# Why Does an Exponential Make ANYTHING a Probability Distribution

I am posting this here because my background in estimation theory and optimization has been developed entirely through my experience in robotics.

TLDR: What makes it so that any time you put something in an exponential you can suddenly call it a probability distribution?

The Deets: I thought of this question while reading A General and Adaptive Robust Loss Function by Jonathan Barron. The paper describes a single parameter loss function that can subsume the most commonly used individual loss functions:

$$f(x, \alpha, c)$$

Where $$x$$ is parameter, $$\alpha$$ is the "single parameter", and $$c$$ is a distance scale that describes the "quadratic bowl" around the origin which all the loss functions share in common. When $$\alpha = 2$$ it is squared error loss, $$\alpha =1$$ it is essentially absolute error loss, and when $$\alpha = 0$$ it is cauchy loss.

My confusion starts when the paper goes on to say "With our loss function we construct a general probability distribution".

$$p(x|\mu,\alpha, c) = \frac{1}{cZ(\alpha)}\exp(-f(x-\mu, \alpha, c))$$

My question is, why does dividing by a normalization constant, and taking the exponential of the negative loss function suddenly make it a pdf? I understand the results, I have used these same ideas in my estimation theory courses, but I just don't get how we can make the jump from loss to pdf OR why it is so intuitive to do so.

Cheers, and happy holidays :)

• The loss function was introduced by Jonathan Barron to solve an optimization problem. There is a model, and adapting the model to the real world produces a loss. The task for the solver is to reduce the costs to a minimum. A probability distribution helps to apply existing search techniques likes gradient descent, so that the minimum value can be found faster. Dec 25 '20 at 18:14
• You would probably have more luck with this question on stats.stackexchange.com or ai.stackexchange.com
– 50k4
Dec 27 '20 at 19:59 