Normal Linear Model: Prediction of original variable from log transformed variable? [duplicate]

Question

Suppose we have observations $(x_1,y_1),\dots, (x_n,y_n)$ which for some reason cannot be modelled reasonably using a Normal Linear Model. Assume we instead model the log transformed response variables as from a Normal Linear Model i.e. $\text{log}(Y_i) \sim N(\alpha +\beta x_i, \sigma^2)$. Say we have estimated $\alpha$ and $\beta$ as $\hat{\alpha}$ and $\hat{\beta}$ respectively. Then we can from a new observation $x_*$ predict $E[\text{log}(y_*)]$ as $\hat{\alpha} + \hat{\beta} x_{*}$.

But say we really were interested in $E[y_{*}]$ instead, can we predict this? It seems to me we cannot just take the exponential function on either side as this cannot be moved into the expected value. Can we say anything about the relationship between $x_*$ and $E[y_{*}]$?

Answer 1

You are right, you can't simply take the exponent. It seems that it is a log normal distribution. If $Log(Y)$ is normally distributed with $\mu, \sigma^2$, then $Y$ is Log normally distributed with these parameters. The expectation of a log normally distributed variable is: $exp({\mu +\frac{\sigma^2}{2}})$

So you will have to learn $\sigma$ as well. You can do so, by model your output as distribution instead of a single number. i.e. in addition to the $\alpha + \beta x_i$, you need to output $\sigma_i$ as well and use likelihood as the loss function. You can read about log normal distribution here