Probability of a given data set given the parameters

[tronter]

Why do we look at the probability of a given data set given the parameters, as opposed to the probability of the parameters given the data set?

So $$y(x) = y(x;a_{1} \ldots a_{M})$$

 

[HallsofIvy]

I'm not sure I understand your question. In "Probability" we always assume we are given some basic probability distribution- i.e. the parameters of some general distribution (such as the normal distribution). The problems in probability then are generally, "given the parameters of the probability distribution, determine the probability of various outcomes (i.e. the data set)".

Probabilty is then used as the theory behind "Statistics" in which we do just the opposite: given an outcome (data set) try to estimate the parameters of the probability distribution.

 

[quadraphonics]

Why do we look at the probability of a given data set given the parameters, as opposed to the probability of the parameters given the data set?

Well, in maximum likelihood estimation, the reason is simple: the parameters are not random variables, and so there is no probability distribution defined on them. Instead, we use the likelihood function, which is similar.

In Bayesian/MAP estimation, on the other hand, the parameters are viewed as random variables, and we do use the probability of the parameters given the data.