If you want to be picky then Mutations and Genetic Drift are not random. If you have sufficient resources and time you could identify the cause of any particular mutation or occurance of genetic drift but for the most part this isn't practical or worthwhile unless you are specifically investigating causes of mutations or instances of genetic drift. Its more convenient to simply state that they are random.

As for natural selection this is not a random process because if it was then evolution simply wouldn't happen. Natural selection describes the process whereby those individuals who are best adapted to their environment have the greatest chance of surviving and passing their genes onto the next generation. If it was random then we wouldn't expect to see the the conservation of genes and the high abundance of genes in a population that increase the fitness of individuals, but we do.

Allan Miller wrote:One discrepancy I see is in the way drift is treated, The "drift-is-systematic" view arises from the fact that a population consisting of 66% allele A and 34% allele B will tend to become enriched in allele A 66% of the time. But what is really going on is that the classification of alleles doesn't matter a damn in drift. You have 2N instances of a locus in a diploid population. With no selective distinction between them, population sampling retains some by descent and discards others.

You are generalizing about drift from a special case, namely neutrality. Considering a simple case (haploid clonal organisms), in a birth-death process with s=0 (neutrality) you of course have the probability of increasing the frequency by 1/N as 0.5 and that of decreasing the frequency by 1/N as 0.5 as well. However if there is selection, you get probabilities of - say - 0.7 and 0.3 (these depend on both the frequency and s). At this point the Expected change is 0.7/N-0.3/N=0.4/N. Drift now consists of -1.4/N with probability 0.3 and 0.6/N with probability 0.7, note that both the values the drift term takes and the probability with which it takes them explicitly depend on s, the only thing that doesn´t is the expected value (which is 0 by definition).

Allan Miller wrote:All of which boils down to multiple ways of saying that each member of an allele population has a bare 1/2N chance of being in the next offspring produced, but if you choose to classify, you can claim a "systematic" bias in favour of the most numerous class, since chances multiply. In NS, the numerator is not exactly 1 for every individual; in drift, it is.

You are using drift as a synonym for neutrality here. In a finite population with s=0 you have drift only. In an infinite population with s=!0 you have only selection. In a finite population with s=!0 you have both drift and selection. In some ways you assume that you can restrict yourself to looking at the first two, where you get drift and selection alone and get the 3rd and most general case by superposing the two. This however does not work in this case.

susu.exp wrote:

Allan Miller wrote:One discrepancy I see is in the way drift is treated, The "drift-is-systematic" view arises from the fact that a population consisting of 66% allele A and 34% allele B will tend to become enriched in allele A 66% of the time. But what is really going on is that the classification of alleles doesn't matter a damn in drift. You have 2N instances of a locus in a diploid population. With no selective distinction between them, population sampling retains some by descent and discards others.

You are generalizing about drift from a special case, namely neutrality.

No, I am defining drift wrt that special case. Drift is sampling error. There is a continuum from lethality through s=0 to essentiality. With -1<s<1, some lives are affected by selection on the allele, and some are not. So what I call drift is that random (unsystematic/equiprobable) component; the unselected lives. For everything from near-neutrality onwards, the expectation is the resultant of two processes: random drift (unsystematic sampling error) and the operation of selective factors, which is systematic (else we wouldn't even be able to give a sign for s, still less a value).

susu.exp wrote:

Allan Miller wrote:All of which boils down to multiple ways of saying that each member of an allele population has a bare 1/2N chance of being in the next offspring produced, but if you choose to classify, you can claim a "systematic" bias in favour of the most numerous class, since chances multiply. In NS, the numerator is not exactly 1 for every individual; in drift, it is.

You are using drift as a synonym for neutrality here. In a finite population with s=0 you have drift only. In an infinite population with s=!0 you have only selection. In a finite population with s=!0 you have both drift and selection. In some ways you assume that you can restrict yourself to looking at the first two, where you get drift and selection alone and get the 3rd and most general case by superposing the two. This however does not work in this case.

Nah, I'm afraid I just can't buy this infinite population crap. Populations occupy physical space; this is real biological stuff in geometric space here; the fact that this renders it mathematically troublesome is not an excuse for just sweeping all that under the carpet, 1930's style. In an infinite population with real spatial/mating constraints with s<>0 you still get a component due to drift. You can't avoid it. The fact that members of the population are NOT in direct contact with every other, and mating is NOT random, are significant facts in the progress of an allele through it. And the infinite population with s<>0 still has a set of members for whom that selective factor happened not to operate in their lives, or where it operated but didn't make a difference. The 'infinite population' paradigm simply sweeps all that away, the Law of Large Numbers being the only constraint worthy of consideration. Any value of s incorporates the frequency of real selective events. That frequency is only 100% for a new lethal allele, and one can separate drift and selection by declaring drift to be what happens when selection doesn't. The chance of an allele under drift alone is 1/2N; the chance under selection contains two factors: 1/2N and an adjustment dependent upon s. Drift is a random component with weakening significance as s increases/decreases.

Allan Miller wrote:No, I am defining drift wrt that special case. Drift is sampling error. There is a continuum from lethality through s=0 to essentiality. With -1<s<1, some lives are affected by selection on the allele, and some are not. So what I call drift is that random (unsystematic/equiprobable) component; the unselected lives. For everything from near-neutrality onwards, the expectation is the resultant of two processes: random drift (unsystematic sampling error) and the operation of selective factors, which is systematic (else we wouldn't even be able to give a sign for s, still less a value).

Try to do this explicity. I did, breaking up the overall sampling (including drift and selection) X into selection E(X) and drift (X-E(X)). Drift is defined as the component for which the expectation is 0. What you are suggesting is some other division, which I´d be interested in seeing spelled out (noting that the one given above is the standard way to break them up). To have drift with a uniform distribution over some interval, you need to twist selection into a term that´s unweildly and doesn´t seem to correspond to anything.

Allan Miller wrote:Nah, I'm afraid I just can't buy this infinite population crap. Populations occupy physical space; this is real biological stuff in geometric space here; the fact that this renders it mathematically troublesome is not an excuse for just sweeping all that under the carpet, 1930's style. In an infinite population with real spatial/mating constraints with s<>0 you still get a component due to drift. You can't avoid it.

Which is precisely why nobody ever cared about building infinite population models with real spatial/mating constraints. The whole point of an infinite population model is that you can look at selection without a drift component.

Allan Miller wrote:That frequency is only 100% for a new lethal allele, and one can separate drift and selection by declaring drift to be what happens when selection doesn't. The chance of an allele under drift alone is 1/2N; the chance under selection contains two factors: 1/2N and an adjustment dependent upon s. Drift is a random component with weakening significance as s increases/decreases.

It´s just the other way around. Selection is defined as that which happens if there´s no drift. Drift is the remaining component. The way you are diving the two up leads to your drift term containing some of what´s usually called selection and vice versa.

susu.exp wrote:

Allan Miller wrote:No, I am defining drift wrt that special case. Drift is sampling error. There is a continuum from lethality through s=0 to essentiality. With -1<s<1, some lives are affected by selection on the allele, and some are not. So what I call drift is that random (unsystematic/equiprobable) component; the unselected lives. For everything from near-neutrality onwards, the expectation is the resultant of two processes: random drift (unsystematic sampling error) and the operation of selective factors, which is systematic (else we wouldn't even be able to give a sign for s, still less a value).

Try to do this explicity. I did, breaking up the overall sampling (including drift and selection) X into selection E(X) and drift (X-E(X)). Drift is defined as the component for which the expectation is 0. What you are suggesting is some other division, which I´d be interested in seeing spelled out (noting that the one given above is the standard way to break them up). To have drift with a uniform distribution over some interval, you need to twist selection into a term that´s unweildly and doesn´t seem to correspond to anything.

It depends what is being aimed at. Evidently, the mathematical treatment has specific needs and adopts appropriate simplifications. But I'm no mathematician, and nor are most biologists. But we do understand experimental error, and these we can cheerfully separate into a random component and a systematic one, which mathematicians seem not to carp about. We also understand individual lives, and individual alleles, and the role selection plays in living systems. The digital elements that make up the mathematical terms are individual organism lives; the stuff of selection the visibility of DNA sequence to selective agents, as perceived through phenotype. There are chunks of sequence that selection cannot distinguish, rendering allele classification immaterial. Certain among such sequences are nonetheless concentrated, by drift. Then there are sequences that selection does distinguish, at least in some lives. This gives them a bit of extra 'push' in the drift-dominated lottery. Now, if that's mathematically unwieldy, then you deal with it differently - but it still corresponds to something: the separate but cumulatively coupled influence of two processes acting on populations, one involving discrimination, and one not.

susu.exp wrote:

Allan Miller wrote:Nah, I'm afraid I just can't buy this infinite population crap. Populations occupy physical space; this is real biological stuff in geometric space here; the fact that this renders it mathematically troublesome is not an excuse for just sweeping all that under the carpet, 1930's style. In an infinite population with real spatial/mating constraints with s<>0 you still get a component due to drift. You can't avoid it.

Which is precisely why nobody ever cared about building infinite population models with real spatial/mating constraints. The whole point of an infinite population model is that you can look at selection without a drift component.

True, but why draw any conclusions from it at all? It continues to be wheeled out to support proposals about real populations, but they are bogus, if N is the only parameter under variance.

susu.exp wrote:

Allan Miller wrote:That frequency is only 100% for a new lethal allele, and one can separate drift and selection by declaring drift to be what happens when selection doesn't. The chance of an allele under drift alone is 1/2N; the chance under selection contains two factors: 1/2N and an adjustment dependent upon s. Drift is a random component with weakening significance as s increases/decreases.

It´s just the other way around. Selection is defined as that which happens if there´s no drift.

You mean that? Parsing, you seem to be saying that drift only applies in the absence of selection - ie drift only occurs where s=0, which is precisely what you complained about me doing, initially.

When I said "drift [...] what happens when selection doesn't", I meant in respect to any individual life - if an allele does not contribute to its own success in that life, then it is nonetheless subject to sampling. If it does contribute to its own success, then it is selected, rather than randomly sampled.

susu.exp wrote:Drift is the remaining component. The way you are diving the two up leads to your drift term containing some of what´s usually called selection and vice versa.

If I depart from standard usage, so be it. I jib at calling sampling error "selection", however. And I would also jib at calling a process which directly prefers a particular sequence "drift". So separating them seems entirely natural and intuitive, even though they both contribute in tandem to the progress of any allele with s<>0.

Allan Miller wrote:It depends what is being aimed at. Evidently, the mathematical treatment has specific needs and adopts appropriate simplifications. But I'm no mathematician, and nor are most biologists. But we do understand experimental error, and these we can cheerfully separate into a random component and a systematic one, which mathematicians seem not to carp about.

Of course mathematicians carp about the difference between systematic and non-systematic error (these are terms from mathematics). There aren´t any simplifications in what I wrote. And finally: I´m not a mathematician. But it turns out that understanding population genetics requires some mathematics, so I did take care to pick up the relevant maths.

Allan Miller wrote:There are chunks of sequence that selection cannot distinguish, rendering allele classification immaterial. Certain among such sequences are nonetheless concentrated, by drift. Then there are sequences that selection does distinguish, at least in some lives. This gives them a bit of extra 'push' in the drift-dominated lottery. Now, if that's mathematically unwieldy, then you deal with it differently - but it still corresponds to something: the separate but cumulatively coupled influence of two processes acting on populations, one involving discrimination, and one not.

Indeed. And the standard way to define the two is precisely the one given in my last post. Drift is the component for which the expectation is 0. Selection sits at the expected value. You on the other hand have argued for selection defined as the median - the two are equivalent for s=0, they are not equivalent for s=!0. Using the median is highly problematic and most importantly not how selection has been used in biology for a century. If you want to present an argument that it´s better to think of selection differently than evolutionary biology has for a long time, you can bring it forth. It´s certainly not one you can rest on the way biologist understand selection.

Allan Miller wrote:True, but why draw any conclusions from it at all? It continues to be wheeled out to support proposals about real populations, but they are bogus, if N is the only parameter under variance.

Can you give an example? Of course there are situations under which assuming N to be infinite is a useful approximation. I agree that it has been overly used (and the whole "random debates", of which this tread is the most recent one have revolved around some people arguing in favor of such models - I´ve held the opposite position and that´s precisely why I´m arguing this way). But I don´t think I´ve encountered any argument that the assumption of an infinite population is useful if N is the only parameter under variance.

Allan Miller wrote:You mean that? Parsing, you seem to be saying that drift only applies in the absence of selection - ie drift only occurs where s=0, which is precisely what you complained about me doing, initially.

You misunderstood me. For N->infinity there is no drift as the actual change in allele frequency is the expected change in allele frequency (law of large numbers). So selection being defined as the expected change is the same as defining it as the result of resampling without sampling error.

Allan Miller wrote:If I depart from standard usage, so be it. I jib at calling sampling error "selection", however.

So do I. The simple - and standard - way is to have selection as the expected change and drift as the remainder. As long as s doesn´t explicitly depend on random variables (usually it does), the selection term is a real number. Drift contains all the higher oder momenta (variance, skew, kurtosis) and they depend on s, hence drift is not independent from fitness. However it has an expected value of 0. Defining drift in some other way means that drift has an expected value that is not 0.

Allan Miller wrote:And I would also jib at calling a process which directly prefers a particular sequence "drift".

Well, the problem here is that drift is defined as the component that basically prefers a particular sequence in the least direct way. Using the median for selection will usually lead to an expected value that is not 0 for drift, which means it very directly prefers a particular allele. Like it or not, you can´t have a drift term that is unaffected by s.

Allan Miller wrote:So separating them seems entirely natural and intuitive, even though they both contribute in tandem to the progress of any allele with s<>0.

Sure. The issue is how to seperate them. And seperating them through defining drift as the component with an expectation of 0 is the most natural way to do it (not the least because fitness is the expected number of offspring. If the mean number of offpring for members of the population matches their fitness, there is no drift).

susu.exp wrote:

Allan Miller wrote:There are chunks of sequence that selection cannot distinguish, rendering allele classification immaterial. Certain among such sequences are nonetheless concentrated, by drift. Then there are sequences that selection does distinguish, at least in some lives. This gives them a bit of extra 'push' in the drift-dominated lottery. Now, if that's mathematically unwieldy, then you deal with it differently - but it still corresponds to something: the separate but cumulatively coupled influence of two processes acting on populations, one involving discrimination, and one not.

Indeed. And the standard way to define the two is precisely the one given in my last post.

Well, I am unconvinced that we have a standard way. The Wikipedia article on drift is interesting; even more so are the talk pages. I'm not about to take your assertion that there is a consensus. (The article has been cited for some "good article" badge, FWIW ). The article contains this little tidbit:

Genetic drift versus natural selection
Although both processes drive evolution, genetic drift operates randomly while natural selection functions non-randomly. This is because natural selection emblematizes the ecological interaction of a population, whereas drift is regarded as a sampling procedure across successive generations without regard to fitness pressures imposed by the environment.

(my italics)

And on the Talk page, we have a discussion regarding Gale's identification of (at least) 2 separate processes: "random genetic drift" ("the narrow view", quoth Kimura) and "stochastic changes in selection intensity". Other authors identify as many as 7 separate processes capable of sheltering under the "stochastic" umbrella. So in my terms, Drift is sampling error alone. Defining Drift in terms of departure from "expected offspring value" (by dint of organisms' inability to produce non-integer offspring?) obviously conflates several stochastic factors into one.

susu.exp wrote:
Drift is the component for which the expectation is 0. Selection sits at the expected value. You on the other hand have argued for selection defined as the median - the two are equivalent for s=0, they are not equivalent for s=!0. Using the median is highly problematic and most importantly not how selection has been used in biology for a century. If you want to present an argument that it´s better to think of selection differently than evolutionary biology has for a long time, you can bring it forth. It´s certainly not one you can rest on the way biologist understand selection.

Oh, that one again! "If you think you know better than evolutionary biologists (for whom I speak) then get your radical new formulation published, matey-boy ...". I'm just joining in a word-game: "is there a way to define random ... etc". And it seems to me that, so long as we compartmentalise the processes appropriately, there is. That's a long way from saying "chuck away your slide-rules; I've just spotted something!".

susu.exp wrote:

Allan Miller wrote:True, but why draw any conclusions from it at all? It continues to be wheeled out to support proposals about real populations, but they are bogus, if N is the only parameter under variance.

Can you give an example? Of course there are situations under which assuming N to be infinite is a useful approximation. I agree that it has been overly used (and the whole "random debates", of which this tread is the most recent one have revolved around some people arguing in favor of such models - I´ve held the opposite position and that´s precisely why I´m arguing this way). But I don´t think I´ve encountered any argument that the assumption of an infinite population is useful if N is the only parameter under variance.

Well, it is standard textbook fare.

"The rate of fixation of selectively neutral alleles is inversely related to the population size: P(fix)=1/2N; P(loss) = 1-1/2N."

So

Genetic drift occurs because the population size is not infinite, allowing chance events (sampling error) to occur.

Drift is routinely restricted to 'small' populations (without much more than a nod to "how small is small?"). But change N and you change the 'work' the population has to do to continue to behave in the idealised manner. Smear a population out over a surface (a planet, say) and the situation changes. Introduce an internal 'motor' (eg the need to find partners) and the situation changes again. In a maximal, planet-covering population, you would still get drift and fragmentation. I know population structure has been modelled to death, but the simplistic point is routinely made, whereas I simply don't see the Law of Large Numbers overriding geometry and other restraints on gene flow.

susu.exp wrote:

Allan Miller wrote:You mean that? Parsing, you seem to be saying that drift only applies in the absence of selection - ie drift only occurs where s=0, which is precisely what you complained about me doing, initially.

You misunderstood me. For N->infinity there is no drift as the actual change in allele frequency is the expected change in allele frequency (law of large numbers). So selection being defined as the expected change is the same as defining it as the result of resampling without sampling error.

See above. You also have to assume constant s, for the LLN to track, which seems a heck of an assumption for the 'smeared-out' population across all environments, and with widely varying local mutants to deal with among its own and every other species in the ecosphere. There is something of the Hoyle-fallacy about this. When N gets large, other factors inevitably supervene upon naive assumptions based on number-crunching.

susu.exp wrote:

Allan Miller wrote:So separating them seems entirely natural and intuitive, even though they both contribute in tandem to the progress of any allele with s<>0.

Sure. The issue is how to seperate them. And seperating them through defining drift as the component with an expectation of 0 is the most natural way to do it (not the least because fitness is the expected number of offspring. If the mean number of offpring for members of the population matches their fitness, there is no drift).

Ah yes, the disconnect between theory and practice, which is why there continues to be a "neutralist-selectionist contoversy". Separating drift and selection after the fact (and distinguishing hitch-hiking, inbreeding and god-knows-what to boot) is a tough nut. As is getting a decent handle on what the "expected number of offspring" actually is. The genetic snowflake that is every sexual individual does not yield up a neat value of s by integrating the allele-fitnesses of every sequence they possess. You can get a rough figure assuming uniform background - but things in the selective milieu have moved on even as you gathered your data. I once knew a bloke who had a profound influence on the winkle population of Anglesey due to his research efforts .

Allan Miller wrote:And on the Talk page, we have a discussion regarding Gale's identification of (at least) 2 separate processes: "random genetic drift" ("the narrow view", quoth Kimura) and "stochastic changes in selection intensity". Other authors identify as many as 7 separate processes capable of sheltering under the "stochastic" umbrella. So in my terms, Drift is sampling error alone. Defining Drift in terms of departure from "expected offspring value" (by dint of organisms' inability to produce non-integer offspring?) obviously conflates several stochastic factors into one.

You do know how sampling error is defined, right?

Allan Miller wrote:Oh, that one again! "If you think you know better than evolutionary biologists (for whom I speak) then get your radical new formulation published, matey-boy ...". I'm just joining in a word-game: "is there a way to define random ... etc". And it seems to me that, so long as we compartmentalise the processes appropriately, there is. That's a long way from saying "chuck away your slide-rules; I've just spotted something!".

Well, in that case you still need to define your terms. So far you haven´t (or you aren´t disagreeing with me).

Allan Miller wrote:Drift is routinely restricted to 'small' populations (without much more than a nod to "how small is small?").

Small is "smaller than infinitely large". Though if 2Ns>10 you can often neglect drift (there are exceptions to this and I´ve previously argued that they represent the usual situation).

Allan Miller wrote:But change N and you change the 'work' the population has to do to continue to behave in the idealised manner. Smear a population out over a surface (a planet, say) and the situation changes. Introduce an internal 'motor' (eg the need to find partners) and the situation changes again. In a maximal, planet-covering population, you would still get drift and fragmentation. I know population structure has been modelled to death, but the simplistic point is routinely made, whereas I simply don't see the Law of Large Numbers overriding geometry and other restraints on gene flow.

Well, the main thing about saying N=ifninite is that you assume no population structure. It´s a highly idealized situation and I frankly don´t get your argument. If you are using an idealized model, you do know that it is idealized and thus does not capture everything. In physics there are a lot of models ignoring friction, in chemistry you are usually neglecting local differences in concentration of reactants. Of course such models are missing some things, but they have the advantage of being easy to understand and thus are used as teaching tools. For some problems you don´t need anything more (as the effects not taken into account are small). For the purpose of our N->infinity model, the population occupies a point in space (of course what we really have to think about is an infinite population of populations. there are infinitely many earths, on each of which a particular population exists. The mean behaviour of allele frequency changes is predicted by selection. The remaining parts of the distribution of behaviours is drift). Apart from being annoying this part is also completely off-topic.

Allan Miller wrote:Ah yes, the disconnect between theory and practice, which is why there continues to be a "neutralist-selectionist contoversy".

a) Whatever the "disconnect between theory and practice" is, it plays no role here. There´s somebody who doesn´t understand the theory and can´t verbalize his position.
b) There´s no neutralist-selectionist controversy. There was some debate on how important neutrality is in the 70s and early 80s and then we got experimental results.

Allan Miller wrote:Separating drift and selection after the fact (and distinguishing hitch-hiking, inbreeding and god-knows-what to boot) is a tough nut. As is getting a decent handle on what the "expected number of offspring" actually is. The genetic snowflake that is every sexual individual does not yield up a neat value of s by integrating the allele-fitnesses of every sequence they possess.

That´s the wrong way around. s is the natural logarithm of the mean fitness of carriers divided by the mean fitness of non-carriers. There´s no formula for the reverse (an nobody ever proposed there was one).

susu.exp wrote:

Allan Miller wrote:And on the Talk page, we have a discussion regarding Gale's identification of (at least) 2 separate processes: "random genetic drift" ("the narrow view", quoth Kimura) and "stochastic changes in selection intensity". Other authors identify as many as 7 separate processes capable of sheltering under the "stochastic" umbrella. So in my terms, Drift is sampling error alone. Defining Drift in terms of departure from "expected offspring value" (by dint of organisms' inability to produce non-integer offspring?) obviously conflates several stochastic factors into one.

You do know how sampling error is defined, right?

I reckon so. Why do you ask?

susu.exp wrote:

Allan Miller wrote:Oh, that one again! "If you think you know better than evolutionary biologists (for whom I speak) then get your radical new formulation published, matey-boy ...". I'm just joining in a word-game: "is there a way to define random ... etc". And it seems to me that, so long as we compartmentalise the processes appropriately, there is. That's a long way from saying "chuck away your slide-rules; I've just spotted something!".

Well, in that case you still need to define your terms. So far you haven´t (or you aren´t disagreeing with me).

Well, I thought I had. For the purposes of discussion, Random = Unsystematic. Drift is unsystematic with respect to phenotype. Changes in allele frequency due to drift are causally unaffected by allele sequence.

Non-random = Systematic. Natural Selection is systematic wrt phenotype - if phenotypes are selectively distinguished, then this introduces a departure from the 'phenotype-blind' case. This introduces change in allele frequency causally linked to allele sequence. If this bias is consistent in sign (regardless of considerations of actual value) then we have a net assessment of benefit/detriment.

Drift (sample error) is the only process in operation when phenotypes are selectively indistinguishable. Otherwise, there is a component due to sample error, and a component due to selective discrimination. The analogy is offered between a fair and a weighted dice. A fair dice is blind to the numbers on the face, but any face has an equal chance. Upon this (popularly 'random') chance is superposed in the weighted case a bias in the direction of one particular face (with a parallel bias against the rest).

susu.exp wrote:

Allan Miller wrote:Drift is routinely restricted to 'small' populations (without much more than a nod to "how small is small?").

Small is "smaller than infinitely large".

So Drift is restricted to populations that exist. Thanks for clearing that up.

susu.exp wrote:Though if 2Ns>10 you can often neglect drift (there are exceptions to this and I´ve previously argued that they represent the usual situation).

I don't think you can ever neglect drift, due to the 'viscosity' of gene flow in nature. This creates local islands, however incompletely isolated, within which sampling error can operate upon a smaller effective population.

susu.exp wrote:

Allan Miller wrote:But change N and you change the 'work' the population has to do to continue to behave in the idealised manner. Smear a population out over a surface (a planet, say) and the situation changes. Introduce an internal 'motor' (eg the need to find partners) and the situation changes again. In a maximal, planet-covering population, you would still get drift and fragmentation. I know population structure has been modelled to death, but the simplistic point is routinely made, whereas I simply don't see the Law of Large Numbers overriding geometry and other restraints on gene flow.

Well, the main thing about saying N=ifninite is that you assume no population structure. It´s a highly idealized situation and I frankly don´t get your argument. If you are using an idealized model, you do know that it is idealized and thus does not capture everything. In physics there are a lot of models ignoring friction, in chemistry you are usually neglecting local differences in concentration of reactants. Of course such models are missing some things, but they have the advantage of being easy to understand and thus are used as teaching tools. For some problems you don´t need anything more (as the effects not taken into account are small). For the purpose of our N->infinity model, the population occupies a point in space (of course what we really have to think about is an infinite population of populations. there are infinitely many earths, on each of which a particular population exists. The mean behaviour of allele frequency changes is predicted by selection. The remaining parts of the distribution of behaviours is drift). Apart from being annoying this part is also completely off-topic.

Annoying it may be, but you brought it up. Having brought it up, it becomes relevant to a discussion on drift. If you are offering it as an illustration a drift-free world, then the facts of the world that are dispensed with in that model become germane. In physics and chemistry you can dispense with friction, viscosity or local concentration when you are dealing with gross behaviour - random factors cancel each other out. But you can't get rid of local phenomena (such as random drift) in a real population. The population has little 'gross behaviour' to speak of; such as there is tends to be emergent upon the behaviour at individual level, which is far more sensitive to local effects than to N.

susu.exp wrote:

Allan Miller wrote:Ah yes, the disconnect between theory and practice, which is why there continues to be a "neutralist-selectionist contoversy".

a) Whatever the "disconnect between theory and practice" is, it plays no role here. There´s somebody who doesn´t understand the theory and can´t verbalize his position.

Well, I'm trying to help you with that.

mjpam, you are using an incorrect understanding of the word random. It doesn't seem nearly as broad as your trying to use it.

A dice roll is random even though I have a greater chance of rolling a 7. If I modified the dice so they were slanted in a certain way and even more often landed on seven it would no longer be random. So just as mutations are random because there is no way to tell when or where they will happen natural selection is not because there is an outside force deciding the outcome ahead of time (a hurricane destroying a stand of trees, or a pathogen killing all the weak antelopes, for example).

EDIT: Hmm, I didn't even see that there was a page 2 and 3. . .

Allan Miller wrote:I reckon so. Why do you ask?

Well, mainly because X-E(X) is sample error, yet you protest that definition.

Allan Miller wrote:Well, I thought I had. For the purposes of discussion, Random = Unsystematic. Drift is unsystematic with respect to phenotype. Changes in allele frequency due to drift are causally unaffected by allele sequence.

Now define unsystematic. If you are using the usual (equiprobability of drift giving positive and negative values), then it is not sample error. The sample error does contain systematic effects.

Allan Miller wrote:Drift (sample error) is the only process in operation when phenotypes are selectively indistinguishable. Otherwise, there is a component due to sample error, and a component due to selective discrimination. The analogy is offered between a fair and a weighted dice. A fair dice is blind to the numbers on the face, but any face has an equal chance. Upon this (popularly 'random') chance is superposed in the weighted case a bias in the direction of one particular face (with a parallel bias against the rest).

The problem with the analogy is that the distribution for a die is symetric and for symetric distributions the mean equals the median. This is not the case for skewed distributions, which we are looking at in population resampling. In this case the mean and the median differ and you are arguing that drift is both resampling minus mean (sample error) and resampling minus median (unsystematic). Drift is defined as the former, and thus tends to be systematic unless you are looking at cases where the distribution is not skewed. That´s the case for neutrality in the Moran Model. Which is precisely why focussing on that special case is such a problem.

Allan Miller wrote:So Drift is restricted to populations that exist. Thanks for clearing that up.

I´ve argued before that it´s the only unstoppable process in real populations. Selection stops if s=0, mutation stops if µ=0, but drift only goes away for N=infinite.

Allan Miller wrote:I don't think you can ever neglect drift, due to the 'viscosity' of gene flow in nature. This creates local islands, however incompletely isolated, within which sampling error can operate upon a smaller effective population.

In some cases you can - but you have to justify it. If your point is that drift is neglected too often, you are preaching to the choir here.

susu.exp wrote:

Allan Miller wrote:I reckon so. Why do you ask?

Well, mainly because X-E(X) is sample error, yet you protest that definition.

Nah, just puzzled how many offspring we actually "expect". We are carriers of 20,000+ genes. Each of those introduces an element to the 'standard expectation' of the individual Human Bean. My expectation depends upon the genetic cards I actually hold (cross-referenced to those held at each corresponding locus in every other individual). If we were all genetically uniform but for the locus of interest, then yes, I could see that we could ascribe separate expectations to the bearers of the distinct allele classes, and one generation later ascribe drift to the deviation from the expectation based upon fitness. But that only works in the idealised situation.

susu.exp wrote:

Allan Miller wrote:Well, I thought I had. For the purposes of discussion, Random = Unsystematic. Drift is unsystematic with respect to phenotype. Changes in allele frequency due to drift are causally unaffected by allele sequence.

Now define unsystematic. If you are using the usual (equiprobability of drift giving positive and negative values), then it is not sample error. The sample error does contain systematic effects.

If I select balls from a bag blindly, I am picking them unsystematically. If I have a peek, and ignore the even numbers, then I am introducing a systematic element to the process of ball selection. Of course, replacement and resampling cause concentration of some class of balls and loss of the rest. Repeat sampling therefore could be viewed as systematic. But it is not sampling per se that is behaving systematically; it is resampling.

susu.exp wrote:

Allan Miller wrote:Drift (sample error) is the only process in operation when phenotypes are selectively indistinguishable. Otherwise, there is a component due to sample error, and a component due to selective discrimination. The analogy is offered between a fair and a weighted dice. A fair dice is blind to the numbers on the face, but any face has an equal chance. Upon this (popularly 'random') chance is superposed in the weighted case a bias in the direction of one particular face (with a parallel bias against the rest).

The problem with the analogy is that the distribution for a die is symetric and for symetric distributions the mean equals the median. This is not the case for skewed distributions, which we are looking at in population resampling. In this case the mean and the median differ and you are arguing that drift is both resampling minus mean (sample error) and resampling minus median (unsystematic). Drift is defined as the former, and thus tends to be systematic unless you are looking at cases where the distribution is not skewed. That´s the case for neutrality in the Moran Model. Which is precisely why focussing on that special case is such a problem.

It is becoming clearer to me why there is disagreement. The sampling of a current population is still drift, and as such is unsystematic. Repeated sampling concentrates those alleles that survived the previous rounds, in a manner that could be described as systematic. Nonetheless, all that is being iterated is that same unsystematic process. The arrow derives from repetition. Single-generation sampling will act in an identical manner on 10,000 completely different alleles or a 50/50 A/B split ... it acts 'at random'. One, 'at random', out of a current set of 10,000 selectively indistinguishable alleles, however we may choose to classify them, will [almost surely] become the ancestor at its locus.

Allan Miller wrote:Nah, just puzzled how many offspring we actually "expect".

If you agree that there is some probability distribution for offspring, then there is an expected value. Yes, figuring the distribution out is rather tricky, but that´s enormously besides the point.

Allan Miller wrote:We are carriers of 20,000+ genes. Each of those introduces an element to the 'standard expectation' of the individual Human Bean. My expectation depends upon the genetic cards I actually hold (cross-referenced to those held at each corresponding locus in every other individual). If we were all genetically uniform but for the locus of interest, then yes, I could see that we could ascribe separate expectations to the bearers of the distinct allele classes, and one generation later ascribe drift to the deviation from the expectation based upon fitness. But that only works in the idealised situation.

I noticed before that you´ve got that the wrong way around. You can not calculate individual fitness from the contribution of individual alleles. Nobody does that and nobody claims this can be done. What you can calculate is s from the various individual fitnesses.

Allan Miller wrote:If I select balls from a bag blindly, I am picking them unsystematically. If I have a peek, and ignore the even numbers, then I am introducing a systematic element to the process of ball selection. Of course, replacement and resampling cause concentration of some class of balls and loss of the rest. Repeat sampling therefore could be viewed as systematic. But it is not sampling per se that is behaving systematically; it is resampling.

That´s not a definition, that´s an example. And the trouble with examples is that they are unclear. Let´s talk about Gnoffos. You may want to know what I mean by that term before you can actually discuss Gnoffos, but rather than tell you what a Gnoffo is, I merely tell you that Diego Maradonna is a Gnoffo. Now, that still doesn´t tell you what I mean by Gnoffo (Humans? Football players? Football coaches? Argentinians? Bearded men?). Now, very simple from Wikipedia on sample error is this one:
"[..] sampling errors which either have a prevalence to be positive or negative. Such errors can be considered to be systematic errors."
No prevalence to be positive or negative is given for X-M(X) with M(X) the median of X. So you can either say drift is X-E(X) and it is sample error, or you can say it´s X-M(X) and unsystematic. But for all cases where E(X)=!M(X), you can not have both.

Allan Miller wrote:It is becoming clearer to me why there is disagreement. The sampling of a current population is still drift, and as such is unsystematic. Repeated sampling concentrates those alleles that survived the previous rounds, in a manner that could be described as systematic. Nonetheless, all that is being iterated is that same unsystematic process. The arrow derives from repetition. Single-generation sampling will act in an identical manner on 10,000 completely different alleles or a 50/50 A/B split ... it acts 'at random'. One, 'at random', out of a current set of 10,000 selectively indistinguishable alleles, however we may choose to classify them, will [almost surely] become the ancestor at its locus.

I´m starting to swear here. NO, NO, NO, 100 times NO. You are again returning to discussing drift only for selectively neutral alleles. The special case for which E(X)=M(X). But that special case is not a general case. There´s no 50/50 split for the drift term when there´s selection (again there are special cases, but in general the split is not 50/50).

susu.exp wrote:

Allan Miller wrote:We are carriers of 20,000+ genes. Each of those introduces an element to the 'standard expectation' of the individual Human Bean. My expectation depends upon the genetic cards I actually hold (cross-referenced to those held at each corresponding locus in every other individual). If we were all genetically uniform but for the locus of interest, then yes, I could see that we could ascribe separate expectations to the bearers of the distinct allele classes, and one generation later ascribe drift to the deviation from the expectation based upon fitness. But that only works in the idealised situation.

I noticed before that you´ve got that the wrong way around. You can not calculate individual fitness from the contribution of individual alleles. Nobody does that and nobody claims this can be done. What you can calculate is s from the various individual fitnesses.

Point is, I am not aiming to calculate anything, so I can put it whichever way round suits my purposes. There is a way to understand evolution without recourse to maths. Or, if there isn't, let's shut up shop, because all discussions would become too esoteric for general consumption. The people whose task it is to try and understand these processes mathematically have a set of tools that enable that job to be done, and some strategies are more useful than others. The role of the other 20,000 alleles in a single life and in successive lives is, nonetheless, relevant to evolutionary processes and outcomes. And as far as the sampling process goes, it is as legitimate to talk about each new offspring sampling the alleles in the population, as it is to talk of each new offspring sampling the genes of the parents.

susu.exp wrote:

Allan Miller wrote:If I select balls from a bag blindly, I am picking them unsystematically. If I have a peek, and ignore the even numbers, then I am introducing a systematic element to the process of ball selection. Of course, replacement and resampling cause concentration of some class of balls and loss of the rest. Repeat sampling therefore could be viewed as systematic. But it is not sampling per se that is behaving systematically; it is resampling.

That´s not a definition, that´s an example. And the trouble with examples is that they are unclear. Let´s talk about Gnoffos [...].

Well, that example demonstrates clearly the distinction I have in mind, hilarious lampoon nothwithstanding. The trouble with definitions is that you can spend months arguing about them (eg: "random").

susu.exp wrote: Now, very simple from Wikipedia on sample error is this one:
"[..] sampling errors which either have a prevalence to be positive or negative. Such errors can be considered to be systematic errors."

You have quoted selectively. The passage in full:

SAMPLING BIAS is a possible source of sampling errors. IT leads to sampling errors which either have a prevalence to be positive or negative. Such errors can be considered to be systematic errors.

ie, sampling bias is systematic, not sampling error. But bias in drift? Well no, not in the way I view it. Drift is in its essence unbiased with respect to a given allele's phenotypic effect. Selection is not. When an allele is selected - when its phenotypic effect influences its survival/reproduction - that introduces a bias. In lives in which it is unselected (even if it has s<>0), the bias does not operate.

susu.exp wrote:
No prevalence to be positive or negative is given for X-M(X) with M(X) the median of X. So you can either say drift is X-E(X) and it is sample error, or you can say it´s X-M(X) and unsystematic. But for all cases where E(X)=!M(X), you can not have both.

Or, I can say drift is the process in effect in all lives other than those when the allele makes a difference, and is unsystematic. There is no consistency in sampling error in either direction, because there is no source of bias. I don't just mean neutral alleles, as you continually read me as saying. The neutral case is simply that where drift is the only force in operation. But no value of s except 1 is deterministic on every body in which the allele resides. If an allele gets replicated, and it was not tested by selection on the way, regardless of its value of s, then you can chalk it up to drift - the random component. If it was tested, then chalk it up to selection - a systematic layer upon the symmetrical background of drift.

susu.exp wrote:

Allan Miller wrote:It is becoming clearer to me why there is disagreement. The sampling of a current population is still drift, and as such is unsystematic. Repeated sampling concentrates those alleles that survived the previous rounds, in a manner that could be described as systematic. Nonetheless, all that is being iterated is that same unsystematic process. The arrow derives from repetition. Single-generation sampling will act in an identical manner on 10,000 completely different alleles or a 50/50 A/B split ... it acts 'at random'. One, 'at random', out of a current set of 10,000 selectively indistinguishable alleles, however we may choose to classify them, will [almost surely] become the ancestor at its locus.

I´m starting to swear here. NO, NO, NO, 100 times NO. You are again returning to discussing drift only for selectively neutral alleles. The special case for which E(X)=M(X). But that special case is not a general case. There´s no 50/50 split for the drift term when there´s selection (again there are special cases, but in general the split is not 50/50).

You missed the point. 50/50 was just one of the 10000! ways the alleles in the population could have been classified. Yes, I keep discussing the neutral case, because that's the ground floor. Selection builds upon that by consistently offering up certain alleles at a rate above or below the neutral assumption. But these alleles still behave as if neutral in every life that they did not influence. In those lives, they are simply 1 humble allele among 10,000 other contenders for sampling. s is a composite built from instances of replication or failure, each of which gives a 1 or a zero to the set of datum points comprising the statistical metric. But nothing 'knows' what s is and acts accordingly. I don't just mean "drift means s=0". But s might as well be zero when selection on that allele does not operate in a life.

Drift is akin to picking for clinical trials by tossing a fair coin. Selection is akin to saying "OK, best of three" when the patient is female. Now, you would say that drift is in effect even in the latter case - there is still a sampling error, and this time there is bias. But any bias present does not derive from the sampling process (random drift), but from selection.

Allan Miller wrote:There is a way to understand evolution without recourse to maths. Or, if there isn't, let's shut up shop, because all discussions would become too esoteric for general consumption.

The way to understand evolution without recourse to maths is somewhat superficial and that holds for understanding drift in particular. Selection and mutation can be easily put into words, drift can´t (that´s the main reason drift always gets the short end of the stick in popular treatments). But we can discuss drift without doing any hard maths and it is worthwile doing it.

Allan Miller wrote:Well, that example demonstrates clearly the distinction I have in mind, hilarious lampoon nothwithstanding. The trouble with definitions is that you can spend months arguing about them (eg: "random").

No, it doesn´t. Because for your ball picking story you again do not take care of making the distinction between median and mean.

Allan Miller wrote:You have quoted selectively. The passage in full:

SAMPLING BIAS is a possible source of sampling errors. IT leads to sampling errors which either have a prevalence to be positive or negative. Such errors can be considered to be systematic errors.

ie, sampling bias is systematic, not sampling error.

You´re misreading this. Bias leads to systematic effects in the sample error. That´s because in a skewed distribution the mean and median are not the same.

Allan Miller wrote:But bias in drift? Well no, not in the way I view it. Drift is in its essence unbiased with respect to a given allele's phenotypic effect. Selection is not.

I agree. Selection is the bias in population resampling. Drift is the sample error. If s=!0, then tere is a bias and the result of that is that the sample error (drift) has a prevalence of being positive or negative, hence it contains a systematic effect.

To give you an example: Let´s roll 30 dice and look at the number of times we roll a 4 or higher. The expected number is 15, and the probability for having less than 15 is equal to the probability of having more than 15. Now, repeat this with counting only the dice with 5 or more. In this case our expected number is 10, and the probability of rolling less is 43.2%, while that of rolling more is 41.5%. This difference in how likely you are to be better or worse than the expectation is what is refered to as systematic error. And when there is selection, then the sample error (drift) has a systematic component.

Allan Miller wrote:Or, I can say drift is the process in effect in all lives other than those when the allele makes a difference, and is unsystematic.

No you can´t. Because it does contain the systematic component.

Allan Miller wrote:There is no consistency in sampling error in either direction, because there is no source of bias.

Natural selection.

Allan Miller wrote:I don't just mean neutral alleles, as you continually read me as saying. The neutral case is simply that where drift is the only force in operation. But no value of s except 1 is deterministic on every body in which the allele resides. If an allele gets replicated, and it was not tested by selection on the way, regardless of its value of s, then you can chalk it up to drift - the random component. If it was tested, then chalk it up to selection - a systematic layer upon the symmetrical background of drift.

Holy shit, there are so many errors there...
a) An allele get´s passed on and you claim that this is due either to selection of drift. That´s just nuts, not to meantion that it makes no sense. Selection is a statistical effect and so is drift. You can not look at an individual and say "there´s selection". Selection only happens at the level of the local population.
b) Drift is not symatrical and does contain a systematic component, if there is selection. It´s only symatrical and unsystematic in the neutral case.
c) s=1 is not deterministic, if you are using Kimuras s (and since you´ve been discussing neutrality it´s the s you should be using). For this s=infinity and s=-infinity are deterministic. s=1 does not have any special properties there.

random/spontaneous = we dont know whether there is a cause or not.
Its is a very convenient device in "science" to assert that the absence of evidence proves an event is "random"
Theists would NEVER be allowed to fill a GAP in the way "science" does.
Science can have "cosmological constants" and "singularities" and "random mutation" and "quantum weirdness" and "uncertainty principles" as assumed FACTS but dont anybody dare to use God as a gap filler.
Lion (IRC)

Lion IRC wrote:random/spontaneous = we dont know whether there is a cause or not.
Its is a very convenient device in "science" to assert that the absence of evidence proves an event is "random"
Theists would NEVER be allowed to fill a GAP in the way "science" does.
Science can have "cosmological constants" and "singularities" and "random mutation" and "quantum weirdness" and "uncertainty principles" as assumed FACTS but dont anybody dare to use God as a gap filler.
Lion (IRC)

You wrote this after reading only the thread title, didn't you?

Is there a way to define "random".........

"Randomness is a concept of non-order or non-coherence in a sequence of symbols or steps, such that there is no intelligible ........blah blah blah......."

What a load of gobbledygook

Lion IRC wrote:Is there a way to define "random".........

"Randomness is a concept of non-order or non-coherence in a sequence of symbols or steps, such that there is no intelligible ........blah blah blah......."

What a load of gobbledygook

This post is of great value to the ongoing discussion and will, I boldly predict, fundamentally change the way we think about mathematics and, in turn, the mathematical basis of science.