Inferential statistics

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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. This is an example of the latter.

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.

Deduction distribution formula

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)]

[f(N,n,I)=\frac}}]

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

[i\approx f(2,1,1)=\frac\pm\frac]

confirming that the number of special items in the sample is either 0 or 1.

Induction distribution formula

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)]

where a+(b±c)=(a+b)±c.

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) gives

[I\approx -1-f(-2-0,-2-1,-1-0)=\frac\pm\frac]

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 [I\approx \frac=\frac] giving no meaning.

Binomial distribution formula

In the limiting case where N is a large number, the deduction distribution of i tends towards the binomial distribution with the probability [P=\frac] as a parameter,

[i\approx nP\left (1\pm\sqrt-1}}\right )]

Example: The population is big, the probability [P=\frac=\frac], and the sample contains one item. n = 1 gives

[i\approx \frac\pm\frac]

confirming that the sample contains 0 or 1 special items, with equal probability.

Beta distribution formula

In the limiting case where N is a large number, the induction distribution of [P=\frac] tends towards the beta distribution

[P\approx\frac}}.]

The frequency probability solution to this problem is [P \approx \frac]. The probability is estimated by the relative frequency.

Example: The population is big and the sample is empty. n = i = 0 gives

[P \approx(50 \pm 29)\%].

The frequency probability solution to this problem is [P \approx \frac=\frac], giving no meaning.

Poisson distribution formula

In the limiting case where [\frac] and [\ n] are large numbers, the deduction distribution of i tends towards the poisson distribution with the intensity [M=\frac] as a parameter,

[i \approx M \pm \sqrt]

Example: The population is big and the sample is big, and the intensity [M=\frac=1] gives

[i\approx 1 \pm 1].

Gamma distribution formula

In the limiting case where [\frac] and [\ n] are large numbers, the induction distribution of [M=\frac] tends towards the gamma distribution with i as a parameter:

[M \approx i+1 \pm \sqrt.]

Example: The population is big and the sample is big but contains no special items. i = 0 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

• Descriptive statistics
• Induction (philosophy)

 

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