I know about multinoulli distribution, but I found a different explanation in a book that I have been reading and I didn't quite get it. It says:
The multinoulli, or categorical, distribution is a distribution over a single discrete variable with $k$ different states, where k is finite. The multinoulli distribution is parameterized by a vector $p ∈[0,1]^-1$, where $p_i$ gives the probability of the $i$-th state. The final, $k$-th state's probability is given by $1−(1^)\cdot $. Note that we must constrain (1^) ≤ 1.
I didn't understand how this represents the final state like this $1−(1^)\cdot $ and anything about the vector.
If someone can provide detailed explanation it will be a real help.
here is a clarifying example using $k = 3$. Suppose that your experiment consists of drawing one ball. There are balls with three diverent colors: blue (b), green (g) and red (r). This corresponds to a multinoulli distribution with $k = 3$. We just have to specify the probabilities for the different outcomes. So we set $$ p_b = \Bbb P (\text{Ball is blue}) = \frac 16 \quad \text{and} \quad p_g = \Bbb P (\text{Ball is green}) = \frac 13. $$
These two probabilities are enough to determine $p_r = \Bbb P (\text{Ball is red})$: $$ p_r = 1 - (p_g + p_b) = \frac 12. $$ Or in more fancy notation if we set $\mathbf{p} = (p_b, p_g)$ and $1^T = (1, 1)^T$ then $$ p_r = 1 - 1^T \cdot \mathbf{p}. $$
Personally, I would not use the last notation.