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If we don't impose any restrictions on the parameter

\(\pi=(\pi_1,\pi_2,\ldots,\pi_k)\)

other than the logically necessary constraints

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

and

\(\pi_1+\pi_2+\ldots+\pi_k=1\)    (2)

then the parameter space is the set of all π-vectors that satisfy (1) and (2). This set is called a simplex. In the special case of k = 3, we can visualize π = (π1, π2, π3) 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):

plot

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 π3 by 1 − π1− π2 and view the parameter space as a triangle:

plot