# 2.3.3 - Parameter space

2.3.3 - Parameter space

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 $$\pi$$-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:

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