous form of probability theory is weird and hard. They tried less weird and less hard ways of doing it, but Kolmogorov (and others) showed that those versions can lead to inconsistency. So they had to get hard-core on us and do it the difficult way.

Okaaay...
So let's see if I can relate this to a simple evolutionary example.

x(t) is a stochastic process such as a single nucleotide mutation in the germline (A single nucleotide A,C,G or may mutate (or not) to an other one) . t is generation time from the set J of generations under consideration. S is the set of elements {A,C,T,G} the nucleotides. s is the set of subsets of S={{},{A},{C},{T},{G},{AC},{AT},{AG}......{ACTG}}
P(A),the presence of an A at the locus, is however dependent on whether an A was previously present or not i.e. P(A|A(t-1)), which looks like a Markov process, but I guess that's what you meant by "(which in general could depend on t, I suppose, but they left that out)"?

P(A|A(t-1))= 1- (slightly less than 1)
P(C|A(t-1))= 0+

In this case the S' (equivalent to the domain of a function?) and s' are equal to S and s, I think.

If we extend this to include development D(x(t),E), where E is the 'environment' of the locus, both genetic and otherwise then we have stochastic process producing an organism. Natural selection,N(), acts on the organism in the context of its environment which influences its probability of survival.
N(D(x(t),E), E')=p

As x(t) is a stochastic process then all processes dependent on x(t) are also stochastic processes.

Have I got the idea?

 

[BillHoyt]

Ok. I see that we are using different definitions of the phrase "predicted with certainty". You seem to mean that, given the statistical traits of a stochastic process x(t), we can predict in advance the probability distribution of x for each t. This is not the standard meaning of the word "predict". For example, you cannot tell me a prior what specific value x will take at t=14.3. That value is clearly random, if x(t) is truly a non-trivial stochastic process. Your enumeration above of the various ways in which the random component of a process can effect the ultimate distribution of the values of a stochastic process in no way means that the values themselves may be predicted. They are just as random as ever.

Excuse me, but I have not argued against "random" in the correct (mathematical) usage. I'm arguing against Justin-speak, anticipating his usual nonsense.

Do you really think that T'ai is a secret creationist? I've often thought of making a very similar argument as his (though worded differently) on this board, and I'm certainly no creationist.

I cannot figure what bugs are up Justin's rump. Not being a proctologist, I don't wish to investigate. He has written both aways about science and both ways about evolution. I expect Justin to be Justin.

[self-protection]
At any rate, I only agree with Tai's current point, and not necessarily with any hypothetical future "gotcha".
[/self-protection]

I understand that. I am trying to put boundary ropes around the definition of "random" before Justin goes into Justin mode. The bomb goes off. That is predictable. The population genome will shift. That is predictable. I'd recommend you contrast your comments about specific values at specific times with Justin's non-commital about either the overall effect (boom) or the genome shift. When I posed the bomb example, Justin demurred. Justin failed to respond to Pixy Misa, who requested elaboration. Justin ignored Pixy completely. One has to wonder why the points being made seem to elude Justin.

 

[CFLarsen]

Um, wrong. Don't think coin flips, think die rolls. But I get to roll one of those geekly icosahedral dice used in role playing games, and you have to roll a standard cubical die. Higher value wins. Which one of us is "fittest"?

Which one of us will win this next roll?

Those are two separate questions, with two entirely separate answers.

In the long run, I'm obviously fitter. In the short run, you may get luckier than I do. Speaking more formally, I can predict that with probability 1 ("certainty"), I will eventually have an arbitrarily good win/loss record. Over any finite time, I can predict "with high confidence" that I will win more often than I will lose. But can I guarantee that I won't lose the next hand coming up? Of course not.

So you can predict what species will win?

Claus, did you even read the words I posted in response to you above?

I didn't say that the "weeding out" is random. In fact, I specifically argued that even if the "weeding out" part of the process is not random at all, then evolution would still be "random" in the technical sense.

This is what you answered to my statement: "But the "weeding out" isn't random":

It doesn't have to be.

That must mean that you think at least sometimes "weeding out" is random. If that's not what you meant, fine.