G-Exponential Families through Probability Coordinates

What the math gives to ML

The paper gives a constructive way to build heavy-tailed likelihoods without changing the exponential function: map the target through a probability chart G, fit an ordinary exponential family in the bounded coordinate, and pull the density back with the chart Jacobian. The useful neural-network property is that bounded sufficient statistics produce bounded likelihood scores, while the chart controls tail behavior independently of the predicted natural parameter. This can become a robust probabilistic regression head whose gradients are less sensitive to extreme targets than Gaussian likelihood training. The most direct test is contaminated regression with Cauchy or Pareto noise.