Question on cumulative probability In an article I'm reading the probability of an observed point falling in bin i is written as:
$P_i=\frac{m_i}{\sum\limits_j m_j}$
where $m_i$ is the number of model points in bin i.
Then, the cumulative probability of obtaining the entire data set $n_i$ is written as:
$P=\prod\limits_i \left( \frac{m_i}{\sum\limits_j m_j} \right)^{n_i}$
where $n_i$ if the observed value of bin i.
My question is: how was equation 2 obtained?
Edit: this is the article in question. I didn't add much information in the question because I wanted to keep it as simple as possible. Equations in the article are (14) and (15).
 A: Assume that you have independent observations. If the first observation falls in bin $i$ then the probability of obtaining that observation is $P_i$.
If the second observation falls in bin $j$, the probability of obtaining those observations in that order is $P_i\cdot P_j$. If the third observation falls into bin $i$ the cumulative probability is now $P_i\cdot P_j\cdot P_i=P_i^2\cdot P_j$, and so on.
Continuing in this way, we find that the cumulative probability is $P_i^{n_i}\cdot P_j^{n_j}\cdots=\prod_i P_i^{n_i}$, which is the second formula.
A: It just used that each points falls independently into  a bin. Remember that for independent events A and B 
$$
P(A \cap B)=P(A)P(B).
$$
Each of the points falling into bin $i$ had a chance of $P_i$ to do so. Because of independence the probability of all the $n_i$ points falling into bin $i$ doing so, is just
$$
{P_i}^{n_i}=({\frac{m_i}{\sum_j m_j}})^{n_i}.
$$ 
Repeat for all the other bins (the events there are independent) and you get the final result. 
What I personally find more confusing is your first sentence. 
The tag maximum-likelihood gives it much needed context, since otherwise it makes no sense to equate short-run frequencies and probabilities. I think the whole point of the argument is taking the short-run frequencies as estimates of the probabilities (i.e. long-run frequencies) maximizes the Likelihood of the observed data and is therefore the MLE for estimating the long-run frequencies. 
