The Principle of Indifference, Discrete, and Continuous Distributions and the 6 / not 6 die.
Bertrand's Paradox is an example that is cited to show that the Principle of Indifference doesn't work for continuous distributions. The dogma is that there are multiple ways to choose "uniform randomness" with continuous distributions and that the various ways aren't invariant in assigning probability. For example the random radius and the random angle methods yield different results.
The dogma also is that discrete distributions are well behaved and you can apply the Principle of Indifference without problem. But are the discrete and continuous cases so different?
Let's say I have a normal 6 sized die spotted 1 to 6. What is the probability that I get a 6?
I can of course write down all the numbers thrown so so I get $53261352315...$ and so on. Since there are six digits each digit gets 1/6 probability.
But I could also write $bar6bar6bar66bar6bar66bar6...$ Now there are just two possible states and each should get 1/2.
But common sense... Well wait! What if the die was marked with a $6$ on one face and $bar6$ on the other five? Who is to say that I shouldn't just look at the indication on the face of the die?
It just doesn't seem to me that the different between the discrete case and the continuous case is so "obviously obvious to the casual observer" anymore. In both cases if you don't use the "right" probability distributions for the real world system that you're trying to model then you get burned.
What am I missing?
probability
asked Nov 26 at 19:40
MaxW
651411
|
show 4 more comments
Bertrand's Paradox is an example that is cited to show that the Principle of Indifference doesn't work for continuous distributions. The dogma is that there are multiple ways to choose "uniform randomness" with continuous distributions and that the various ways aren't invariant in assigning probability. For example the random radius and the random angle methods yield different results.
The dogma also is that discrete distributions are well behaved and you can apply the Principle of Indifference without problem. But are the discrete and continuous cases so different?
Let's say I have a normal 6 sized die spotted 1 to 6. What is the probability that I get a 6?
I can of course write down all the numbers thrown so so I get $53261352315...$ and so on. Since there are six digits each digit gets 1/6 probability.
But I could also write $bar6bar6bar66bar6bar66bar6...$ Now there are just two possible states and each should get 1/2.
But common sense... Well wait! What if the die was marked with a $6$ on one face and $bar6$ on the other five? Who is to say that I shouldn't just look at the indication on the face of the die?
It just doesn't seem to me that the different between the discrete case and the continuous case is so "obviously obvious to the casual observer" anymore. In both cases if you don't use the "right" probability distributions for the real world system that you're trying to model then you get burned.
What am I missing?
probability
asked Nov 26 at 19:40
MaxW
651411
With the six-sided die, there is a natural physical interpretation of what uniform means. Perhaps less so for the question "will there be a sunrise tomorrow?" or for the chords in Bertrand's paradox
– Henry
Nov 26 at 19:47
What's the question?
– Mike Hawk
Nov 26 at 20:19
@MikeHawk - Ok, fair enough. I added question "What am I missing?" to the post.
– MaxW
Nov 26 at 20:35
@Henry - Yes, but "natural" was supposed to be unambiguous in the discrete case. So for my die with $6$ & $bar6$ it seems "natural" to just write down what I see. I have to analyze the problem more deeply to figure out that the probability is 1/6, which is exactly where Bertand's chord paradox gets you.
– MaxW
Nov 26 at 20:40
The natural interpretation @Henry had in mind is for the sides themselves to be equally likely to land face up. If they're not, what makes one side more likely? The fact you drew a $6$ on it?
– J.G.
Nov 26 at 23:09
|
show 4 more comments
Bertrand's Paradox is an example that is cited to show that the Principle of Indifference doesn't work for continuous distributions. The dogma is that there are multiple ways to choose "uniform randomness" with continuous distributions and that the various ways aren't invariant in assigning probability. For example the random radius and the random angle methods yield different results.
The dogma also is that discrete distributions are well behaved and you can apply the Principle of Indifference without problem. But are the discrete and continuous cases so different?
Let's say I have a normal 6 sized die spotted 1 to 6. What is the probability that I get a 6?
I can of course write down all the numbers thrown so so I get $53261352315...$ and so on. Since there are six digits each digit gets 1/6 probability.
But I could also write $bar6bar6bar66bar6bar66bar6...$ Now there are just two possible states and each should get 1/2.
But common sense... Well wait! What if the die was marked with a $6$ on one face and $bar6$ on the other five? Who is to say that I shouldn't just look at the indication on the face of the die?
It just doesn't seem to me that the different between the discrete case and the continuous case is so "obviously obvious to the casual observer" anymore. In both cases if you don't use the "right" probability distributions for the real world system that you're trying to model then you get burned.
What am I missing?
probability
asked Nov 26 at 19:40
MaxW
651411
Bertrand's Paradox is an example that is cited to show that the Principle of Indifference doesn't work for continuous distributions. The dogma is that there are multiple ways to choose "uniform randomness" with continuous distributions and that the various ways aren't invariant in assigning probability. For example the random radius and the random angle methods yield different results.
The dogma also is that discrete distributions are well behaved and you can apply the Principle of Indifference without problem. But are the discrete and continuous cases so different?
Let's say I have a normal 6 sized die spotted 1 to 6. What is the probability that I get a 6?
I can of course write down all the numbers thrown so so I get $53261352315...$ and so on. Since there are six digits each digit gets 1/6 probability.
But I could also write $bar6bar6bar66bar6bar66bar6...$ Now there are just two possible states and each should get 1/2.
But common sense... Well wait! What if the die was marked with a $6$ on one face and $bar6$ on the other five? Who is to say that I shouldn't just look at the indication on the face of the die?
It just doesn't seem to me that the different between the discrete case and the continuous case is so "obviously obvious to the casual observer" anymore. In both cases if you don't use the "right" probability distributions for the real world system that you're trying to model then you get burned.
What am I missing?
probability
probability
asked Nov 26 at 19:40
MaxW
651411
asked Nov 26 at 19:40
MaxW
651411
edited Nov 27 at 7:35
asked Nov 26 at 19:40
MaxW
651411
asked Nov 26 at 19:40
MaxW
651411
asked Nov 26 at 19:40
MaxW
651411
651411
With the six-sided die, there is a natural physical interpretation of what uniform means. Perhaps less so for the question "will there be a sunrise tomorrow?" or for the chords in Bertrand's paradox
– Henry
Nov 26 at 19:47
What's the question?
– Mike Hawk
Nov 26 at 20:19
@MikeHawk - Ok, fair enough. I added question "What am I missing?" to the post.
– MaxW
Nov 26 at 20:35
@Henry - Yes, but "natural" was supposed to be unambiguous in the discrete case. So for my die with $6$ & $bar6$ it seems "natural" to just write down what I see. I have to analyze the problem more deeply to figure out that the probability is 1/6, which is exactly where Bertand's chord paradox gets you.
– MaxW
Nov 26 at 20:40
The natural interpretation @Henry had in mind is for the sides themselves to be equally likely to land face up. If they're not, what makes one side more likely? The fact you drew a $6$ on it?
– J.G.
Nov 26 at 23:09
|
show 4 more comments
With the six-sided die, there is a natural physical interpretation of what uniform means. Perhaps less so for the question "will there be a sunrise tomorrow?" or for the chords in Bertrand's paradox
– Henry
Nov 26 at 19:47
What's the question?
– Mike Hawk
Nov 26 at 20:19
@MikeHawk - Ok, fair enough. I added question "What am I missing?" to the post.
– MaxW
Nov 26 at 20:35
@Henry - Yes, but "natural" was supposed to be unambiguous in the discrete case. So for my die with $6$ & $bar6$ it seems "natural" to just write down what I see. I have to analyze the problem more deeply to figure out that the probability is 1/6, which is exactly where Bertand's chord paradox gets you.
– MaxW
Nov 26 at 20:40
The natural interpretation @Henry had in mind is for the sides themselves to be equally likely to land face up. If they're not, what makes one side more likely? The fact you drew a $6$ on it?
– J.G.
Nov 26 at 23:09
With the six-sided die, there is a natural physical interpretation of what uniform means. Perhaps less so for the question "will there be a sunrise tomorrow?" or for the chords in Bertrand's paradox
– Henry
Nov 26 at 19:47
With the six-sided die, there is a natural physical interpretation of what uniform means. Perhaps less so for the question "will there be a sunrise tomorrow?" or for the chords in Bertrand's paradox
– Henry
Nov 26 at 19:47
What's the question?
– Mike Hawk
Nov 26 at 20:19
What's the question?
– Mike Hawk
Nov 26 at 20:19
@MikeHawk - Ok, fair enough. I added question "What am I missing?" to the post.
– MaxW
Nov 26 at 20:35
@MikeHawk - Ok, fair enough. I added question "What am I missing?" to the post.
– MaxW
Nov 26 at 20:35
@Henry - Yes, but "natural" was supposed to be unambiguous in the discrete case. So for my die with $6$ & $bar6$ it seems "natural" to just write down what I see. I have to analyze the problem more deeply to figure out that the probability is 1/6, which is exactly where Bertand's chord paradox gets you.
– MaxW
Nov 26 at 20:40
@Henry - Yes, but "natural" was supposed to be unambiguous in the discrete case. So for my die with $6$ & $bar6$ it seems "natural" to just write down what I see. I have to analyze the problem more deeply to figure out that the probability is 1/6, which is exactly where Bertand's chord paradox gets you.
– MaxW
Nov 26 at 20:40
The natural interpretation @Henry had in mind is for the sides themselves to be equally likely to land face up. If they're not, what makes one side more likely? The fact you drew a $6$ on it?
– J.G.
Nov 26 at 23:09
The natural interpretation @Henry had in mind is for the sides themselves to be equally likely to land face up. If they're not, what makes one side more likely? The fact you drew a $6$ on it?
– J.G.
Nov 26 at 23:09
|
show 4 more comments
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With the six-sided die, there is a natural physical interpretation of what uniform means. Perhaps less so for the question "will there be a sunrise tomorrow?" or for the chords in Bertrand's paradox
– Henry
Nov 26 at 19:47
What's the question?
– Mike Hawk
Nov 26 at 20:19
@MikeHawk - Ok, fair enough. I added question "What am I missing?" to the post.
– MaxW
Nov 26 at 20:35
@Henry - Yes, but "natural" was supposed to be unambiguous in the discrete case. So for my die with $6$ & $bar6$ it seems "natural" to just write down what I see. I have to analyze the problem more deeply to figure out that the probability is 1/6, which is exactly where Bertand's chord paradox gets you.
– MaxW
Nov 26 at 20:40
The natural interpretation @Henry had in mind is for the sides themselves to be equally likely to land face up. If they're not, what makes one side more likely? The fact you drew a $6$ on it?
– J.G.
Nov 26 at 23:09