[vermarish]

Right. But if you make the notation slightly more explicit, then the integral of L(data, params) over data is 1. This follows from the independence assumption.

So we ARE working with a probability function. Its output can be interpreted as probabilities. It's just that we're maximizing L = P(events | params) with respect to params.

[kgwgk]

The likelihood function is a function of params for a fixed value of data and it is not a probability function.

There is another function - a function of data for fixed params - which is a probability density. That doesn’t change the fact that the likelihood function isn’t.

[331c8c71]

The independence has nothing do with the integral being 1 to be honest. You could write a model where the observations are not independent but the (multivariate) integral over their domain will still be 1.

[vermarish]

But for such a model, the joint pdf would not be written simply as a product of each individual pdf. That's what independence provides.