ISF Logo   IS Forum
Forum Index Register Members List Events Mark Forums Read Help

Go Back   International Skeptics Forum » General Topics » Science, Mathematics, Medicine, and Technology
 


Welcome to the International Skeptics Forum, where we discuss skepticism, critical thinking, the paranormal and science in a friendly but lively way. You are currently viewing the forum as a guest, which means you are missing out on discussing matters that are of interest to you. Please consider registering so you can gain full use of the forum features and interact with other Members. Registration is simple, fast and free! Click here to register today.
Reply
Old 2nd August 2018, 09:12 AM   #201
SOdhner
Graduate Poster
 
Join Date: Apr 2010
Location: Arizona
Posts: 1,725
Originally Posted by Porpoise of Life View Post
I have a sneaking suspicion Fudbucker is going to use this thread to argue that there's a 50/50 chance of there being a Creator...
Nah, he's arguing for him being a Boltzmann Brain. The thing is, even if this is resolved since he's refusing to consider alternatives it's invalid. Really once you start asking this question it's not "which of these two hotels am I in" because it's not just two options. If I'm taking Boltzmann Brains seriously I have an infinite number of other things it could be as well. So not only are there infinite rooms in the hotels but there's infinite hotels.
SOdhner is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 2nd August 2018, 08:08 PM   #202
psionl0
Skeptical about skeptics
 
psionl0's Avatar
 
Join Date: Sep 2010
Location: 31°57'S 115°57'E
Posts: 12,947
Originally Posted by Darat View Post
Slightly disagree with how you put that. The assumption shouldn't be that's it 50/50 when you have no information, you shouldn't make any assumptions as to the probabilities as it is for you a total guess. You simply don't know.
I was going to ask "how about if you toss a coin?" since you don't have any information on which to base a guess.

However, I don't think I need to go that far. The sample space is 2 hotels and you have to select one of them.
__________________
"The process by which banks create money is so simple that the mind is repelled. Where something so important is involved, a deeper mystery seems only decent." - Galbraith, 1975
psionl0 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 2nd August 2018, 11:49 PM   #203
GlennB
Loggerheaded, earth-vexing fustilarian
 
GlennB's Avatar
 
Join Date: Sep 2006
Location: Arcadia, Greece
Posts: 23,303
Originally Posted by psionl0 View Post
I was going to ask "how about if you toss a coin?" since you don't have any information on which to base a guess.

However, I don't think I need to go that far. The sample space is 2 hotels and you have to select one of them.
Yes, but you know the distribution of types of room in those hotels, then wake up in a room of a certain type. When that information was somehow discounted as useful I lost track of the analysis here.
__________________
"Even a broken clock is right twice a day. 9/11 truth is a clock with no hands." - Beachnut
GlennB is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 2nd August 2018, 11:51 PM   #204
Darat
Lackey
Administrator
 
Darat's Avatar
 
Join Date: Aug 2001
Location: South East, UK
Posts: 82,536
Originally Posted by psionl0 View Post
I was going to ask "how about if you toss a coin?" since you don't have any information on which to base a guess.

However, I don't think I need to go that far. The sample space is 2 hotels and you have to select one of them.
That works when you know there are only 2 choices, in the general case my comment was about you wouldn't know there are only 2 hotels to choose from. This goes back to the where this thread came from, trying to claim there is a 50/50 chance of us being a Boltzmann brain or the "external" world exists. The point being is that we don't know those are the only two possible choices.
__________________
I wish I knew how to quit you
Darat is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 12:21 AM   #205
psionl0
Skeptical about skeptics
 
psionl0's Avatar
 
Join Date: Sep 2010
Location: 31°57'S 115°57'E
Posts: 12,947
Originally Posted by GlennB View Post
Yes, but you know the distribution of types of room in those hotels, then wake up in a room of a certain type. When that information was somehow discounted as useful I lost track of the analysis here.
"That information" became discountable when it was revealed that each hotel had an infinite number of rooms and we realized that we had NO information about how the rooms were grouped nor how a room was selected.

Originally Posted by Darat View Post
That works when you know there are only 2 choices, in the general case my comment was about you wouldn't know there are only 2 hotels to choose from.

That information was the first thing given in the OP.
Originally Posted by Fudbucker View Post
The only clue as to my whereabouts is the fact that I know I'm in one of two sets of infinite hotel rooms:
__________________
"The process by which banks create money is so simple that the mind is repelled. Where something so important is involved, a deeper mystery seems only decent." - Galbraith, 1975
psionl0 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 01:29 AM   #206
Dave Rogers
Bandaged ice that stampedes inexpensively through a scribbled morning waving necessary ankles
 
Dave Rogers's Avatar
 
Join Date: Jan 2007
Location: Cair Paravel, according to XKCD
Posts: 27,169
Originally Posted by GlennB View Post
Yes, but you know the distribution of types of room in those hotels, then wake up in a room of a certain type. When that information was somehow discounted as useful I lost track of the analysis here.
The problem is that you don't know the distribution of types of rooms if the total is infinite. That's the bit I didn't get at first. The statement "There are an infinite number of rooms, of which one in a billion is opulent and the rest are dingy" is in fact meaningless starting from the words "of which"; the apparent proportion can be changed in an infinite set just by ordering it differently.

If the question had been phrased as "The receptionist in the Motel 6 places every billionth customer in an opulent room, and the rest in dingy rooms; in the Ritz it's the other way round," then the question would have made sense (and, in fact, I read it as if it had been phrased that way). But that requires knowledge of how one came to be in the room, which the original question is trying to exclude.

Dave
__________________
Me: So what you're saying is that, if the load carrying ability of the lower structure is reduced to the point where it can no longer support the load above it, it will collapse without a jolt, right?

Tony Szamboti: That is right
Dave Rogers is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 02:26 AM   #207
Darat
Lackey
Administrator
 
Darat's Avatar
 
Join Date: Aug 2001
Location: South East, UK
Posts: 82,536
Originally Posted by psionl0 View Post
...snipo...



That information was the first thing given in the OP.
May come as a shock to you but not every single post in a thread may be discussing a point in the opening post at all times....
__________________
I wish I knew how to quit you
Darat is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 03:58 AM   #208
psionl0
Skeptical about skeptics
 
psionl0's Avatar
 
Join Date: Sep 2010
Location: 31°57'S 115°57'E
Posts: 12,947
Originally Posted by Darat View Post
May come as a shock to you but not every single post in a thread may be discussing a point in the opening post at all times....
__________________
"The process by which banks create money is so simple that the mind is repelled. Where something so important is involved, a deeper mystery seems only decent." - Galbraith, 1975
psionl0 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 05:29 AM   #209
Clive
Critical Thinker
 
Join Date: Dec 2008
Posts: 367
Can you see an invisible pink unicorn if it's standing in front of you in broad daylight, is about the same size as a horse, and your eyesight is 20/20?

What is the difference between a hotel with an infinite number of rooms and an invisible pink unicorn?

What is the probability of god existing if an infinite force can move an immovable object?

Etc.
Clive is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 05:37 AM   #210
Roboramma
Penultimate Amazing
 
Roboramma's Avatar
 
Join Date: Feb 2005
Location: Shanghai
Posts: 10,903
We should never consider the implications of infinities, the square root of a negative number, negative numbers themselves, irrational numbers (how can you have a fraction that can't be expressed as a ratio of whole numbers???) zero, etc.
__________________
"... when people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together."
Isaac Asimov
Roboramma is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 07:00 AM   #211
caveman1917
Philosopher
 
Join Date: Feb 2015
Posts: 5,840
Originally Posted by GlennB View Post
Yes, but you know the distribution of types of room in those hotels, then wake up in a room of a certain type. When that information was somehow discounted as useful I lost track of the analysis here.
Originally Posted by Darat View Post
That works when you know there are only 2 choices, in the general case my comment was about you wouldn't know there are only 2 hotels to choose from. This goes back to the where this thread came from, trying to claim there is a 50/50 chance of us being a Boltzmann brain or the "external" world exists. The point being is that we don't know those are the only two possible choices.
Originally Posted by psionl0 View Post
"That information" became discountable when it was revealed that each hotel had an infinite number of rooms and we realized that we had NO information about how the rooms were grouped nor how a room was selected.
Originally Posted by Dave Rogers View Post
The problem is that you don't know the distribution of types of rooms if the total is infinite. That's the bit I didn't get at first. The statement "There are an infinite number of rooms, of which one in a billion is opulent and the rest are dingy" is in fact meaningless starting from the words "of which"; the apparent proportion can be changed in an infinite set just by ordering it differently.

If the question had been phrased as "The receptionist in the Motel 6 places every billionth customer in an opulent room, and the rest in dingy rooms; in the Ritz it's the other way round," then the question would have made sense (and, in fact, I read it as if it had been phrased that way). But that requires knowledge of how one came to be in the room, which the original question is trying to exclude.

Dave
In the general case you choose the maximum entropy distribution given certain constraints (those constraints would be the information you have, so in case you have no information there are no constraints).

Given no constraints, in the case of a finite universe the maximum entropy distribution is uniform, each outcome is equally likely. This gets us P(Ritz) = P(Motel 6) = 0.5. In the case of an uncountably infinite universe the maximum entropy distribution is also uniform. In the case of a countably infinite universe though there is no maximum entropy distribution.[*] Hence why we can't assign a distribution over the rooms, but we can assign a distribution over the hotels. In other words, the problem is not that we have no information but that the specific case of "no information + countably infinite universe" doesn't give us a maximum entropy distribution.

Compare with W.D. Clinger's argument that we could get a uniform distribution if the rooms were finite or uncountably infinite but not if they are countably infinite. Slightly more precise would be that we're not necessarily looking for a uniform distribution as such but for the maximum entropy distribution, which just happens to be uniform under no constraints (though might not be uniform if there are constraints).

* For 3 points: Prove that there does not exist a maximum entropy distribution (under no constraints) over a countably infinite universe.
__________________
"Ideas are also weapons." - Subcomandante Marcos
"We must devastate the avenues where the wealthy live." - Lucy Parsons
"Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!" - Mikhail Bakunin

Last edited by caveman1917; 3rd August 2018 at 07:02 AM.
caveman1917 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 06:45 PM   #212
W.D.Clinger
Illuminator
 
W.D.Clinger's Avatar
 
Join Date: Oct 2009
Posts: 3,402
caveman1917 has raised a couple of interesting points, which I do not fully understand but would like to understand.

The technical parts of this post are encapsulated within spoilers.

Originally Posted by caveman1917 View Post
In the general case you choose the maximum entropy distribution given certain constraints (those constraints would be the information you have, so in case you have no information there are no constraints).

Given no constraints, in the case of a finite universe the maximum entropy distribution is uniform, each outcome is equally likely. This gets us P(Ritz) = P(Motel 6) = 0.5.
That much I understand.

Originally Posted by caveman1917 View Post
In the case of an uncountably infinite universe the maximum entropy distribution is also uniform.
Here I question whether entropy is even defined over arbitrary uncountable state spaces. I think we need to assume we can take integrals.


There are actually two issues here. Assuming the axiom of choice, all standard set theories imply an infinite well-ordering of increasingly larger uncountable cardinals. I suspect caveman1917 is using "uncountably infinite" to mean having the cardinality of the real numbers, and I will assume that is so throughout the rest of this post.

The second issue is the question of how entropy is defined over a state space having the cardinality of the real numbers. For a finite or countably infinite state space, I believe the entropy is defined as

S = - kBi pi log pi
where kB is Boltzmann's constant and pi is the probability of the ith configuration.

Summations over a countably infinite set don't necessarily converge, but the summation above must converge because all of the probabilities are between 0 and 1, inclusive, and they must add up to 1.

When we move to an uncountably infinite state space whose cardinality is that of the real numbers, I believe we must replace that summation by an integral:

S = - kB ∫ px log px dx
Integrals are not defined over arbitrary uncountable sets, so we must make an additional assumption: that the integral given above is well-defined. That will be true if the state space is the real numbers and px is integrable, but it can also be true if the state space is the compact interval I = [0,1], or the Cartesian product I2 of that interval with itself, and so on. Some of those spaces are so weird that physicists seldom consider them as a possibility.

Furthermore the integral is itself defined using some particular measure. In probability theory, as in most areas of mathematics and physics, we usually use Lebesgue measure, but Lebesgue measure is defined only on Euclidean spaces and on similar spaces (such as some manifolds). Lebesgue measure is not defined on all uncountable state spaces, nor is it even defined on all state spaces whose cardinality is that of the real numbers. In particular, Lebesgue measure is not defined on uncountable sets of hotel rooms.

So I'd like to understand how caveman1917 is defining the entropy of probability distributions whose state space is an uncountable set of hotel rooms.


Even if we restrict our attention to Lebesgue-integrable state spaces and probability distributions, I suspect the maximal entropy distribution is not well-defined, if it exists at all.


Consider, for example, the entire real number line using Lebesgue measure. I can believe that all compact subsets of the real numbers have a maximal entropy distribution, but I don't see how that could be true for the entire real number line:

∫ px dx = 1 = ∑nnn+1 px dx
from which it follows that ∫nn+1 px dx is greater for some intervals than for others. If we have two intervals of unit length for which that integral is different, we can define a similar probability distribution that differs from the original only because the probability distributions of those two intervals are exchanged. If the original distribution has maximal entropy, then so does the distribution obtained by exchanging the probabilities for those two intervals, which proves there is no unique distribution with maximal entropy.

The next spoiler proves there is no maximal entropy distribution for a countably infinite state space. I believe that proof is easily modified to show there is no maximal entropy distribution for the real number line, using the same idea as in the previous paragraph.


Originally Posted by caveman1917 View Post
In the case of a countably infinite universe though there is no maximum entropy distribution.[*]
...snip...
* For 3 points: Prove that there does not exist a maximum entropy distribution (under no constraints) over a countably infinite universe.


With a countably infinite state space, the probabilities pi cannot all be equal: If all the pi were zero, they wouldn't add up to 1 as required by the laws of probability. If all the pi were equal to some positive value c, then they would add up to more than 1, again contrary to the laws of probability.

So there must exist some pair of probabilities pi and pj with pi < pj. Replacing those two probabilities with their average increases the entropy, so the original probability distribution did not have maximal entropy.
W.D.Clinger is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 09:04 PM   #213
caveman1917
Philosopher
 
Join Date: Feb 2015
Posts: 5,840
Originally Posted by W.D.Clinger View Post
Here I question whether entropy is even defined over arbitrary uncountable state spaces. I think we need to assume we can take integrals.


There are actually two issues here. Assuming the axiom of choice, all standard set theories imply an infinite well-ordering of increasingly larger uncountable cardinals. I suspect caveman1917 is using "uncountably infinite" to mean having the cardinality of the real numbers, and I will assume that is so throughout the rest of this post.

The second issue is the question of how entropy is defined over a state space having the cardinality of the real numbers. For a finite or countably infinite state space, I believe the entropy is defined as

S = - kBi pi log pi
where kB is Boltzmann's constant and pi is the probability of the ith configuration.

Summations over a countably infinite set don't necessarily converge, but the summation above must converge because all of the probabilities are between 0 and 1, inclusive, and they must add up to 1.

When we move to an uncountably infinite state space whose cardinality is that of the real numbers, I believe we must replace that summation by an integral:

S = - kB ∫ px log px dx
Integrals are not defined over arbitrary uncountable sets, so we must make an additional assumption: that the integral given above is well-defined. That will be true if the state space is the real numbers and px is integrable, but it can also be true if the state space is the compact interval I = [0,1], or the Cartesian product I2 of that interval with itself, and so on. Some of those spaces are so weird that physicists seldom consider them as a possibility.

Furthermore the integral is itself defined using some particular measure. In probability theory, as in most areas of mathematics and physics, we usually use Lebesgue measure, but Lebesgue measure is defined only on Euclidean spaces and on similar spaces (such as some manifolds). Lebesgue measure is not defined on all uncountable state spaces, nor is it even defined on all state spaces whose cardinality is that of the real numbers. In particular, Lebesgue measure is not defined on uncountable sets of hotel rooms.

So I'd like to understand how caveman1917 is defining the entropy of probability distributions whose state space is an uncountable set of hotel rooms.
I was indeed using uncountable in the sense of the cardinality of the reals, not cardinalities above it. Specifically I meant uncountable universe where "universe" means base set of a probability space, so we can assume integrability of p etc. It is true that that none of those things are defined over an uncountable set of hotel rooms but I was assuming the existence of a map from the set of hotel rooms to some space which has the required properties, such as a real interval.

For the entropy, just replacing the sum with an integral doesn't work, it stops being invariant under coordinate transformations. A better form is

S = -∫ p(x) log(p(x)/m(x)) dx

where m plays the role of the measure, a function proportional to the limiting density of discrete points when extending the discrete case to the continuous case. See Jaynes 1957 for the specific procedure of taking such limit. Note here that we could write the entropy for the finite case similarly as

S = -∑i pi log (pi/mi)

where mi is a constant which factors out.

But we don't actually need the entropy, we're just looking for the maximum entropy distribution so all we really need is an ordering of distributions according to entropy. Which gives us for the discrete case

K(p, q) = ∑i pi log (pi/qi)

and for the continuous case

K(p, q) = ∫ p(x) log(p(x)/q(x)) dx

It's interesting to note here that the absolute entropy is a special case of the relative entropy with q being uniform under some conditions. In the discrete case let

qi = mi / ∑i mi

where q is uniform since m is constant. In the continuous case let

q(x) = m(x) / ∫ m(x) dx

where m(x) is the measure. Both of these only work under the conditions that in the discrete case the set is finite and in the continuous case that the total measure is finite. In both these cases q is the uniform distribution, which is the maximum entropy distribution under no constraints, so we can interpret the absolute entropy of a distribution as the relative entropy to the least-constrained maximum entropy distribution.

Quote:
Even if we restrict our attention to Lebesgue-integrable state spaces and probability distributions, I suspect the maximal entropy distribution is not well-defined, if it exists at all.


Consider, for example, the entire real number line using Lebesgue measure. I can believe that all compact subsets of the real numbers have a maximal entropy distribution, but I don't see how that could be true for the entire real number line:

∫ px dx = 1 = ∑nnn+1 px dx
from which it follows that ∫nn+1 px dx is greater for some intervals than for others. If we have two intervals of unit length for which that integral is different, we can define a similar probability distribution that differs from the original only because the probability distributions of those two intervals are exchanged. If the original distribution has maximal entropy, then so does the distribution obtained by exchanging the probabilities for those two intervals, which proves there is no unique distribution with maximal entropy.

The next spoiler proves there is no maximal entropy distribution for a countably infinite state space. I believe that proof is easily modified to show there is no maximal entropy distribution for the real number line, using the same idea as in the previous paragraph.
A maximum entropy distribution needn't always exist, but it will not just depend on the support of the distribution but also on the constraints. For example, as you say, there is no maximum entropy distribution over the reals with no constraints. There is also no maximum entropy distribution over the natural numbers with no constraints. But there can be maximum entropy distributions over those spaces when there are constraints, see for example here. Either way, if it exists it is well-defined through the expressions for relative entropy above.

It's interesting that it is specifically the cases with an infinite measure (interpreting the m from above as the measure in both the discrete and continuous case) where neither the absolute entropy can be seen as a relative entropy nor where an unconstrained maximum entropy distribution exists.

Quote:

With a countably infinite state space, the probabilities pi cannot all be equal: If all the pi were zero, they wouldn't add up to 1 as required by the laws of probability. If all the pi were equal to some positive value c, then they would add up to more than 1, again contrary to the laws of probability.

So there must exist some pair of probabilities pi and pj with pi < pj. Replacing those two probabilities with their average increases the entropy, so the original probability distribution did not have maximal entropy.
Well yes, obviously you can do it, it was more meant as a possibly interesting problem for others in the thread to get themselves started in probability theory. The 3 points are in the sense of "3 points if this question was posed on a freshman probability theory exam rated on a total of 20 points."
__________________
"Ideas are also weapons." - Subcomandante Marcos
"We must devastate the avenues where the wealthy live." - Lucy Parsons
"Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!" - Mikhail Bakunin

Last edited by caveman1917; 3rd August 2018 at 10:53 PM.
caveman1917 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd August 2018, 10:12 PM   #214
caveman1917
Philosopher
 
Join Date: Feb 2015
Posts: 5,840
Huh, I guess that in a way it is then true that the problem here is infinity, but not in the sense of cardinality but in the sense of measure. Infinite measure + no information = problem (no maximum entropy distribution), in both the discrete and the continuous case.
__________________
"Ideas are also weapons." - Subcomandante Marcos
"We must devastate the avenues where the wealthy live." - Lucy Parsons
"Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!" - Mikhail Bakunin
caveman1917 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 03:41 AM   #215
W.D.Clinger
Illuminator
 
W.D.Clinger's Avatar
 
Join Date: Oct 2009
Posts: 3,402
Originally Posted by caveman1917 View Post
I was indeed using uncountable in the sense of the cardinality of the reals, not cardinalities above it. Specifically I meant uncountable universe where "universe" means base set of a probability space, so we can assume integrability of p etc. It is true that that none of those things are defined over an uncountable set of hotel rooms but I was assuming the existence of a map from the set of hotel rooms to some space which has the required properties, such as a real interval.

...snipped for brevity...
I appreciate your explanation. Thank you.

Originally Posted by Porpoise of Life View Post
I have a sneaking suspicion Fudbucker is going to use this thread to argue that there's a 50/50 chance of there being a Creator...
Originally Posted by theprestige View Post
Context matters. I'm fairly confident Fudbucker is working on an Answer of the Gaps that meets a rhetorical goal he's pursuing in other threads.
Originally Posted by SOdhner View Post
Nah, he's arguing for him being a Boltzmann Brain. The thing is, even if this is resolved since he's refusing to consider alternatives it's invalid. Really once you start asking this question it's not "which of these two hotels am I in" because it's not just two options. If I'm taking Boltzmann Brains seriously I have an infinite number of other things it could be as well. So not only are there infinite rooms in the hotels but there's infinite hotels.

For those who may be wondering what all this technical stuff means for Fudbucker's original question and for any arguments that might be based on that or similar questions: It means game theory and Bayes' Theorem simply cannot be applied to infinite hotel rooms and similar problems in the absence of information or (typically unjustified) assumptions about the a priori probabilities or probability measure. As caveman1917 summarized:

Originally Posted by caveman1917 View Post
Huh, I guess that in a way it is then true that the problem here is infinity, but not in the sense of cardinality but in the sense of measure. Infinite measure + no information = problem (no maximum entropy distribution), in both the discrete and the continuous case.
W.D.Clinger is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 04:00 AM   #216
Darat
Lackey
Administrator
 
Darat's Avatar
 
Join Date: Aug 2001
Location: South East, UK
Posts: 82,536
Glad to see you agree with what I posted!
__________________
I wish I knew how to quit you
Darat is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 04:55 AM   #217
W.D.Clinger
Illuminator
 
W.D.Clinger's Avatar
 
Join Date: Oct 2009
Posts: 3,402
I think I need to respond to this, lest someone take my lack of response as support for Darat's claim.

Originally Posted by Darat View Post
Glad to see you agree with what I posted!
Darat's claim that game theory cannot deal with infinities was incorrect.

Game theory is applicable when we have relevant information. As has been pointed out several times, it was at first possible to interpret Fudbucker's original question as asking us to assume certain a priori probabilities, which would have allowed a straightforward solution using game theory despite the infinity of rooms.

In subsequent posts, Fudbucker appears to have clarified his question in ways that suggest to me he did not intend for us to assume the a priori probabilities that would have made his question answerable. It has been suggested that Fudbucker did so because he wanted to make a game-theoretic argument in some other thread where a priori probabilities were clearly not available. I have no opinion on that.
W.D.Clinger is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 07:09 AM   #218
Darat
Lackey
Administrator
 
Darat's Avatar
 
Join Date: Aug 2001
Location: South East, UK
Posts: 82,536
Originally Posted by W.D.Clinger View Post
I think I need to respond to this, lest someone take my lack of response as support for Darat's claim.


Darat's claim that game theory cannot deal with infinities was incorrect.

Game theory is applicable when we have relevant information. As has been pointed out several times, it was at first possible to interpret Fudbucker's original question as asking us to assume certain a priori probabilities, which would have allowed a straightforward solution using game theory despite the infinity of rooms.

In subsequent posts, Fudbucker appears to have clarified his question in ways that suggest to me he did not intend for us to assume the a priori probabilities that would have made his question answerable. It has been suggested that Fudbucker did so because he wanted to make a game-theoretic argument in some other thread where a priori probabilities were clearly not available. I have no opinion on that.
Deary me - some people really do need to get a sense of humour - and of course keep track of a discussion in a thread.....
__________________
I wish I knew how to quit you
Darat is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 09:29 AM   #219
caveman1917
Philosopher
 
Join Date: Feb 2015
Posts: 5,840
Originally Posted by W.D.Clinger View Post
I appreciate your explanation. Thank you.
I'm wondering how much the idea can be generalized. If m and n are sigma-finite measures on some measurable space, with m absolutely continuous with respect to n, then the Radon-Nikodym derivative exists and we can define a similar relative entropy

K(m, n) = ∫ log(dm/dn) dm

which, if it exists, gives us an ordering again. So it seems that it can be generalized to any collection of sigma-finite measures which are absolutely continuous with respect to each other.

ETA: But both the counting measure over the integers and the Lebesgue measure over the reals are sigma-finite, and any probability measure p is going to be absolutely continuous with respect to them, so I must be missing something here as to why it won't work in those cases.

ETA2: Of course it works just fine in those cases as well, it just happens to be the case that there is no maximal element in our ordered set of distributions. Silly me.
__________________
"Ideas are also weapons." - Subcomandante Marcos
"We must devastate the avenues where the wealthy live." - Lucy Parsons
"Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!" - Mikhail Bakunin

Last edited by caveman1917; 4th August 2018 at 09:39 AM.
caveman1917 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 11:15 AM   #220
caveman1917
Philosopher
 
Join Date: Feb 2015
Posts: 5,840
I think I'm starting to wrap my head around it. Let M = (X, ∑) be some measurable space. Earlier on I was talking about the unconstrained maximum entropy distribution p which was correct for probability distributions since p(X) = 1 by definition of a probability space, and is hence not a constraint as such. If we however consider a general measure space then the total measure needn't be equal to 1 and hence p(X) = 1 becomes a constraint.

Let μ be a sigma-finite base measure on M, which will usually be the counting measure in the discrete case and the Lebesgue measure in the continuous case, but it could be any measure. Let's extend the notion of a maximum entropy distribution to a maximum entropy measure as per my previous post. Let S be the collection of sigma-finite measures on M including μ which are pair-wise absolutely continuous with respect to each other. Then, assuming the necessary integrals exist, we do get an unconstrained maximum entropy measure ν ∈ S - up to proportionality - in all cases.

The problem is then that if μ(X) is infinite then so will ν(X) and it hence can't be normalized to a probability measure. So the unconstrained maximum entropy measure does always exist, and is equal - up to proportionality - to the counting measure or Lebesgue measure as the case may be. Or differently, it is the constraint that ν(X) = k for some strictly positive k which stops the maximum entropy measure from existing if μ(X) is infinite, but that is a necessary constraint in order to be able to normalize ν to a probability measure.

ETA: interesting that this seems to give us a justification for why we conventionally use the counting measure or Lebesgue measure, they happen to be the maximum entropy measure.
__________________
"Ideas are also weapons." - Subcomandante Marcos
"We must devastate the avenues where the wealthy live." - Lucy Parsons
"Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!" - Mikhail Bakunin

Last edited by caveman1917; 4th August 2018 at 11:26 AM.
caveman1917 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 01:53 PM   #221
Dabbler
New Blood
 
Join Date: Jul 2018
Posts: 3
Enjoyed reading this thread and seeing everyone's perspectives.

If I've learned anything from these types of problems, it's that whenever you see wordings such as "you wake up in uncertain conditions/with amnesia, what is the probability of xxxx".......then run for the hills! (Sleeping Beauty problem, anyone?)

Most of these the problems seem to be ill-defined, and the OP problem appears to fall into this category. Particularly, as noted in Post #147 by W.D.Clinger, the sentence

"In the Motel 6 set, for every opulent room, there are a billion dingy ones."


is ambiguous. It needs to be made more clear. It could be saying the order of the countably infinite* rooms goes opulent, followed by a billion dingy, followed by opulent, followed by a billion dingy, etc, which is not to be confused with merely saying "for every opulent room, there are a billion dingy ones". In the case of sequential ordering, it makes more sense to conclude P(opulent|Motel 6) could be something like (1/1,000,000,001), since if we assume that somehow you could find yourself in a particular equally-likely random room, with a set room number, then you could be certain that the set of room numbers containing your room number, and the billion following numbers (which should be equally likely under this assumption, however impossible this assumption is), contains exactly one opulent number- translating the infinite to a finite case, so to speak. Or bypassing all this, perhaps the highlighted phrase above, as W.D.Clinger
noted, could simply be defined as expressing the conditional probabilities in question.

*Some are talking about this being a hidden assumption, and indeed it is, but I would argue from the language of the problem, this appears warranted. "Infinite hotel" brings to mind infinite hallways of sequential rooms, or infinite floors, etc. If someone is in a particular room, there would be a set "next" room next door (there is no next real number after 1). But anyway we could just add this as an assumption....yet another reason why vague physical impossibilities dont translate well to mathematical problems.

Perhaps the Motel 6 owner could have a policy to simply put all the guests in opulent rooms, in which case P(opulent|Motel 6) could be 1, but this of course would be additional information unknown to the guest from the given info, and so wouldnt factor into their credence.

The question as given in the OP cant be answered clearly because it is not well-defined, but I suppose the best way to estimate probabilities involved would be to somehow repeat the experiment over and over again (just pretend it could be), and record the relative frequency of times people are right/wrong when saying Motel 6/Ritz, after waking up in an opulent room (empirical definition of probability).

Last edited by Dabbler; 4th August 2018 at 02:27 PM.
Dabbler is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 02:06 PM   #222
Dabbler
New Blood
 
Join Date: Jul 2018
Posts: 3
Originally Posted by psionl0 View Post
It is not right to talk about the "size" of a countably infinite set. All you can say is that there is a 1 to 1 correspondence between the elements of a countably infinite set and the set of counting numbers (1, 2, 3, . . .).
Originally Posted by This is The End View Post
This is correct. It makes no sense mathematically to say they have the same "size".
We can simply define

"A has the same size as B" as

"there exists a one-to-one correspondence between the elements of A and B".

Mathematically, then, it would make sense to talk about infinite sets having the same size. I think someone brought it up earlier, but the above definition fits nicely with our notion of "same size" of finite sets. If there are two collections of books, each containing 7 books, I can conclude these collections are the same size by pairing off books (exactly) from each collection. Or I can take the longer route and count the number of books in each (which is really just using the one-to-one correspondence notion between the books and the counting numbers, or say my fingers, but you are doing it twice)
Dabbler is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 05:54 PM   #223
W.D.Clinger
Illuminator
 
W.D.Clinger's Avatar
 
Join Date: Oct 2009
Posts: 3,402
Welcome to the ISF, Dabbler.

Originally Posted by Dabbler View Post
We can simply define

"A has the same size as B" as

"there exists a one-to-one correspondence between the elements of A and B".

Mathematically, then, it would make sense to talk about infinite sets having the same size.
You have stated that definition exactly as it is stated and used by logicians and mathematicians.

When we want to be stuffy or need to be careful, as when there is some other notion of size under discussion (such as measure), we say two sets have the same cardinality or are in one-to-one correspondence. Most of the time, though, we just say they have the same size.

If we are willing to assume the axiom of choice, we can prove a law of trichotomy for that notion of cardinality: either two sets A and B are the same size, or one of the sets is strictly smaller than the other (i.e. A has the same cardinality as a proper subset of B but does not have the same cardinality as B). If we don't want to assume the axiom of choice or any equivalent of it, then we have to face the possibility that A and B may be incomparable in size, much as you can have two unequal sets where neither is a subset of the other.

Last edited by W.D.Clinger; 4th August 2018 at 05:59 PM. Reason: added missing comma
W.D.Clinger is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th August 2018, 09:23 PM   #224
psionl0
Skeptical about skeptics
 
psionl0's Avatar
 
Join Date: Sep 2010
Location: 31°57'S 115°57'E
Posts: 12,947
Originally Posted by Dabbler View Post
We can simply define

"A has the same size as B" as

"there exists a one-to-one correspondence between the elements of A and B".
The problem is that when the word "size" is used, the definition isn't included so a conversation about the "size" of infinity gets confusing fast.

Better to stick with "cardinality".
__________________
"The process by which banks create money is so simple that the mind is repelled. Where something so important is involved, a deeper mystery seems only decent." - Galbraith, 1975
psionl0 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 9th August 2018, 02:06 PM   #225
jt512
Graduate Poster
 
Join Date: Sep 2011
Posts: 1,738
Originally Posted by Fudbucker View Post
Now, how does that apply to the subjective probability that you are in the Ritz? Suppose someone puts a gun to your head and says, "Make a guess which hotel you're in". What's the optimum strategy? 50/50?
Assuming for the moment that the condition of your room provides no information about the hotel you are in, then the optimum betting strategy would use your prior probabilities.
jt512 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 9th August 2018, 07:43 PM   #226
whoanellie
Critical Thinker
 
Join Date: Apr 2012
Posts: 326
Originally Posted by psionl0 View Post
You can't calculate probabilities where infinities are concerned because dividing infinity by infinity doesn't make sense.
Although it is not fashionable today, frequentist statistics and the frequentist interpretation of probability explicitly assume there is some (real or imagined) infinite set of events/experiments/measurements. Infinity and probability are not incompatible. By definition, if the true ratios are 1:1e9 and 1e9:1 as stated in the original post then those are the ratios that one would observe as the 1 person in one of an infinite number of hotel rooms was repeated a very large number of times (approaching infinity).
whoanellie is online now   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 10th August 2018, 12:09 AM   #227
jt512
Graduate Poster
 
Join Date: Sep 2011
Posts: 1,738
Whoanellie, you need additional assumption to do either frequentist or Bayesian statistics. Usually one would assume that each sample is an independent and identically distributed Bernoulli random variable. This assumption obviates the need to put a distribution on the infinite set itself.
jt512 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 10th August 2018, 02:24 AM   #228
psionl0
Skeptical about skeptics
 
psionl0's Avatar
 
Join Date: Sep 2010
Location: 31°57'S 115°57'E
Posts: 12,947
Originally Posted by whoanellie View Post
By definition, if the true ratios are 1:1e9 and 1e9:1 as stated in the original post then those are the ratios that one would observe as the 1 person in one of an infinite number of hotel rooms was repeated a very large number of times (approaching infinity).
A large number which, in the limit, approaches infinity is not the same as infinity itself.

As a matter of fact, if the hotels actually had infinity rooms each then the stated ratios are meaningless.
__________________
"The process by which banks create money is so simple that the mind is repelled. Where something so important is involved, a deeper mystery seems only decent." - Galbraith, 1975
psionl0 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 10th August 2018, 07:55 AM   #229
whoanellie
Critical Thinker
 
Join Date: Apr 2012
Posts: 326
Originally Posted by jt512 View Post
Whoanellie, you need additional assumption to do either frequentist or Bayesian statistics. Usually one would assume that each sample is an independent and identically distributed Bernoulli random variable. This assumption obviates the need to put a distribution on the infinite set itself.
I am not assuming anything. I am simply asserting that probability does make sense even when one is dealing with infinities. That's how frequentist statistics works. Frequentist statistics does not require an assumption of a Bernoulli variable.

Last edited by whoanellie; 10th August 2018 at 07:56 AM. Reason: add link
whoanellie is online now   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 10th August 2018, 08:04 AM   #230
whoanellie
Critical Thinker
 
Join Date: Apr 2012
Posts: 326
Originally Posted by whoanellie View Post
I am not assuming anything. I am simply asserting that probability does make sense even when one is dealing with infinities. That's how frequentist statistics works. Frequentist statistics does not require an assumption of a Bernoulli variable.
Originally Posted by psionl0 View Post
A large number which, in the limit, approaches infinity is not the same as infinity itself.

As a matter of fact, if the hotels actually had infinity rooms each then the stated ratios are meaningless.
There are more than one way to think of the size of a countable infinity. The concept of natural density is relevant to this discussion.
whoanellie is online now   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 10th August 2018, 08:09 AM   #231
JoeMorgue
Self Employed
Remittance Man
 
JoeMorgue's Avatar
 
Join Date: Nov 2009
Location: Florida
Posts: 11,956
Originally Posted by whoanellie View Post
There are more than one way to think of the size of a countable infinity.
There's an infinite number of ways to think about the size of any infinite from your friend and mine the Aleph-Naught to infinite nested uncountable infinite sets of infinities.

Because infinity isn't a number. It doesn't have a size. It's a concept defined by it's undefinablity.

The size of infinity is like the corner of a circle. It doesn't have one.

All discussions of infinity run into the Supertask Paradox in one form or another; give us an infinite concept and ask us what happens at the end of it.
__________________
"Ernest Hemingway once wrote that the world is a fine place and worth fighting for. I agree with the second part." - Detective Sommerset, Se7en
JoeMorgue is online now   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 10th August 2018, 09:17 AM   #232
jt512
Graduate Poster
 
Join Date: Sep 2011
Posts: 1,738
Originally Posted by whoanellie View Post
I am not assuming anything. I am simply asserting that probability does make sense even when one is dealing with infinities. That's how frequentist statistics works. Frequentist statistics does not require an assumption of a Bernoulli variable.
You miunderstand frequentist probability. As psion alluded to, p = limit as n approaches infinity of p(n) does not imply that p is defined at infinity. As numerous people in this thread have explained, a proportion of a countably infinite set is meaningless.

What I was getting at is that you can construct an infinite sequence (of say hypothetical hotel rooms) Xi, i=1,2,... , by treating the Xi as iid Bernoulli(p) random variables. Then, no matter which Xi (hotel room) you choose (wake up in), there is a probability p that Xi=1 (is opulent). So constructed, the sequence in a sense emulates a set with proportion p of elements that have the characreristic of interest even though the concept of proportion is undefined. Note, too, that this construction circumvents the problem of it being impossible to impose a uniform distribution on the set.

Last edited by jt512; 10th August 2018 at 09:58 AM.
jt512 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 10th August 2018, 06:59 PM   #233
whoanellie
Critical Thinker
 
Join Date: Apr 2012
Posts: 326
Originally Posted by jt512 View Post
As psion alluded to, p = limit as n approaches infinity of p(n) does not imply that p is defined at infinity. As numerous people in this thread have explained, a proportion of a countably infinite set is meaningless.
This is where the concept of natural density comes in handy.

see also https://www.rug.nl/research/portal/f...8949/02_c2.pdf
whoanellie is online now   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 11th August 2018, 01:23 AM   #234
jt512
Graduate Poster
 
Join Date: Sep 2011
Posts: 1,738
Whoanellie, thanks, but the only relevant question is whether you can back up your claim rigorously. You wrote:

Originally Posted by whoanellie View Post
Although it is not fashionable today, frequentist statistics and the frequentist interpretation of probability explicitly assume there is some (real or imagined) infinite set of events/experiments/measurements. Infinity and probability are not incompatible. By definition, if the true ratios are 1:1e9 and 1e9:1 as stated in the original post then those are the ratios that one would observe as the 1 person in one of an infinite number of hotel rooms was repeated a very large number of times (approaching infinity).

Please state your sample space S, your sigma-algebra F of subsets of S, and a probability function that would enable the repeated sampling you talk about above. If you think „natural density“ is helpful with this feel free to incorporate it.


Sent from my iPhone using Tapatalk
jt512 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 11th August 2018, 11:16 AM   #235
whoanellie
Critical Thinker
 
Join Date: Apr 2012
Posts: 326
Originally Posted by jt512 View Post
Whoanellie, thanks, but the only relevant question is whether you can back up your claim rigorously. You wrote:




Please state your sample space S, your sigma-algebra F of subsets of S, and a probability function that would enable the repeated sampling you talk about above. If you think „natural density“ is helpful with this feel free to incorporate it.


Sent from my iPhone using Tapatalk
I don't know anything about sigma-algebraa, but I don't believe they are relevant to my point. There are different interpretations of probability and it appears that one's perspective on the original question is largely determined by the interpretation of probability that one favors. The links I've posted above were intended to back up that point.
whoanellie is online now   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 11th August 2018, 01:33 PM   #236
caveman1917
Philosopher
 
Join Date: Feb 2015
Posts: 5,840
Originally Posted by whoanellie View Post
I don't know anything about sigma-algebraa, but I don't believe they are relevant to my point.
Please see the definition of a probability space. Without a probability space your point is not in probability theory, and you have a no-go I'm afraid.

Quote:
There are different interpretations of probability and it appears that one's perspective on the original question is largely determined by the interpretation of probability that one favors. The links I've posted above were intended to back up that point.
I've read your link on natural density, or rather read the first part and skimmed the rest. It basically goes like this:

Let N be the set of natural numbers, and let {A, B} be a partition of N such that both A and B are infinite. Then P(A) is defined as limn->inf |An| / (|An| + |Bn|) where Xn = {x in Nn | x in X} where Nn is the set of the first n natural numbers.

The problem here is in the meaning of "first n natural numbers" (the "n->inf") part. Consider my reordering argument as per earlier in the thread. Let A be the set of of natural numbers divisible by 1e9 and B the set of natural numbers not divisible by 1e9. Suppose I now reorder N such that I first take the first number in A, then the first number in B, then the next number in A, then the next number in B, and so on. If I then take your limit above I get P(A) = 0.5.

This might become clearer if we take a look at what's going on when we take that limit, we have a sequence of finite subsets of N with the limiting set being N such that each subset is a strict superset of the previous one:

{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, ... -> N

We then have a function P from finite subsets of N to the unit interval which gives us the fraction of numbers in that subset divisible by 1e9. We apply P to the sequence above:

P({1}) = 0, P({1, 2}) = 0, P({1, 2, 3}) = 0, P({1, 2, 3, 4}) = 0, ... -> P(N) = 1/1e9.

And then define P(N) by that limit and get a natural density of 1/1e9. But we can take such limit in many ways. Consider the following where di and ndi are the i-the natural number divisible or non-divisible by 1e9 respectively:

{d1}, {d1, nd1}, {d1, nd1, d2}, ... -> N

and apply P:

P({d1}) = 1, P({d1, nd1}) = 1/2, P({d1, nd1, d2}) = 2/3, ... -> P(N) = 1/2

and we get P(N) = 1/2.

There are many ways to approach N with a sequence of its subsets.

ETA: So that's not saying that you can't define P(N) in such a way, but there are plenty of other ways to do it which get different results. More importantly, in terms of the hotel rooms, I could change the result of the experiment by doing nothing other than switching the room numbering around a bit (ie use a different bijection between the set of rooms and the set of natural numbers, giving a different ordering on the set of rooms).
__________________
"Ideas are also weapons." - Subcomandante Marcos
"We must devastate the avenues where the wealthy live." - Lucy Parsons
"Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!" - Mikhail Bakunin

Last edited by caveman1917; 11th August 2018 at 02:13 PM.
caveman1917 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 11th August 2018, 02:50 PM   #237
caveman1917
Philosopher
 
Join Date: Feb 2015
Posts: 5,840
Originally Posted by whoanellie View Post
This is where the concept of natural density comes in handy.

see also https://www.rug.nl/research/portal/f...8949/02_c2.pdf
Originally Posted by caveman1917 View Post
Please see the definition of a probability space. Without a probability space your point is not in probability theory, and you have a no-go I'm afraid.
I read your second link (or rather, again, read the first part until I figured out what was going on and then skimmed the rest). And, as per my post above, it is indeed possible to define P(N) in various ways by changing the Kolmogorov axioms of probability or changing other conditions (which puts the solutions outside of probability theory proper). The problem isn't that it's impossible to define P(N) but that there are many ways to do it, leading to different results, with there not being clear reason to prefer one over another. And all of them have problems, for example the class of solutions you're proposing fails the reordering argument.

The experiment was someone waking up in a hotel room, looking at the wealth of the room, and then making a guess as to which hotel the subject is in. Suppose that during the night I go through hallway and do naught but switch the numbering of the rooms around, that should not change the guess made by the subject - yet in your solution(s) it does. Above Dabbler made the point that we can assume a physical succesor function of the hotel rooms, in that - given for example an infinite hallway of rooms - for each room there is, physically, a next room. But it fails the same problem, that during the night I could physically switch the rooms around, whether we consider it switching the room numbering around or physically switching the rooms around is immaterial to the point that this operation should not change the guess made by the subject when waking up and seeing the room wealth.

ETA: actually, if we define Motel 6 and Ritz as per the OP (distinguished by the natural density of opulent rooms to dingy ones - assuming that, indeed, natural density was meant in the ambiguous statement) then these hotels are indistinguishable expect for the ordering of their rooms. In a way they're the same hotel, they just chose to order their rooms differently. So if the guest isn't given information as to the ordering of the rooms, which doesn't seem to be the case when only given the wealth of his or her room, then the hotels are indistinguishable to the guest and the guess should be 50/50 (or whatever the prior probabilities were).
__________________
"Ideas are also weapons." - Subcomandante Marcos
"We must devastate the avenues where the wealthy live." - Lucy Parsons
"Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!" - Mikhail Bakunin

Last edited by caveman1917; 11th August 2018 at 03:29 PM. Reason: opulent isn't a colour property but a wealth property, learn some English every day :)
caveman1917 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 11th August 2018, 04:21 PM   #238
Fudbucker
Philosopher
 
Join Date: Jul 2012
Posts: 8,268
Under the conditions described in the OP, there should be some optimal strategy for choosing what to say if someone puts a gun to my head and asks me which hotel I'm in. If there is no strategy, then the default strategy would be to flip a coin.

Last edited by Fudbucker; 11th August 2018 at 04:26 PM.
Fudbucker is online now   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 11th August 2018, 05:14 PM   #239
W.D.Clinger
Illuminator
 
W.D.Clinger's Avatar
 
Join Date: Oct 2009
Posts: 3,402
Originally Posted by Fudbucker View Post
Under the conditions described in the OP, there should be some optimal strategy for choosing what to say if someone puts a gun to my head and asks me which hotel I'm in. If there is no strategy, then the default strategy would be to flip a coin.
Optimal under what conditions?

Optimal solutions/strategies don't always exist.

Consider, for example, the following problem: What real value for x maximizes f(x) subject to the constraint g(x) > 0? If we don't know what f(x) is, the question is unanswerable. In most cases, it is also unanswerable if we know f(x) but don't know g(x).

There has been some debate concerning whether your original post asked us to assume certain prior probabilities (in which case your question is easily answered) or told us only to assume certain correspondences (in which case your question has no answer, because the existence of those correspondences implies nothing about probabilities). There has also been some speculation that you did not want us to assume prior probabilities, because prior probabilities would not be available for an argument you are alleged to want to make in another thread. If you want us to continue to consider your question, you should give us a clear statement concerning what you want us to assume.
W.D.Clinger is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 11th August 2018, 05:56 PM   #240
caveman1917
Philosopher
 
Join Date: Feb 2015
Posts: 5,840
It's interesting that, the Ritz and Motel 6 being indistinguishable except for room ordering, and interpreting the room ordering as the receptionist's strategy for filling up the hotel as more and more guests arrive (giving us a specific sequence of subsets to take a limit on, namely the sequence of sets of occupied rooms as more and more guests arrive) then we can consider a similar problem.

Suppose there is only 1 hotel with an infinite hallway of rooms, for which there are two receptionists each with a different filling strategy. RRitz puts 1e9 - 1 guests in opulent rooms and then one guest in a dingy room, and rinse and repeat. RMotel 6 does the opposite. Given that you awoke in an opulent room, which receptionist was on call when you checked in?

All of these are basically the same thing: receptionist's filling strategy, room ordering, sequence of subsets of hotel rooms to take limits on.
__________________
"Ideas are also weapons." - Subcomandante Marcos
"We must devastate the avenues where the wealthy live." - Lucy Parsons
"Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!" - Mikhail Bakunin

Last edited by caveman1917; 11th August 2018 at 05:59 PM.
caveman1917 is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Reply

International Skeptics Forum » General Topics » Science, Mathematics, Medicine, and Technology

Bookmarks

Thread Tools

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Forum Jump


All times are GMT -7. The time now is 04:20 PM.
Powered by vBulletin. Copyright ©2000 - 2018, Jelsoft Enterprises Ltd.

This forum began as part of the James Randi Education Foundation (JREF). However, the forum now exists as
an independent entity with no affiliation with or endorsement by the JREF, including the section in reference to "JREF" topics.

Disclaimer: Messages posted in the Forum are solely the opinion of their authors.