any examples of 'modeless' continuous probability distributions?

Originally posted by humy
I take that as a logical contradiction of the normal way we define 'up' and 'down' thus it is logically impossible.
I suppose you could arbitrary contrive a different definition of up and down which allows that without contradiction. But then we merely wouldn't agree with the same definition thus we would not be talking about exactly the same thing.

This is the main objection I have to the whole 'logical' / 'causal' divide. It seems that you can arbitrarily decide what is an agreed definition and what is not.

Originally posted by twhitehead
No, I am not.

Here is the scenario: you have a coin that has three possible symbols on it, 'heads','tails' or 'snakes'. What is the probability of throwing 'heads'? Answer: 1/3. And our sample space has three entries each with equal probability.
That is the prior probability.
Now I tell you that this particular coin does not have a 'snakes' symbol.
...[text shortened]... u would have that as undefined.

What is the probability of throwing a 'square circle' symbol?

And our sample space has three entries each with equal probability.
That is the prior probability.
Now I tell you that this particular coin does not have a 'snakes' symbol.
The posterior probability of throwing 'heads' is now 1/2.

No it isn't!
If 1/3 was a prior probability with three entries by application of the principle of indifference (which you must apply if it is to be a "prior probability" here), then 1/2 probability must be a prior probability, NOT a posterior probability, with two entries by application of the principle of indifference. And the fact you considered the 1/3 probability first is totally irrelevant! So the is no call for this "We are in a new world and ..."

Posterior probabilities come form updating prior probabilities (actually, my research as come up with an extremely subtle exception to that, but that is not relevant here).
But you haven't updated any probability! (at least not in the sense of updating to derive a posterior probability )
The world where the coin has three sides has 1/3 probability of that throw has nothing to do with how you derived the other 1/2 probability in that other world; what if the coin in that first world had four sides and thus that probability was 1/4? would it made any difference to that 1/2 probability in the second world? answer, no! so you haven't updated anything here to get a posterior probability thus that 1/2 is a prior probability. (except, as I said, my research as come up with an extremely subtle exception to that very type of inference. But that exception doesn't apply here and isn't relevant here! ).

Originally posted by twhitehead
This is the main objection I have to the whole 'logical' / 'causal' divide. It seems that you can arbitrarily decide what is an agreed definition and what is not.

NO! It isn't just a matter of definition!
I think the best way I can explain this is to talk in terms of "impossible" rather than "possible":

Exactly what makes something impossible can be different for different cases. consider:

flawless maths saying: 2 + 2 = 7

Clearly 2 + 2 = 7 is 'impossible' in flawless maths. But for what KIND of reason is it 'impossible'? Is it that we know from empirical observation that 2 + 2 = 7 is false? do we really NEED empirical observation to know that? Or can we know it is false merely by knowing there a logical contradiction in 2 + 2 = 7 ?
I would say the latter.

Now consider:

All birds are black.

clearly we know that is impossible to be true because we know from empirical observation that NOT all birds are black. But is the any deductive logical contradiction in all birds being black? What if we never observed a non-black bird? Can we deduce by considering possible logical contradictions that, actually, birds necessarily cannot all be black?
I don't think so.

So the two cases are impossible for two very different KINDS of reasons; the first one logically impossible and the second one causally impossible. This is not just a matter of "definition"; the difference between the two reasons is real and fundamental.

Failure to make the distinction between the two has resulted in many people thinking they have found a solution to the problem of induction when they haven't because they have equated the two kinds of 'possible'; By equivocating the two you can contract an argument that sounds correct but is logically flawed.
This is an extremely common error made in philosophy as my bother, who is a professor in philosophy, would tell you as he had many of his students make this error in their assays.

Originally posted by humy
No it isn't!
If 1/3 was a prior probability with three entries by application of the principle of indifference (which you must apply if it is to be a "prior probability" here), then 1/2 probability must be a prior probability, NOT a posterior probability, with two entries by application of the principle of indifference. And the fact you considered the 1/3 pr ...[text shortened]... bability first is totally irrelevant! So the is no call for this "We are in a new world and ..."

I realise there is room for confusion, but that too helps to highlight my point. If the sample space changes, then it has effects. You on the other hand are claiming that the sample space does not change. So if you were to do the scenario, you would keep the three possibilities in, and assign a zero probability to throwing snakes.

Posterior probabilities come form updating prior probabilities
Which fits better with your model where you would update the probability of throwing 'snakes' to zero. But I say it is better to change your sample space and make the probability of throwing 'snakes' undefined.

A different situation where where the sample space does not change would not violate either model. If I told you that 'snakes' are only half as likely as 'heads' then you would update the probabilities. What you would not do is consider what happens to the fourth symbol 'birds'.

Originally posted by humy
This is not just a matter of "definition"; the difference between the two reasons is real.

And I say it is not real. I say that in one you are arbitrarily ignoring some of the definitions in play. In the second case the impossibility arises because the world is defined as having white birds which logically contradicts the statement. That the definition of the world is imposed upon us is not that important.

Originally posted by twhitehead
I realise there is room for confusion, but that too helps to highlight my point. If the sample space changes, then it has effects. You on the other hand are claiming that the sample space does not change. So if you were to do the scenario, you would keep the three possibilities in, and assign a zero probability to throwing snakes.

[b]Posterior probabil ...[text shortened]... the probabilities. What you would not do is consider what happens to the fourth symbol 'birds'.

If the sample space changes, then it has effects.

Neither of your two sample spaces changed. The first one continued to have the snake on its coin while the second one continues without the snake on its coin.

You on the other hand are claiming that the sample space does not change.

I am saying neither of your two sample spaces changed.

So if you were to do the scenario, you would keep the three possibilities in, and assign a zero probability to throwing snakes.

in which of your two sample spaces? your first one with the snake or your second one without?

They are in two different worlds, right? So, unless you, say, randomly picked between the two worlds and you don't know which world you picked, you have two different sample spaces. Even if they are in the same world, they would be for two different coins thus, again, two different sample spaces unless, of course, you, say, arrange to randomly pick one of those two coins blindfolded and then tossed it; but then there would be zero probability (or undefined if you are right) of throwing a snake in the event that you had unknowingly picked the coin without the snake before throwing it. Say that happened; would the probability of throwing a snake still be undefined and left out your sample space? In that situation, probability of throwing a snake after you selected the coin would actually be zero (or undefined if you are right) but you wouldn't know that if you was blindfolded so what rational reason would you have for defining out the snake out of your sample space?

-but I admit, with that very last question, not meaning to be shifty or evasive, I think I may now be inadvertently subtly shifting the subject matter of our conversation here to a subtly different problem from the exact one you were actually just discussing.

Originally posted by humy
They are in two different worlds, right?

I appears I misunderstood what prior and posterior probabilities mean, so I could be wrong about a number of things.

I am not sure in your post what you are referring to where as I have presented several scenarios without proper numbering.

Lets try again:
Scenario 1)
Time a)
You have a coin.
You know for a fact that coins can only have three possible symbols on them, head, tails and snakes.
You do not know how frequent these symbols are or if any coins actually have these symbols. But probability theory tells us to apply the principle of indifference, so we assume randomness and estimate the probabilities to be 1/3 for each.
Time b)
New information comes to light. You discover that snake symbols are a myth and don't actually exist on coins. You have two ways of working out the new probability:
i) You can keep the sample space intact and assign a probability of zero to 'snakes'. You then divide the 1/3 from the snakes equally between the remaining possibilities and bring them up to 1/2.
ii) You create a new sample space without the snakes in and re apply the principle of indifference.

Your claim that zero probabilities are valid would seem to favour technique (i) . My claim is that it is not helpful and technique (ii) seems more sensible. You seemed to concur.

Scenario 2)
Time a)
As in scenario 1.
Time b)
You learn that snake symbols are only half as likely as heads or tails.
You do not change the sample space but you can recalculate the probabilities.

Scenario 3) (based on my new understanding of prior and posterior
Time a)
As in scenarios 1 and 2.
Time b)
You flip 100 coins and 50 turn up heads and 50 turn up tails.
You use probability theory to work out how likely it is that a new throw will turn up each of the three symbols. We have both prior probability and posterior probabilities.
The sample spaces still contain 3 symbols all of which are both logically and causally possible. None of them have probability zero.

Why is that you earlier claimed that:
"If that is true, there wouldn't be such thing as prior probability."

What you should include or exclude in your sample space depends in part on what you know.

Originally posted by twhitehead
I appears I misunderstood what prior and posterior probabilities mean, so I could be wrong about a number of things.

I am not sure in your post what you are referring to where as I have presented several scenarios without proper numbering.

Lets try again:
Scenario 1)
Time a)
You have a coin.
You know for a fact that coins can only have three po ...[text shortened]... you earlier claimed that:
"If that is true, there wouldn't be such thing as prior probability."

Why is that you earlier claimed that:
"If that is true ( i.e. before you can talk of probability, you must set out the rules of what is causally possible), there wouldn't be such thing as prior probability."

Remember, what is causally possible takes full account of what we know about natural law (such as the laws of physics ) but what is logically possible takes no account of natural law whatsoever so absolutely anything goes as long as it doesn't logically contradict itself (magic, Santa, tooth fairy ... )
So, to know what is causally possible, you have to first have knowledge of natural law. But You cannot ever gain knowledge of natural law without first having posterior probability since you cannot rationally assign the probability of any said natural law without first applying prior probabilities of theories to observations to obtain posterior probabilities of those theories that give reasonable degree of rational certainty that those laws are true.

But now which comes first for that? The prior probability of the posterior probability? Obviously, it cannot be the posterior probability because posterior probabilities came form prior probabilities (with an exception I know of but that is not relevant here ) . So it must be prior probabilities that come first. But then, if you cannot even rationally "talk of probability" without "the rules of what is causally possible" i.e. deciding what is causally possible, then not even prior probabilities can come first because prior probabilities alone without posterior probabilities don't give you knowledge of natural law and thus don't tell you what is causally possible; you need posterior probabilities for that. So, because of this chicken-and-egg problem, now you are left with no rational means to 'get-the-ball running' to ever even get to the start-post to rationally start to talk about ANY probability; and that includes prior probabilities.

Originally posted by twhitehead
I appears I misunderstood what prior and posterior probabilities mean, so I could be wrong about a number of things.

I am not sure in your post what you are referring to where as I have presented several scenarios without proper numbering.

Lets try again:
Scenario 1)
Time a)
You have a coin.
You know for a fact that coins can only have three po ...[text shortened]... you earlier claimed that:
"If that is true, there wouldn't be such thing as prior probability."

i) You can keep the sample space intact and assign a probability of zero to 'snakes'...
...
Your claim that zero probabilities are valid would seem to favour technique (i)

I now will logically contradicts myself here by contradicting part of what I said earlier but:
No, It would favour (ii) because we now know it is zero probability so it would be idiotic to continue having that now redundant superfluous snake outcome explicitly stated in our said sample space.

When I read read VERY carefully through
https://en.wikipedia.org/wiki/Sample_space
And other links, although I'm still not totally sure, I now think I was probably wrong about events with zero probability allowed in the sample space, since I cannot find anywhere there nor anywhere else even the slightest hint of a distinction made, in respect to sample spaces, of logically possible and causally possible.
If I was wrong then, strange that that real maths expert I spoke to totally agreed with me!

However, It makes no difference to the impossibility (such of tossing snakes in you example) having zero probability; So zero probability is not in any sample space; so what? That just means the probability of 0 exists outside any sample space.
And the domain of my OP function, NOT to be confused with the sample space of my OP function, contains x=0 and f(0)=0 indicating correctly that x=0 has zero probability, not undefined probability. The fact (if it is a fact ) that 0 is not in the sample space of my function would be irrelevant!

So the maths expert was right about;

"Events that are impossible have probability zero. "

But I think he probably made a mistake when he said:

"There is no requirement that events in a sample space have nonzero probability."

-perhaps he mixed up in his head the difference between 'sample space' and 'domain'?

Originally posted by humy
So, to know what is causally possible, you have to first have knowledge of natural law. But You cannot ever gain knowledge of natural law without first having posterior probability since you cannot rationally assign the probability of any said natural law without first applying prior probabilities of theories to observations to obtain posterior probabilities of those theories that give reasonable degree of rational certainty that those laws are true.

I disagree.
First some basic definitions.
Correct me if I am wrong, so that we are on the same page:
My understanding is that a prior probability exists as an assumed probability of events based on partial knowledge. We then run some experiments that essentially test the probability. We combine our initial estimate with the results of the experiment and come up with a new estimate we call the posterior probability.
Note:
This mechanism can never ever result in certainty that something is impossible. At best it can lead one to believe that something is highly improbable.

Now, suppose I say I have a coin that has 'heads' on one side and 'tails' on the other. It has no snakes. If I throw the coin, 'snakes' will be an impossible result.
Your description above seems to suggest that I know that snakes is an impossible result through my knowledge of natural laws and that my knowledge of natural laws was obtained via an extensive repetitive process starting with the assumption that everything is random and then making observations about the frequency of events and using Bays theorem or other probability calculations to keep obtaining new posterior probabilities.

I say no. I say that the impossibility of 'snakes' has nothing whatsoever to do with natural law but is merely part of the question, a definition if you will. No real coins were observed or harmed in the making of the question. I say that throwing 'snakes' is a logical impossibility in my situation.

In the case of the work tunnelling to the centre of the earth, you may have a case. My belief that it cannot is based largely on observation and knowledge of tunnelling worms, but if this is so then I cannot rightly assign a probability of exactly zero for it tunnelling to the centre of the earth. It is not actually impossible, just improbable. It may even be more probable than the chance of picking 42 at random from the real numbers.

Originally posted by humy
And the domain of my OP function, NOT to be confused with the sample space of my OP function, contains x=0 and f(0)=0 indicating correctly that x=0 has zero probability, not undefined probability.

My understanding is that f is supposed to be a function that maps on to the sample space. Yours does not. I cannot find a definition that supports such usage. At best I found mention of mapping the empty set (translated as a non-event, or not flipping the coin) to a zero probability.

Originally posted by twhitehead
I disagree.
First some basic definitions.
Correct me if I am wrong, so that we are on the same page:
My understanding is that a prior probability exists as an assumed probability of events based on partial knowledge. We then run some experiments that essentially test the probability. We combine our initial estimate with the results of the experiment an ...[text shortened]... ble. It may even be more probable than the chance of picking 42 at random from the real numbers.

My understanding is that a prior probability exists as an assumed probability of events based on partial knowledge.

The 'loose' meaning of prior probability, which is unfortunately the one it is conventionally given, is that it is the probability before you have looked at some evidence, not necessarily before you have looked at ALL the previous evidence in the past before looking at that new evidence. Unfortunately that often causes confusing because that means that probability is a prior probability in respect to that new evidence but posterior probability in respect to any old evidence before observing that new evidence.

To avoid that confusion, I had been giving 'prior probability' a less conventional 'strict' meaning of prior probability where it means the probability before you have looked at ANY evidence relevant to the theory; because only that strict 'true prior probability' tell you nothing about what is or is not causally possible.

Perhaps we can both agree to, for now on, call that 'strict' prior probability a 'true prior', to avoid possible confusion.

We then run some experiments that essentially test the probability.

Was it tested before in the past? Do we take into account prior knowledge we have of the actual external world that was gained by past observations?

We combine our initial estimate with the results of the experiment and come up with a new estimate we call the posterior probability.

Was the said 'prior probability' a posterior probability in respect to knowledge gained earlier in the past?

This mechanism can never ever result in certainty that something is impossible.

Not ABSOLUTE certainty. But if we simply say we cannot ever have certainty, period; we would then be in the uncomfortable position that we must reject ALL scientific facts (because we cannot know for a 'fact' that it is impossible for a given theory to be possible to be true ) except those that are just tautologies such as mathematical facts.

Now, suppose I say I have a coin that has 'heads' on one side and 'tails' on the other. It has no snakes. If I throw the coin, 'snakes' will be an impossible result.

LOGICALLY impossible, therefore causally impossible, yes.

Your description above seems to suggest that I know that snakes is an impossible result ...

LOGICALLY impossible, therefore causally impossible, yes.

through my knowledge of natural laws ...

NO, through it being LOGICALLY impossible! so natural laws have nothing to do with it here!
It is LOGICALLY impossible to select out of 'heads' or 'tails' 'snake'.

I say no. I say that the impossibility of 'snakes' has nothing whatsoever to do with natural law

EXACTLY!
THEREFORE it isn't impossible because it violates natural law but rather because it is LOGICALLY impossible!

I say that throwing 'snakes' is a logical impossibility in my situation.

You understand!

Originally posted by humy
The 'loose' meaning of prior probability, which is unfortunately the one it is conventionally given, is that it is the probability before you have looked at some evidence,

It doesn't seem either loose or unfortunate to me. It seems like the most useful definition as it is the sort of situation that will be most often encountered in probability calculations.

To avoid that confusion, I am here giving 'prior probability' a less conventional 'strict' meaning of prior probability where it means the probability before you have looked at ANY evidence relevant to the theory; because only that strict 'true prior probability' tell you nothing about what is or is not causally possible.
I think that that not only will cause confusion as it is a non-standard definition, but I will try to keep it in mind when you talk of prior probability.

THEREFORE it isn't impossible because it violates natural law but rather because it is LOGICALLY impossible!
That would seem to contradict the definitions presented earlier for logically possible vs causally possible. Some of your references said 'if you can imagine it then it is logically possible' or 'if a world can exist in which it is possible, then it is logically possible'.
I can imagine a world in which coins have snakes on them.

Originally posted by humy
Not ABSOLUTE certainty. But if we simply say we cannot ever have certainty, period; we would then be in the uncomfortable position that we must reject ALL scientific facts (because we cannot know for a 'fact' that it is impossible for a given theory to be possible to be true ) except those that are just tautologies such as mathematical facts.

We should not strictly call them 'facts'. Nor should we reject them, but accept that we do not have absolute certainty of their validity.
So earlier in the thread when you talked of something having a probability of absolute zero, were you allowing for the kind of uncertainty here or not?
In your OP, does the zero refer to the kind of uncertainty here? ie at x=0 has the possibility of it happening been ruled out by repeated experiments? If I recall correctly you said it was ruled out by definition, similar to my coins examples. In your overtaking cars example it was ruled out by definition. So it was logically impossible to have a car overtake another with a relative velocity of zero. I still say that assigning a probability of zero to such an event is incoherent.