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I was going through a lecture slides on maximum likelihood estimation (MLE) and bumped into something I couldn't understand. The lecturer was using MLE to estimate a function $$ y_i=Q(\alpha , \beta ,x_i). $$ Now during MLE estimation (estimating parameters $\alpha$ and $\beta$), the term $$ y_i-Q(\alpha ,\beta,x_i)\quad (1) $$ is directly plugged into Gaussian distribution equations as a random variable. $$ L=\prod\limits_{i=1}^n {\frac{1}{\sigma_i \sqrt{2 \pi}}} \exp{\left(-\frac{{(y_i-Q(\alpha,\beta,x_i))}^2}{2 \sigma_i^2}\right)} $$ Shouldn't $(1)$ be a result of the distribution with random variable as $x$, assuming that the variable $y$ has some noise present in it ?

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    $\begingroup$ Most certainly, $y_i=Q(\alpha , \beta ,x_i) +u_i \Rightarrow y_i-Q(\alpha , \beta ,x_i) =u_i$ and not just "some noise", but noise for which we assume that it follows the normal distribution and that it is independently distributed (otherwise the likelihood function wouldn't be just the product of the individual densities). $\endgroup$ – Alecos Papadopoulos Sep 29 '13 at 10:59
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To close this one:

Lecture slides usually are not drafted in order to tell the whole story on their own, and so in many cases require verbal input from the lecturer to provide a complete picture. In this specific case, probably the lecturer would:

1) Inform the audience that $y_i=Q(\alpha , \beta ,x_i)$ is the "deterministic" version of a relation that comes from theoretical concerns that usually assume away various irregularities that are present in the real world, and so, that for estimation purposes the equation will be supplied by an "error term"

2) Consequently, inform the audience that an assumption is made that this error term is a random variable following the normal distribution, with mean $0$ and variance $\sigma^2_i$ (heteroskedastic and so non-identically distributed), and also, that it is independently distributed across $i$... which would permit him to show the likelihood function presented at the end of the question here.

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