Tractable Distribution
A tractable distribution is a distribution which has closed-form analytical solutions for key functions and properties, especially for the…
- probability density function (PDF)
- cumulative density function (CDF)
- statistical moments (mean, variance, …) Also, integrals can be solved exactly, so marginalizations or expectations can be computed.
Calculations involving tractable distributions can be computed efficiently and sampling is quickly performed.
In contrast, intractable distributions have no closed-form solutions and their integrals require numerical and approximations methods, like Monte Carlo techniques. Computations are therefore much slower and sampling is not very efficient.
Examples for intractable distributions:
- True posterior in [[Bayesian Inference]]: The posterior \(p_{\theta}(z|x)\) of the latent distribution \(p(z)\) is usually intractable and requires approximation techniques using [[Markov-Chain Monte Carlo]] methods or Variational Autoencoders.
- Multimodal Distributions: Distributions with multiple peaks are often complicated to model for. They can be approximated for using [[Gaussian Mixture Model]]
- Complicated Mixtures: Mixtures of distributions involving non-Gaussian components can be intractable.