I'm trying to predict a probability with a neural network, but having trouble figuring out which loss function is best. Cross entropy was my first thought, but other resources always talk about it in the context of a binary classification problem where the labels are $\{0, 1\}$, but in my case I have an actual probability as the target. Is one of these options clearly best, or maybe are they all valid with just minor differences around the extreme 0/1 regions?
Assuming $x$ is the output of the final layer of my model.
Cross Entropy:
$target * -log(sigmoid(x)) + (1 - target) * -log(1 - sigmoid(x))$
Mean Squared Error with Sigmoid:
$(sigmoid(x) - target)^2$
Mean Squared Error with Clamp:
$(x - target)^2$
When I use the output I clamp the values between $[0, 1]$.