Seonglae ChoOnly a handful of specific integrals are analytically tractable
- Substitution
- Coordinate changing such as Polar coordinates
- Transform into Probability Distribution form to use the fact that integral is 1
Future prediction is integration, and rule decomposition is differentiation. Derivation is the process of obtaining local rules (direction/velocity/slope), while Integral is the process of advancing the state forward according to those rules. In other words, because differentiation is definable, it provides guidance on how to handle each specific data sample or each instance of reality. Inference is integration; it approximates the uncertain future prediction. Therefore, training is differentiation and inference is integration.
Integral Notion
The stochasticity trick of Monte Carlo method (Importance sampling)
Assume you have a difficult integral to compute
The Monte Carlo estimator for performs better than sampling from the original distribution when it has lower variance. For comparison, the variance of is , while the variance of is - with lower variance being preferable.
In short, Monte Carlo methods enable us to estimate any integral by random sampling. InBayesian Statistics, Evidence is also form of integral so it becomes tractable.