2.3.3 - Parameter space

If we don't impose any restrictions on the parameter


other than the logically necessary constraints

\(\pi_j \in [0,1],j=1,\ldots,k\) (1)


\(\pi_1+\pi_2+\ldots+\pi_k=1\) (2)

then the parameter space is the set of all \(\pi\)-vectors that satisfy (1) and (2). This set is called a simplex. In the special case of k = 3, we can visualize \(\pi = \left(\pi_1, \pi_2, \pi_3\right)\) as a point in three-dimensional space. The simplex S is the triangular portion of a plane with vertices at (1, 0, 0), (0, 1, 0) and (0, 0, 1):

More generally, the simplex is a portion of a (k − 1)-dimensional hyperplane in k-dimensional space. Alternatively, we can replace

\(\pi_k\text{ by }1-\pi_1-\pi_2-\cdots-\pi_{k-1}\)

because it's not really a free parameter and view the simplex in (k − 1)-dimensional space. For example, with k = 3, we can replace \(\pi_3\) by \(1 − \pi_1− \pi_2\) and view the parameter space as a triangle: