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This is more of a question about my understanding of the concept; suppose we have a normal linear model $Y = X\boldsymbol{\beta}+\epsilon$, where $\epsilon \sim N(0,\sigma^{2}I_{n})$ and $\boldsymbol{\beta}$ is the vector of unknown parameters.

When we estimate the unknown parameter $\boldsymbol{\beta}$ using the maximum likelihood approach, are we basically finding the $\boldsymbol{\hat{\beta}}$ that maximises the probability of obtaining a set of realisations ($y_{1},...,y_ {n}$) given our covariates? E.g. if we postulate that two variables have such a relationship and we know the realisations of the response and also the covariates, we want to maximise the probability that the model spurts out the values of the realisations?

python_learner
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  • @Tim Thank you -- but is the 'probability of obtaining the realisations given the covariates' not interchangeable with likelihood? – python_learner May 05 '17 at 20:08
  • Check https://stats.stackexchange.com/questions/2641/what-is-the-difference-between-likelihood-and-probability?noredirect=1&lq=1 as it answers this question. – Tim May 05 '17 at 20:36

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