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When linear regression is formulated probabilistically using MLE, turns out that what we used to get as output, is actually the mean of $P(y|x)$, the latter is normal distributed around the correct output. My question is, can we also predict the standard deviation as well? so, for each $x$, I get mean $y$ plus $\sigma^2$.

My rough idea is, in addition to $h_\theta(x)$, we need $g_\theta(x)$ as our variance, and then $P(y|x) = e^{-\frac{(h_\theta(x_i)-y_i)^2}{g_\theta(x_i)^2}}$

Any reference for such derviation ?

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  • $\begingroup$ The standard deviation of what quantity? Note that the standard deviation of the error term in the normal linear model is assumed to be constant. $\endgroup$
    – Michael M
    Commented Sep 17, 2019 at 5:12
  • $\begingroup$ @MichaelM Excellent point! Yes, the variance of the noise is assumed to be constant y=wx+epsilon. But can we make it variable and dependent on x (heteroskedatecity) and ask for it to be predicted? $\endgroup$
    – John Deterious
    Commented Sep 17, 2019 at 5:17

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