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This article seems to be missing a great deal of description/explanation. It looks to me like this is a stub, with added examples. How do I put up a template (or something) to indicate that this article needs work? DonkeyKong the mathematician (in training) 02:24, 14 June 2006 (UTC)
User:Michael Hardy states that the article needs cleanup. Could you please be more specific? Bo Jacoby 15:28, 21 June 2006 (UTC)
I see what you mean. It seems to talk too much about deduction. The point is that the induction formula is a flexion of the deduction formula. So induction cannot be explained isolated from deduction. Perhaps the title should be changed. Bo Jacoby 08:25, 22 June 2006 (UTC)
I usually work on narrower topics, but I'm going to pay some attention to this page over the next few days. I've deleted material that appeared to say that frenquentist statistics relies ONLY on maximum likelihood estimation (for starters, this would actually suggest that frequentists don't use unbiased estimation!).
So I've marked this for cleanup again.
Wikipedia has not done nearly as well with statistics as with mathematics generally. Michael Hardy 18:15, 25 August 2006 (UTC)
I support this suggestion.
A possible lead-in on the statistics page could be:
(... discuss historical meaning of 'statistics' ....)
Nowadays, statistics generally means either
If the material cut from the article is added, perhaps it should be kept. Otherwise, merge it. Dr. Payne 18:06, 11 December 2006 (UTC)
This is an example of the latter [i.e. of Bayesian inference].
From a population containing N items of which I are special, a sample containing n items of which i are special can be chosen in
ways (see multiset and binomial coefficient).
Fixing (N,n,I), this expression is the unnormalized deduction distribution function of i.
Fixing (N,n,i) , this expression is the unnormalized induction distribution function of I.
The two most important parameters of a probability distribution are: the mean value and the standard deviation . The plus-minus sign, ± , is used to separate the mean from the deviation.
The mean value ± the standard deviation of the deduction distribution is used for estimating i knowing (N, n, I)
where a(b ± c) = ab ± ac. Note that f defines two functions of three variables.
Example: The population contains two items one of which is special, and the sample contains one item. (N, n, I) = (2, 1, 1) gives
confirming that the number of special items in the sample is either 0 or 1.
The mean value ± the standard deviation of the induction distribution is used for estimating I knowing (N,n,i)
where a+(b±c)=(a+b)±c.
Thus deduction is translated into induction by means of the involution
Example: The population contains a single item and the sample is empty. (N,n,i)=(1,0,0) gives
confirming that the number of special items in the population is either 0 or 1.
Note that the frequency probability solution to this problem is giving no meaning.
In the limiting case where N is a large number, the deduction distribution of i tends towards the binomial distribution with the probability as a parameter,
Example: The population is big, the probability , and the sample contains one item. n = 1 gives
confirming that the sample contains 0 or 1 special items, with equal probability.
In the limiting case where N is a large number, the induction distribution of tends towards the beta distribution
The frequency probability solution to this problem is . The probability is estimated by the relative frequency.
Example: The population is big and the sample is empty. n = i = 0 gives
The frequency probability solution to this problem is , giving no meaning.
In the limiting case where and are large numbers, the deduction distribution of i tends towards the poisson distribution with the intensity as a parameter,
Example: The population is big and the sample is big, and the intensity gives
In the limiting case where and are large numbers, the induction distribution of tends towards the gamma distribution with i as a parameter:
Example: The population is big and the sample is big but contains no special items. i = 0 gives
The frequency probability solution to this problem is which is misleading. Even if you have not been wounded you may still be vulnerable.
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content assessment scale. It is of interest to the following WikiProjects: | |||||||
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This article seems to be missing a great deal of description/explanation. It looks to me like this is a stub, with added examples. How do I put up a template (or something) to indicate that this article needs work? DonkeyKong the mathematician (in training) 02:24, 14 June 2006 (UTC)
User:Michael Hardy states that the article needs cleanup. Could you please be more specific? Bo Jacoby 15:28, 21 June 2006 (UTC)
I see what you mean. It seems to talk too much about deduction. The point is that the induction formula is a flexion of the deduction formula. So induction cannot be explained isolated from deduction. Perhaps the title should be changed. Bo Jacoby 08:25, 22 June 2006 (UTC)
I usually work on narrower topics, but I'm going to pay some attention to this page over the next few days. I've deleted material that appeared to say that frenquentist statistics relies ONLY on maximum likelihood estimation (for starters, this would actually suggest that frequentists don't use unbiased estimation!).
So I've marked this for cleanup again.
Wikipedia has not done nearly as well with statistics as with mathematics generally. Michael Hardy 18:15, 25 August 2006 (UTC)
I support this suggestion.
A possible lead-in on the statistics page could be:
(... discuss historical meaning of 'statistics' ....)
Nowadays, statistics generally means either
If the material cut from the article is added, perhaps it should be kept. Otherwise, merge it. Dr. Payne 18:06, 11 December 2006 (UTC)
This is an example of the latter [i.e. of Bayesian inference].
From a population containing N items of which I are special, a sample containing n items of which i are special can be chosen in
ways (see multiset and binomial coefficient).
Fixing (N,n,I), this expression is the unnormalized deduction distribution function of i.
Fixing (N,n,i) , this expression is the unnormalized induction distribution function of I.
The two most important parameters of a probability distribution are: the mean value and the standard deviation . The plus-minus sign, ± , is used to separate the mean from the deviation.
The mean value ± the standard deviation of the deduction distribution is used for estimating i knowing (N, n, I)
where a(b ± c) = ab ± ac. Note that f defines two functions of three variables.
Example: The population contains two items one of which is special, and the sample contains one item. (N, n, I) = (2, 1, 1) gives
confirming that the number of special items in the sample is either 0 or 1.
The mean value ± the standard deviation of the induction distribution is used for estimating I knowing (N,n,i)
where a+(b±c)=(a+b)±c.
Thus deduction is translated into induction by means of the involution
Example: The population contains a single item and the sample is empty. (N,n,i)=(1,0,0) gives
confirming that the number of special items in the population is either 0 or 1.
Note that the frequency probability solution to this problem is giving no meaning.
In the limiting case where N is a large number, the deduction distribution of i tends towards the binomial distribution with the probability as a parameter,
Example: The population is big, the probability , and the sample contains one item. n = 1 gives
confirming that the sample contains 0 or 1 special items, with equal probability.
In the limiting case where N is a large number, the induction distribution of tends towards the beta distribution
The frequency probability solution to this problem is . The probability is estimated by the relative frequency.
Example: The population is big and the sample is empty. n = i = 0 gives
The frequency probability solution to this problem is , giving no meaning.
In the limiting case where and are large numbers, the deduction distribution of i tends towards the poisson distribution with the intensity as a parameter,
Example: The population is big and the sample is big, and the intensity gives
In the limiting case where and are large numbers, the induction distribution of tends towards the gamma distribution with i as a parameter:
Example: The population is big and the sample is big but contains no special items. i = 0 gives
The frequency probability solution to this problem is which is misleading. Even if you have not been wounded you may still be vulnerable.