Is probability verifiable?
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29-12-2013, 01:16 AM
Is probability verifiable?
This is my very first post ...

As far as I know, there are two kinds of interpretation about probability, the frequentist version and the bayesian version. The question I have here is not which side is right, they both have their applications, and in my opinion this is just a matter of perspective.

Take the birthday 'paradox' as an example, what does the probability of there is at least one pair of people with the same birthday really mean? How do we verify that probability? For example, if n=23, then the probability is 50.7%. Could we, for instance, take many samples of groups of 23 people and count the frequency of groups having such a pair of people? Alternatively, in the bayesian sense, since we are using that probability to represent our belief, how do we justify it?
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29-12-2013, 09:06 AM
RE: Is probability verifiable?
(29-12-2013 01:16 AM)womendezuguo Wrote:  This is my very first post ...

As far as I know, there are two kinds of interpretation about probability, the frequentist version and the bayesian version. The question I have here is not which side is right, they both have their applications, and in my opinion this is just a matter of perspective.

Take the birthday 'paradox' as an example, what does the probability of there is at least one pair of people with the same birthday really mean? How do we verify that probability? For example, if n=23, then the probability is 50.7%. Could we, for instance, take many samples of groups of 23 people and count the frequency of groups having such a pair of people? Alternatively, in the bayesian sense, since we are using that probability to represent our belief, how do we justify it?

The birthday example is not at all Bayesian. It is a simple calculation of known probabilities. In principle, we could take many groups of 23 people and test it.

Are you limiting Bayesian probability to belief? That is the subjectivist view. I lean toward the objectivist view.

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30-12-2013, 08:17 AM
RE: Is probability verifiable?
(29-12-2013 09:06 AM)Chas Wrote:  
(29-12-2013 01:16 AM)womendezuguo Wrote:  This is my very first post ...

As far as I know, there are two kinds of interpretation about probability, the frequentist version and the bayesian version. The question I have here is not which side is right, they both have their applications, and in my opinion this is just a matter of perspective.

Take the birthday 'paradox' as an example, what does the probability of there is at least one pair of people with the same birthday really mean? How do we verify that probability? For example, if n=23, then the probability is 50.7%. Could we, for instance, take many samples of groups of 23 people and count the frequency of groups having such a pair of people? Alternatively, in the bayesian sense, since we are using that probability to represent our belief, how do we justify it?

The birthday example is not at all Bayesian. It is a simple calculation of known probabilities. In principle, we could take many groups of 23 people and test it.

Are you limiting Bayesian probability to belief? That is the subjectivist view. I lean toward the objectivist view.

I think this footnote on the wiki page answers my question about the birthday problem pretty well:
http://en.wikipedia.org/wiki/Birthday_pr...irthdays-3

Also, is the objectivist view a consequence of Cox's Theorem? And, if we take the subjectivist view, how do we verify the results of our calculations?

I came across this question after I learned about the BCJR decoding scheme recently which use probabilities to represent the posteriori probabilities of the value of each message bit. The math makes sense, but I had the feeling that probability and statistics by and large are somewhat contrived and less intuitive than say real analysis or algebra.
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30-12-2013, 08:59 AM
RE: Is probability verifiable?
(29-12-2013 01:16 AM)womendezuguo Wrote:  This is my very first post ...

As far as I know, there are two kinds of interpretation about probability, the frequentist version and the bayesian version. The question I have here is not which side is right, they both have their applications, and in my opinion this is just a matter of perspective.

Take the birthday 'paradox' as an example, what does the probability of there is at least one pair of people with the same birthday really mean? How do we verify that probability? For example, if n=23, then the probability is 50.7%. Could we, for instance, take many samples of groups of 23 people and count the frequency of groups having such a pair of people? Alternatively, in the bayesian sense, since we are using that probability to represent our belief, how do we justify it?

For small sample sizes there would be quite a bit of variance (after all 50% is still a large margain) but as the number of samples approaches infinity it would pair out that about 50% of the samples contained two people with the same birthday.

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13-01-2014, 01:13 PM
RE: Is probability verifiable?
(30-12-2013 08:17 AM)womendezuguo Wrote:  
(29-12-2013 09:06 AM)Chas Wrote:  The birthday example is not at all Bayesian. It is a simple calculation of known probabilities. In principle, we could take many groups of 23 people and test it.

Are you limiting Bayesian probability to belief? That is the subjectivist view. I lean toward the objectivist view.

I think this footnote on the wiki page answers my question about the birthday problem pretty well:
http://en.wikipedia.org/wiki/Birthday_pr...irthdays-3

Also, is the objectivist view a consequence of Cox's Theorem? And, if we take the subjectivist view, how do we verify the results of our calculations?

I came across this question after I learned about the BCJR decoding scheme recently which use probabilities to represent the posteriori probabilities of the value of each message bit. The math makes sense, but I had the feeling that probability and statistics by and large are somewhat contrived and less intuitive than say real analysis or algebra.

I have met someone with whom I shared the same birth day and year.

That is, someone else BESIDES my twin (who really shouldn't count WRT this "problem"...

But really, all one would have to know is the exact number of people born on the same day as them, and divide that into the world population. You might have to fiddle a little bit with estimates of attrition rates and maybe some other detail stuff, I reckon, but it should give a pretty good ball-park figure.

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14-01-2014, 12:14 AM
RE: Is probability verifiable?
Yes, it is verifiable.

Give me your argument in the form of a published paper, and then we can start to talk.
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14-01-2014, 11:08 PM
RE: Is probability verifiable?
Maybe I was using a too narrow definition of "verifiable", that is, verifiable or provable mathematically.
Although I still get an uneasy feeling when I see some of the calculations in engineering papers that deliberately ignore some terms or when I was offered a mere intuitive explanation for results that I believe should have a mathematically sound argument. Many people seems to be willing to make the jump that if a method works in many situations it will likely to work elsewhere even when there is no mathematical guarantee.
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15-01-2014, 02:43 AM
RE: Is probability verifiable?
It's verifiable mathematically and in practice. What meaning of verifiable is left to add after you have done both of these things?

Give me your argument in the form of a published paper, and then we can start to talk.
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16-01-2014, 04:26 AM (This post was last modified: 16-01-2014 04:50 AM by womendezuguo.)
RE: Is probability verifiable?
(15-01-2014 02:43 AM)Hafnof Wrote:  It's verifiable mathematically and in practice. What meaning of verifiable is left to add after you have done both of these things?

I think I agree, maybe the birthday problem was a bad example. Also, I was confusing approximations with exact results.

However, although many probabilistic methods are indeed mathematically sound, but there are always assumptions about the data that may not apply, or maybe the method may not work in certain circumstances, but is not specified how (much) it fails or why.

One example is when using neural networks to learn about complicated features in machine learning, there is always the danger of getting a local optimum. Although random initialisation and maybe other numerical hacks can reduce the chances, we are still given no guarantee and even when we do get the global optimum, we can't know for sure.

But I guess many of my problems are mainly theoretical, and it is always possible and necessary to do testing before applying a theoretical result in practice. Also, many theoretical concerns are really not so important in practice. People in academia and engineering sometimes worry about completely different things.
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16-01-2014, 08:04 AM
RE: Is probability verifiable?
(16-01-2014 04:26 AM)womendezuguo Wrote:  
(15-01-2014 02:43 AM)Hafnof Wrote:  It's verifiable mathematically and in practice. What meaning of verifiable is left to add after you have done both of these things?

I think I agree, maybe the birthday problem was a bad example. Also, I was confusing approximations with exact results.

However, although many probabilistic methods are indeed mathematically sound, but there are always assumptions about the data that may not apply, or maybe the method may not work in certain circumstances, but is not specified how (much) it fails or why.

One example is when using neural networks to learn about complicated features in machine learning, there is always the danger of getting a local optimum. Although random initialisation and maybe other numerical hacks can reduce the chances, we are still given no guarantee and even when we do get the global optimum, we can't know for sure.

But I guess many of my problems are mainly theoretical, and it is always possible and necessary to do testing before applying a theoretical result in practice. Also, many theoretical concerns are really not so important in practice. People in academia and engineering sometimes worry about completely different things.

That is specifically stochastic modeling and doesn't necessarily reflect on probability theory as a whole.

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