Statistical Methods in Astronomy--Lecture 3: Maximum-likelihood

Brief Definition of Maximum-likelihood by Alberto Fernandez-Soto

Given D [a data set (y_i)] and a model M [with parameters (a_i)] we have to estimate the likelihood of the model M (i.e., how likely it is that M is the real representation of the world behind the data set D).

Unfortunately this is unfeasible (there is _only_one_real_model_, and it's no good talking about the likelihoods of different models). So, we turn the question upside down and say that we measure how likely it is to get the data set D from the model M (THIS WE CAN DO EASILY!)... the assumption then is that one thing is the same as the other... for which we don't have the slightiest clue of a proof...

I have read, more or less understood and tried to digest this from Numerical Recipes. It's the closest to a 'definition' of Maximum Likelihood Methods that I can get, apart from the mathematical one:

Given observed points (y_i) we want to fit a model that describes the probability function f(y). We parameterize it as f(a_j,y), where (a_j) are the parameters that define a model. We can calculate:

L(a_j)=Prod_i [f(a_j,y_i)]

In this situation, the values of a_j that maximize the function L define the 'maximum likelihood estimate' of the function f(y). This is only a particular case, but it is the one people mostly use, I think.

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1998 Apr 23
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