I want to learn a logistic regression model but my outcome variable can take continuous values in [0;1], not only binary labels {0;1}.
My model thus still needs a logistic link (to bound my predictions in [0;1]) but the log loss is not appropriate in my case. I cannot formulate my problem as a series of Bernoulli trials as in standard binary LR.
Any suggestion for n appropriate loss in my case?
Best Answer
If $Y$ cannot be exactly 0 or 1, you can just take the logit transformation of $Y$ and use ordinary least squares. Such a model reasonably thinks of errors as being on the logit scale (scale of the linear predictor in the linear ols model). You cannot use nonlinear least squares because that would assume errors on the original $Y$ scale, resulting in predictions outside $[0,1]$. I'm not sure what your reference to log loss means. The transformed ols model would be assuming Gaussian errors.