# How to measure a deviation from the theoretical model in terms of both frequencies and probabilities

Consider for instance, that I perform an experiment that consists in observing events. Let there be $$N$$ different events in total. I have a theoretical model that tells me which events should appear with which probability. These are the theoretical probabilities $$p_t^i$$, $$i=1,\dots,N$$.

After having observed sufficiently many events, I see that their frequencies $$f_e^i$$, as well as the empirical probabilities $$p_e^i$$ differ from the predicted ones.

Now I wish to determine the events that most strongly deviate from the theoretical model. The first idea is to consider the events, whose empirical probabilities exceed the theoretical ones by the largest amount, i.e., the events s.t. $$p_e^i/p_t^i\rightarrow \max$$. However, I see that the events with the maximal ratio of probabilities have very small frequencies. That is, these events are very rare and not really interesting.

So far, I filter out the events with small frequencies and sort the remaining ones according to the ration of empirical to theoretical probabilities. However, I'd like to approach this problem in a more formal way.

Thus I wonder, if it is possible to formulate --in a systematic way-- a criterion that would take into account both the frequency of an event and its deviation in terms of probability? What would be the appropriate framework for dealing with such problems?