In logistic regression I assume a set of examples of which I know with security the output. Suppose binary classification. The example $i$ has probability p equal to $1$ or $0$ to belong to the category.

But, how about instead of using deterministic examples outputs, using probability examples outputs.

For example, if this is the set of examples:

$\mathbf{x}_1$ $\mathbf{x}_2$ ... $\mathbf{x}_i$ ...

$y_1$ $y_2$ ... $y_i$ ...

Instead of set the $y$ values to $0$ or $1$, assign them a known or assumed probability $p$ that the example belongs to the category. With this it could be assumed previously known information.

The cross entropy cost function, would perform well under this.

Is this method used? Is there another method to do something like this?

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    $\begingroup$ Could any of your probabilities p be exactly 0 or 1, or would they all be in the open interval (0,1)? $\endgroup$ – EdM Sep 5 '17 at 21:24

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