rollulus gave a good summary of Laplace transforms and what they do. For some more context, they appear regularly in applied probability (e.g. finance, insuran

rollulus gave a good summary of Laplace transforms and what they do. For some more context, they appear regularly in applied probability (e.g. finance, insurance, physical models including dams). A typical problem is dealing with sums of non-negative random variables. Let's say you want the distribution of n independent copies of a non-negative random variable with distribution function F. The hard way is the n-fold convolution or essentially evaluating an n-dimensional integral. The easy way is using the Laplace transform of F and simply raising it to the power of n. The result isn't always invertible analytically, but you can almost always invert it numerically and this is why techniques like the one outlined in the paper are so important. This is a fantastic post and I thoroughly recommend reading it and the 2019 paper that summarises all their work for several reasons: 1. Very clear exposition of previous work and their own. 2. Clear evaluation metrics. 3. They've even made it easy for you to replicate their work and results.