Inferential statistics
From Free net encyclopedia
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. This is an example of the latter.
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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
- <math> {I \choose i}{{N-I} \choose {n-i}} </math>
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)
- <math>i \approx f(N,n,I)</math>
where
- <math>f(N,n,I)=\frac{nI\pm\sqrt{\frac{nI(N-n)(N-I)}{N-1}}}{N}.</math>
The mean value ± the standard deviation of the induction distribution is used for estimating I knowing (N,n,i)
- <math>I \approx -1-f(-2-n,-2-N,-1-i).</math>
Thus deduction is translated into induction by means of the involution
- <math>(N,n,I,i) \leftrightarrow (-2-n,-2-N,-1-i,-1-I).</math>
Example
The population contains a single item and the sample is empty. (N,n,i)=(1,0,0). The induction formula gives
- <math>I\approx -1-f(-2,-3,-1)=\frac{1}{2}\pm\frac{1}{2}</math>
confirming that the number of special items in the population is either 0 or 1.
(The frequency probability solution to this problem is <math>I\approx \frac{Ni}{n}=\frac{0}{0}</math> 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 <math>P=\frac{I}{N}</math> as a parameter,
- <math>i\approx nP\left (1\pm\sqrt{\frac{\frac{1}{P}-1}{n}}\right )</math>
and the induction distribution of <math>\ P</math> tends towards the beta distribution
- <math>P\approx\frac{i+1\pm\sqrt{\frac{(i+1)(n-i+1)}{n+3}}}{n+2}.</math>
(The frequency probability solution to this problem is <math>P \approx \frac{i}{n}</math>: 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 <math>P \approx(50 \pm 29)\%</math>.
(The frequency probability solution to this problem is <math>P \approx \frac{i}{n}=\frac{0}{0}</math> giving no meaning.)
Poisson and Gamma
In the limiting case where <math>\frac{N}{n}</math> and <math>\ n</math> are large numbers, the deduction distribution of i tends towards the poisson distribution with the intensity <math>M=\frac{nI}{N}</math> as a parameter,
- <math>i \approx M \pm \sqrt{M}</math>
and the induction distribution of M tends towards the gamma distribution
- <math>M \approx i+1 \pm \sqrt{i+1}.</math>
Example
The population is big and the sample is big but contains no special items. i = 0. The gamma distribution formula gives <math>M\approx 1 \pm 1</math>.
(The frequency probability solution to this problem is <math>M\approx 0</math> which is misleading. Even if you have not been wounded you may still be vulnerable).