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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).