Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 189

As expected, the width of the confidence interval depends both on the
number of experiments N and on the noise through the scattering of the data
around the mean value, as expressed by the summation under the square root.
The larger the number of experiments, the smaller the confidence interval,
hence the more reliable the estimation of the expectation value µ by the
mean m . Conversely, the larger the variability of the results, the larger the
confidence interval, hence the less reliable the estimation of µ by m .

2.10.2 Hypothesis Testing: An Example

N measurements {g i } have been performed, which can be modeled as inde-
pendent realizations of a random Gaussian variable of mean µ and standard
deviation σ . One would like to know, with a risk α of getting a wrong answer,
whether the mean of the distribution has a given value µ 0 . Thus, the null
hypothesis H 0 is: µ = µ 0 , and the alternative hypothesis H 1 is µ

= µ 0 .Ifthe

null hypothesis is true, then variable

N ( N

µ 0
i =1 ( G i −

M −

Z = − 1)

M ) 2

is a Student variable with N 1 degrees of freedom.

A realization of that random variable can be computed,

−

N ( N

µ 0
i =1 ( G i −

m −

Z = − 1) ,

m ) 2

where m is the mean of the measurements. The values of z 1 and z 2 such that
Pr( z<z 1 )= α/ 2 and Pr( z>z 2 )= α/ 2 can easily be computed. Then the
null hypothesis can be rejected if z is outside the interval [ z 1 , z 2 ].
In that particular case, the hypothesis test consists in checking whether
the assumed value of the mean µ 0 is within the confidence interval computed
in the previous section, and rejecting the null hypothesis if it is outside the
confidence interval.

2.10.3 Pearson, Student and Fisher Distributions
2.10.3.1 χ 2 (Pearson) Distribution

If a random variable S is the sum of the squares of N random independent
Gaussian variables, then it has a χ 2 (or Pearson) distribution with N degrees
of freedom. It can be shown that E ( S )= N and that var( S )=2 N .

2.10.3.2 Student Distribution

If Y 1 is a normal variable, and if Y 2 is a random variable, which is independent
from Y 1 and which has a χ 2 (Pears on) dis tribution with N degrees of freedom,

the random variable Z =( Y 1 ) / ( Y 2 /N ) has a Student distribution with N
degrees of freedom.