I have the problem to understand the following simple Maximum (Log) Likelihood example. Let $X$ be a discrete variable with domain $\{1,\dots,K\}$ and the discrete distribution is parametrized
$P(X=k;\pi) = \pi_k$
With parameters $\pi = (\pi_1,\dots,\pi_K)$ that are constrainted to fulfill $\sum_k \pi_k = 1$ and there is some data $D = \{x_i\}_{i = 1}^n$
What is the log likelihood $\mathcal{L}(\pi)$ of the data under the model?
I have applied the definition which gives me:
$\mathcal{L}(\pi) = \log P(x_{1:n};\pi) = \sum_{i = 1}^n \log P(x_i;\pi)$
At first I thought that it must sum up to 1 and the liklihood is 0, but this does not make sense, this the sum is over the data set which can have different occurences of different $X=k$ values, also the $\log$ is applied every time. The only thing I can think of is that $\mathcal{L}(\pi) \leq 0$ since there is no value which is $\geq$ 1.