I have given some examples of things that lie in both categories. Gas pressure being due to the number and momentum of particles hitting the container, will vary as you measure it, but to such a small extent that it is only technically random, but not practically so.
1. This is the central point of contention. I would agree that it is not wrong to call any individual particle in the gas 'technically random', or just 'random'. But the moments of the gas are neither practically nor technically random. As you say the number and the momentum of the particles is the cause of pressure. In an idealized closed container(no particle leakage,inelastic collisions,& rigid walls) the number of particles and their momentum will be strictly constant. (Due to the fact that the container is closed and due to conservation of momentum) So
technically it is non-random and non-varying.
For example, pressure is:
P = Nm(v
rms)
2/3V
N the number of particles.(constant due to closure)
V
rms is the mean of the distribution(constant due to conservation of momentum)
m is the sum over the mass of the system(constant due to closure and conservation)
V is volume, (system wide constant)
I will note that
practically there will be variation. Practically, the gas will leak energy/momentum into the walls of the container and it will radiate the energy out of the system from there. This will cause a steady decline in temperature.
2. It sounds like you're saying that the system is 'technically random' because there may be some variation in the insignificant digits of the measured moments of a gas. This is equivalent to saying everything is technically random. The is no physical entity that can be measured to infinite precision and there will always be 'quantum variation'. Thus the definition you put forth is devoid of meaning.(A definition that applies to everything equally fails to communicate information, see Shannon information entropy, if you are confused on this point.)
3. Also, no measurement will be to infinite precision. Discussions of any physical quantity need to be put in the context of some experimental method. If our instrument cannot measure down to an accuracy that we see any variation, then it is fair to call that system non-random and deterministic, insofar as our experiment is concerned. If we can measure a small variation in the measurement, we need to be careful how we phrase our statements about the system. We might be say the system is not a completely deterministic, that it has no variation in the significant digits, and that it has variation in the insignificant digits. If there is no pattern in this variation(uniform and uncorrelated) we might say there is random variation in the insignificant digits of our measurement. It would be folly to call the whole system random or even variable if the significant digits of our measurement are constant.
Compare that to a simple random number generator, where a noise source is compared to a reference voltage, and depending on the result a logical high ("1") or logical low ("0") is output. The resulting string of 1s and 0s is random in both senses, not just technically but also in practice.
Practically yes, technically maybe. Actually, at one time it was not uncommon to find systematic variation in a physical measurement used for a random number generator that was thought secure. That is actually part of the reason that software moved to pseudo-random number generators. They ended up being
less random than noise based physical measurements. Although, admittedly technology has improved and cryptographic necessity has brought back the importance of random numbers generated from physical systems these days.
I want to get the definition of random out of the way, since it is useless to discuss how it applies to evolution as long as we are on different. Second I haven't appointed myself arbiter, the examples I gave about aren't ones I supplied.
What I refer to is your tendency to shift to the claim that "well whatever points you're making the system is still
technically random". The points I've been making apply to both your definitions, so you can't logically regress to another nearly identical definition that you name more authoritatively.
Not true, random sampling is different that simple random sampling. Wikipedia doesn't do a bad job of explaining it. From the end of the first paragraph ...
This process and technique is known as Simple Random Sampling, and should not be confused with Random Sampling.
And you can look at both definitions at wikipedia, and note that one of the examples they give of random sample is a
stratified sample.
Actually, you can find "gaussian random distribution" in text books and on the web. If you look up "gaussian random" in google you will find articles, several technical, on gaussian random distribution or generating gaussian random numbers.
The point I'm trying to make is not that it is wrong to have random in the name, but to point out that its presence in the name doesn't add to your claim. For example, you seem to make a big deal about the fact the that the
word 'random' is in a 'gaussian random distribution'. As you so aptly mention further down in your post, it is not the distribution that is random. In fact, a gaussian distribution is entirely deterministic.
G
pdf = (1/sqrt(2*pi))*exp(-x
2/2)
(This is with a mean of 0 and a variance of 1)
The fact is that a large number of measurements may often be
modeled with a Gaussian distribution due to the central limit theorem. We might call this a population or a sample distribution, depending on context.
Stats book talk about random numbers having a non-uniform distribution, stochastic books do, my computer science text book on numerical methods discussing generating non-uniformly distributed random numbers using the transform and acceptance-rejection methods.
Yet a random number generator that isn't uniformly distributed is considered non-random in computer science. A cryptographic random number generator that isn't uniformly distributed is no random number generator at all. Stats books talk about 'random variables' having non-uniform distributions and we've already mentioned that they consider distributions like the dirac-delta to be distributions over random variables as well.
If we want this conversation to go until infinity we can both go and mine all our textbooks and all the scientific literature for examples that support both our cases. The bottom line is that the use of the term isn't going to be consistent from author to author, researcher to researcher, field to field, etc... So which usage should we choose? I would assert strongly that the definition of practitioners in the field of evolution is most applicable, and we've already heard the top minds in the field denying randomness in evolution.
2. As I pointed out, random sample and simple random sample do not mean the same thing.
Yes this was addressed above, I'm not sure how this denies that fact that deviations: correlations, skews, etc... are a deviation from randomness.
3. Sure, if there is no variance in the population being sampled, then the outcome of sampling will be determined.
Exactly the point. For example, in the last post you mention 'Systematic Random Sampling', but insofar as SRS is a deterministic process, it is non-random. And you've just agreed that the object being sampled is not necessarily random either.
So why is the word 'random' in the name at all? What is its meaning? It is talking about the capacity of the technique to create neither bias, nor correlation in the sampled distribution. If the samples themselves are uniformly distributed and uncorrelated(in the appropriate variable space of the problem) they will do exactly that. In that way a
random sampling technique can form an accurate impression of the object being studied. This corresponds well to the definition I put forth. Whereas the definition of random as non-zero variance doesn't seem to explain this usage at all.
Exactly which technique needs to be used depends a lot of the object of study. We might use a systematic technique if we know the objects of interest are in a unordered(randomly ordered) list. We might use stratified sampling, when we believe there is a numerical or representational bias between groups. (For example in a survey by phone certain groups may be more likely to have a telephone, so if we want to eliminate this skew in our sampling, we would do well to sample the different groups separately and then recombine appropriately.)
4. Of course I didn't use the word random there. It's stupid to use a word whose definition we are discussing in a position that would lead to ambiguity. And since random doesn't mean uniform as pointed out previously, it would be down right wrong. There was no manipulating, because random was the wrong word for the occassion.
I thought it interesting. A uniform and uncorrelated number is exactly what you were talking about. Under my definition there would be no ambiguity. It certainly would be called random in any technical description. It only becomes ambiguous when the advocated definition is so broad that the original word is depleted of meaning.
5. I don't think you said what you wanted to there, as simple-random sample and random sample are not the same.
I do understand the difference. 'simple random sample' v 'all other types of random sampling' is a false dichotomy. If you didn't understand my previous explanation, my other one above may be clearer. The basic idea is that the technique ought to be selected to eliminate bias and correlation from the sampling process itself. Practically other variables will be optimized as well when deciding on a technique. For example, cost, speed, practicality, error. A systematic sample can be much easier to implement if its usage is not problematic.
Poisson distribution can be population distributions or random distributions.
I don't think you are using these terms quite correctly, or at least you are twisting 'random distribution' to somehow put it into opposition to 'population distribution' to support your definition. You were much clearer in the previous post when you distinguished between 'probability distribution' (or 'sample distribution') and 'sample distribution'
But I get that a population and sample can both have a Poisson distribution.
I don't agree with Mijo on that, though I haven't followed that line of argument much.
When one definies a probability function over the reals, the function is the probability density function, and you can get the probability of the result being on any interval by intergrating the probability density over that interval. If the density function is delta-dirac(x-k), then any interval not including k will have probability 0. Any interval including k will have probability 1. That seems about as certain as can be.
Hey now. You're arguing my points now. That the moments(or partial moments) of a system are constant and regular. This is integral is
exactly what we do when we calculate the characteristic moments of a gas(density, velocity,temperature, pressure, etc...) and they are
exact.
The point is that if you are looking at any
particular solution of a sample distribution over the reals, you cannot
necessarily be sure that the value will be the expected, regardless of the distribution. Only
almost sure. What this means is that if we use a non-zero variance definition of random, there is not even a theoretical way to be certain a system is non-random. Although, I would tend to agree that this sort of digression is a bit of a red-herring.
The distribution of thermal noise is approximately gaussian. If you sample the voltage on a resistor, the probability distribution of that single sample is gaussian.
Yep thats noise alright. This is just a regression of the gas issue discussed above. The voltage is a function of temperature, and that follows from the random motion of individual molecules. It is governed by the central limit theorem and is not random(remember OHM's law). The single sample would only be random if it didn't tend toward that limit.(as per my stated definition of random)
No, there is no assumption that they are unbiased in the way they are formed. However, if you wanted to measure such a process and get an accurate picture, an unbiased sampling is the best way to go about finding the processes bias.
This is covered in detail above.
No more so that the term technical and practical are indistinguishable. I've given examples above and elsewhere.
You distinguish by saying that 'practically random' has more 'variance than technically random' . That is the only commonality in your distinction. That is just another way for you to say "I reserve to draw the distinction wherever suites my purposes"
Technically something that is random can have any distribution that doesn't have a variance of 0. Determistic system can have similar distributions. What is important in the technical defintion is that one can get different results for the same starting conditions.
Two bad meaningless definitions. I've addressed them both, but I've hit the first a little more explicitly. So I'm just going to address the second one here.
1. The terms 'different' and 'same' are ambiguous. Since no starting conditions are ever the same, there is no identical repeatability and no way to to tell if a system is random or not under this definition. Also the contrary point is true for the result. All results are necessarily different. You couldn't even design an experiment that can test this in theory as you can never measure to infinite precision whether the starts are the same and the ends are different.
2. This claims that random is synonymous with non-deterministic, clearly they are not. We can actually construct a thought experiment that proves this point. Imagine a phenomenon. We're in thought experiment land, though, so lets imagine we can control for every variable and actually get identical conditions. Our first measurement will be 1 or 0 with some probability and every subsequent measurement in the system for that session will be the negation of the previous value:
[latex]$$System: S(t) $$[/latex]
[latex]$$p(S(0)) = .5$$[/latex]
[latex]$$S(t-1) = 0 \Rightarrow S(t) = 1$$[/latex]
[latex]$$S(t-1) = 1 \Rightarrow S(t) = 0$$[/latex]
So is this system random or deterministic? We've stipulated that we control for all variables so we're in the same conditions, yet our first measurement may be 1 or it may be 0, we don't know. Yet after we measure that first value the system behaves in a completely reliable way. The answer seems to be that neither explanation is entirely adequate and we would more accurately call this intermediate system a mixed system or a non-deterministic(also non-random) system. You might argue that this thought experiment is unrealistic, that it is impossible because it postulates an uncaused event and the observer divides the sequences of measurements into sessions, but this is not too different from a quantum experiment testing bell's inequality, where the observer collapses the waveform, so an initial probabilistic measurement determines the system. It necessarily excludes this same-start different result definition.
1. I get a clear answer as well.
I think you miss by point. What I'm saying is that there are tons of systems that seem to fall between random and determinate. The definition I have creates a spectrum(on two dimensions) by placing uniform and uncorrelated
on one extreme and singular(dirac) and correlated on the other extreme. This organizes our conceptual space over a continuous field.(RxR) It allows us to situate the intermediate examples in the appropriate locations in the space. This is an advantage. Your definition has a certain binary quality such that it is unable to properly situate intermediate cases so that they agree with our intuitions. This is a disadvantage.
2. How are you calculating the correlation, by the standard equation I assume?
No I'm talking about lack of correlation in a more fundamental sense. In all senses really. For example there is a certain class of pseudo-random number generators that seemed to show lack of correlation and uniformity. In other words it seemed to produce random numbers. What was later discovered was that if the numbers were plotted in a high-dimensional space they would form clear and identifiable bands. So I don't think any single definition of correlation is sufficient.
3. Non-uniform distributions will generate correlations of 0 as well as uniform ones. They would just have to be independent of the sampling process. If your numbering samples in the order you take them it would be sufficient that the process be independent of time.
If (un)correlation followed from uniformity or uniformity from (un) correlation, I wouldn't require two constraints to describe random. I would just mention the more primary. Only both together create random.
4. Deterministic systems can produce correlations of 0. If you periodically asked me for a number, and I give alternating 1s and 0s you will get an incredibly low correlation to the sample N. Compare that to a random (your definition) string of bits.
See my example above, that any single correlation algorithm is inadequate.
If you plot your alternating 1s and 0s over the appropriate choice of basis a correlation will show.
Just to conclude, I'm getting kind of bored of this discussion.
In the interests of expediency, I'd like to try to work at a compromise. Really since a definition is based upon usage and common usage is based upon a consensus understanding of language, as long as either one of us(or mijo) is willing to assert their definition regardless of evidence to the contrary we're never going to be done. So it might be best for us to start looking for a compromise definition and stop asserting the universal truth of our definition. Since usage will vary(as I mention above) it seems kind of silly to insist that there is one correct definition. The definition is whatever people agree is should be and different groups will have come to different agreements. So what definition practical, technical or not, is appropriate to this discussion?
