Probabilistic interpretation from sigmoid functions Why do we interpret the results of logistic regression as probabilities? Passing the output of any regression procedure through a sigmoid function results in a probabilistic interpretation with respect to classification. Why is that so? Given that the output is between 0 and 1, is it enough to interpret the results as probabilities?
 A: A probability is bounded between 0 and 1 (inclusive), and a sigmoid curve is a convenient curve that can be forced to respect those bounds. It is not the only one, but the sigmoid curve has proven to be the most popular one. So, not all probability models have a sigmoid cure, though many do.
Moreover, not all models with a sigmoid curve model probabilities. For example, models that model a dependent variable that is a fraction also often use a sigmoid curve.
What makes logit, probit, and similar models model a probability is the fact that they model the conditional mean of an indicator variable, which is the conditional proportion of $1$s, which in turn is interpreted as the probability of having a $1$ on that indicator variable.
A: 
Why do we interpret the results of logistic regression as
  probabilities?  

Because the logistic regression model can be viewed as arising from a linear regression latent variable model, where the error term of this linear regression is assumed to follow the standard logistic distribution. See for example this post.

Given that the output is between 0 and 1, is it enough to interpret
  the results as probabilities?  

No. The "output" must come from a function that satisfies the properties of a distribution function in order for us to interpret it as probabilities. These properties are:  
1) The function $F$ under consideration must be non-decreasing and right-continuous ("cadlag")  
2) $\lim_{x\rightarrow -\infty}F(x) =0$  
3) $\lim_{x\rightarrow \infty}F(x) =1$
The "sigmoid function" satisfies these properties.
