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Two schools of inferential statistics are Frequency Probability using Maximum Likelihood estimation, and Bayesian Inference . This 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

: {I \choose i}

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'')
:i \approx f(N,n,I)

where
:f(N,n,I)= rac{nI\pm\sqrt{ rac{nI(N-n)(N-I)}{N-1}}}{N}.

The Mean Value ± the Standard Deviation of the ''induction'' distribution is used for estimating ''I'' knowing (''N,n,i'')
:I \approx -1-f(-2-n,-2-N,-1-i).

Thus deduction is translated into induction by means of the Involution

:(N,n,I,i) \leftrightarrow (-2-n,-2-N,-1-i,-1-I).


Example

The population contains a single item and the sample is empty. (''N,n,i'')=(1,0,0). The induction formula gives
:I\approx -1-f(-2,-3,-1)= rac{1}{2}\pm rac{1}{2}
confirming that the number of special items in the population is either 0 or 1.

(The Frequency Probability solution to this problem is I\approx rac{Ni}{n}= rac{0}{0} 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 P= rac{I}{N} as a parameter,

:i\approx nP\left (1\pm\sqrt{ rac{ rac{1}{P}-1}{n}} ight )

and the induction distribution of \ P tends towards the Beta Distribution

:P\approx rac{i+1\pm\sqrt{ rac{(i+1)(n-i+1)}{n+3}}}{n+2}.

(The is Estimated by the Relative Frequency .)


Example

The population is big and the sample is empty. ''n=i=''0. The Beta Distribution formula gives P \approx(50 \pm 29)\%.

(The Frequency Probability solution to this problem is P \approx rac{i}{n}= rac{0}{0} giving no meaning.)


Poisson and Gamma

In the limiting case where rac{N}{n} and \ n are large numbers, the Deduction Distribution of ''i'' tends towards the Poisson Distribution with the intensity M= rac{nI}{N} as a parameter,

:i \approx M \pm \sqrt{M}

and the induction distribution of ''M'' tends towards the Gamma Distribution

:M \approx i+1 \pm \sqrt{i+1}.


Example

The population is big and the sample is big but contains no special items. ''i'' = 0. The Gamma Distribution formula gives M\approx 1 \pm 1.

(The Frequency Probability solution to this problem is M\approx 0 which is misleading. Even if you have not been wounded you may still be vulnerable).


SEE ALSO