We were discussing how to compare two models. For example, the number of data points must be larger than the number of parameters. if we increase the number of parameters, it will result in goodness of fit and better Chi-square. Consequently the model with higher number of parameters does better.This is articulated as a metric called the mean squared error which is chi-squared divided by degrees of freedom. MSE uses them as numerator and denominator so it represents a trade-off between over-parameterized model and a lower chi-square. A model with a fewer parameters but lower chi-square is preferred.

#codingexercise

Find the Eulerian number E(n,m). This is the number of permutations of the numbers from 1 to n in which exactly m elements are greater than the previous elements.

For example, n = 3, m = 1 and there are 4 permutations in which exactly 1 element is greater than the previous element. These are :

1 2 3  => 1,2 and 2,3 resulting in count 2 ( invalid)

1 3 2 => 1,3 resulting in count 1 (valid)

2 1 3 => 1,3 resulting in count 1 (valid)

2 3 1 => 2,3 resulting in count 1 (valid)

3 1 2 => 1,2 resulting in count 1 (valid)

int GetEulerian(int n, int m)
{
    if (m >= n || n == 0)
        return 0;

    if (m == 0)
        return 1;

    return (n - m) * GetEulerian(n - 1, m - 1) +
           (m + 1) * GetEulerian(n - 1, m);

}

we can easily try this for n = 3, m = 1
(3,1)
2 x (2,0) + 2 x (2,1)
2 + 2 x ( 1x (1,0) +2x (1,1))
2 + 2
4
The recursive formula  uses the notion that either the m reduces or doesn't as n reduces.