"A measurable function of a random variable is a random variable"?

[Meridian]

No - I've nothing against the axiom of choice. But while I can prove non-measurable sets exist, needing AC to construct them makes the individual sets pretty hard to imagine!

 

[ddt]

No - I've nothing against the axiom of choice. But while I can prove non-measurable sets exist, needing AC to construct them makes the individual sets pretty hard to imagine!

But a true mathematician can perform magic uncountably many things in just the blink of an eye!

When it comes to imagining sets, try to imagine which points on a line are rational, which are algebraic, or try to imagine the Cantor set .

 

[GreyICE]

But a true mathematician can perform magic uncountably many things in just the blink of an eye!

When it comes to imagining sets, try to imagine which points on a line are rational, which are algebraic, or try to imagine the Cantor set .

I just looked up "Cantor Set" on Wikipedia.

I'm sticking to engineering.

 

[zosima]

In another thread, someone pointed out to me that functions such as f(X)=0X+c and f(X)=X^0+c, where X is a random variable and c is a real number, are not random variables because they yield the same value for ever value of the random variable X.

If I'm the person you're talking about then you're incorrectly characterizing what I said. You should look back through the transcript. I never said that the output of the function would not also be considered a random variable under some technical definition. What I maintained, which is what I had maintained throughout the discussion, was whether a system is modeled with random variables or not is independent of whether the system itself is random in any meaningful sense of the word.

The fact that constant functions can be functions of random variables proves the point. We have a function of a random variable and yet it models a deterministic system.

ETA: The whole discussion was in the context of your assertions that a system that took a random variable as an input could always have any output. This is clearly not the case for a constant function.

 

[mijopaalmc]

If I'm the person you're talking about then you're incorrectly characterizing what I said. You should look back through the transcript. I never said that the output of the function would not also be considered a random variable under some technical definition. What I maintained, which is what I had maintained throughout the discussion, was whether a system is modeled with random variables or not is independent of whether the system itself is random in any meaningful sense of the word.

The fact that constant functions can be functions of random variables proves the point. We have a function of a random variable and yet it models a deterministic system.

ETA: The whole discussion was in the context of your assertions that a system that took a random variable as an input could always have any output. This is clearly not the case for a constant function.

The discussion on Borel-measurable function of random variables is posted below:

OK, I realize that there is a valid distinction between the process itself and the mathematical models that describe it and I agree that "random" and "non-random" may only meaningfully describe the mathematical models. However, if you are going to describe a mathematical model with random portions, then mathematically speaking to whole system is random. This is a non-negotiable point as we are describing a mathematical process and mathematically speaking any Borel-measurable function (which includes all elementary functions learned from primary school to elementary vector calculus) of a random variable is itself a random variable and the mathematical definition of a random process is a family of random variable defined of the same probability space. This is also why I am often frustrated with articulett when she insist that definition of random I am using makes algebra random, because the variables in the kind of algebra she is talking are not the random variables of probability and statistics, because they are merely values that you plug into a function and not functions in their own right as random variables are.

Wrong, but please do try again.

f(x) = 0*x+c

Obviously, you don't know what a trivial case is.

Sure I do, it is a counter-example to a bad definition, that takes no effort. Now if I had to think about it my counter-example would be pathological, but still acceptable. Just a post ago this was 'non-negotiable' , but now you make exceptions for what you term trivial examples. This is just another way of saying, "you're wrong".



As to my petulance. I can't have you just making up mathematics as you go along, and then try to trick people into believing it by trying to sound authoritative. As to all the other silliness you chose to make up, I'm not worrying about it. You've given up on trying to have a discussion, and I wouldn't want to be overdressed to the party.

But just as a friendly warning I would recommend you avoid straying towards mathematics. You do remember that business with 'almost surely' where you demonstrated you didn't understand the cardinality of a set don't you?

Not only is it useless, it is also wrong. Mijo conceded my counter-example; going from this point being 'non-negotiable' to negotiable. Mijo's statement only holds true insofar as all the examples that disprove his definition are ignored.
Then again that is not so different than the strategy Mijo applies to all the points he makes. So I suppose it is not surprising.

I did not actually concede your "counter-example"; I just never adequately responded to it. [...]

#1 As far as I am concerned running away from a strong objection to a fallacious claim is concession, or perhaps somewhat worse than concession. This is particularly true for you insofar as you have a habit of showing complete amnesia to the points that 'sink your battleship'.

[...]

Your example is in fact a random variable, just as all measurable functions of random variables are. In essence, just because one event has a probability of 1 doesn't make the distribution non-random because other events still exist they just happen with a probability of zero.

#1 This is the 'almost surely' business... we went over this before and you still seem to be missing the point. This claim about the possibility existing with a vanishingly small probability is only true if the function I provided( f(x)=0*x+c) involves a random variable defined over a field with an uncountably large number of elements(like the reals or the complex numbers) if the distribution is defined over a finite field or a countably infinite field(like the rationals or the natural numbers) then the probability of the outcome is exactly 0. Thus my counter example is definitely correct if defined over a finite field, like the non-negative integers modulo 7.

#2 Since evolution involves a discrete number of individuals, it will never correspond to the idealized models of the reals or the complex numbers. It will always exhibit behavior over a finite field. So your objection above, specifically doesn't apply to evolution.

#3 This ignores the fact that a probability distribution over the reals with a dirac delta in it, is not random at all, but simply not-deterministic or not probable. If a model involves a significant random variable the term that is often used technically is stochastic. Stochastic is not freely interchangeable with random. Models with random components may be Stochastic, but not all Stochastic models have random components. The components may be probabilistic, but behave in ways that deviate significantly from random.

[...]

Could you show me where I made the claim (with in that discussion or the entire thread) "that a system that took a random variable as an input could always have any output"?

Note: I have edited the posts to keep them as on-topic for this thread as possible. I urge anyone who is interested to review the entire thread for the appropriate context.

 

[zosima]

I should note in the quotes Mijo presented, what you saw me doing is talking a lot of smack. Which is generally what one does when they've made no progress in an argument for 700+ posts. Mijo seems to infer from my claim that 'he is wrong', that I made sort of specific mathematical claim.

Could you show me where I made the claim (with in that discussion or the entire thread) "that a system that took a random variable as an input could always have any output"?

Note: I have edited the posts to keep them as on-topic for this thread as possible. I urge anyone who is interested to review the entire thread for the appropriate context.

I thought you'd never ask.
Here are the highlights:

That is a distinction without a difference. Something that is non-random is deterministic and vice versa.

You, like sol invictus, are relying on uncertainty in the initial conditions to declare that my definition is meaningless. However, my point is that, in the case of a stochastic process, each possible value in the distribution of initial condition yields at least two distinct outcome, not just one, as would be the case with a deterministic system.

No, you didn't actually read what I wrote or what I referred to (which seems to be a huge problem with those who argue that evolution is non-random.

A sure event (e.g., hitting the dart board universe) is deterministic, because one, and only one, outcome exists). An almost sure event (e.g., hitting a specific point or line the dart board universe) is random, because strictly more than one outcome exists. You can never not hit the dart board universe (deterministic event), but you could get really lucky and hit a specific point or line on the dart board universe (random event)

Do try and actually read before you post.

Note that in the example above, when Mijo says 'strictly more than one event' he is including events that have exactly 0 probability in his conception.

zosima-
The problem is that statistical hypothesis testing (even in maximum parsimony methods, especially when the number of taxa is above 8) used in evolutionary biology has a much wider usage of random that the aforementioned one, and it is therefore inconsistent to state that evolution by natural selection is non-random while making the assumptions of randomness necessary to perform any number of statistical hypothesis tests.

Uh....the other definitions by which evolutionary biologists claim evolution is not random are not consistent with the understanding of randomness needed to meaningfully practice statistics and to use statistical analysis to demonstrate that evolution does occur.

The problem is that you are assuming that there is only one correct usage of 'random variable' and 'random process', that mathematical definitions are applicable to non-mathematical objects, and that anyone who uses the the word 'random' outside of a mathematical context is somehow being inconsistent. Something can be a random process in one sense and not a random process in another, with no contradiction. Moreover, something can be a random process and yet still not random. This confusion results from conflation of the sense of the term from one field to a field where it does not directly apply.

So it seems like we've finally resolved the 1000+ post evolution thread, in a completely different thread. The definition of random process you apply is only meaningful in the context of mathematics. Insofar as evolution is not a mathematical entity* you were incorrect to apply that definition to evolution. That probably explains why 30-40 people told you exactly that 30-40 times each. So I guess my statements that you were incorrect were not incorrect after all. I wouldn't actually recommend anyone spend that much time reading the evolution thread. It is an awful hive of skum and trollery, much like the Mos Eisely spaceport. (Unless you really feel like verifying our statements. You'll find I'm understating its awfulness)

*even though some aspects of it may involve mathematics.

 

[mijopaalmc]

wrong forum