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Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory
Inferential statistics or statistical induction comprises the use of statistics to make inferences concerning some unknown aspect (usually a parameter) of a population.
Two schools of inferential statistics are frequency probability using maximum likelihood estimation, and Bayesian inference. The following is an example of the latter.
Deduction and induction
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.
Mean ± standard deviation
The mean value ± the standard deviation of the deduction distribution is used for estimating i knowing (N,n,I)
where
The mean value ± the standard deviation of the induction distribution is used for estimating I knowing (N,n,i)
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). The induction formula gives
confirming that the number of special items in the population is either 0 or 1.
(The frequency probability solution to this problem is giving no meaning.)
Limiting cases
Binomial and Beta
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,
and 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. The beta distribution formula gives .
(The frequency probability solution to this problem is giving no meaning.)
Poisson and Gamma
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,
and the induction distribution of M tends towards the gamma distribution
Example
The population is big and the sample is big but contains no special items. i = 0. The gamma distribution formula gives .
(The frequency probability solution to this problem is which is misleading. Even if you have not been wounded you may still be vulnerable).
See also
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