Sampling techniques are used to tackle the complex computations which arise in Bayesian calculations. These calculations involve probability integrals in high-dimensional spaces which cannot be solved analytically. Sampling methods are used to draw samples from probability distributions and use these samples to perform further important computations.
| Lecture 12 - Stochastic Simulation Algorithms for Bayesian Computations. Drawing samples from a pdf. Monte Carlo integration of important Bayesian integrals. Introduction to Stochastic Simulation Algorithms for Bayesian Computations. The concept of the representation of the posterior pdf with samples is discussed and compared to previously discussed asymptotic approximations. Next, the question of how to draw samples from a probability distribution is addressed and demonstrated with examples. Next, Monte Carlo integration is described to approximate probability integrals with finite sums using the drawn samples. Important properties of the sums are derived using the Law of Large Numbers and the Central Limit Theorem. Finally, the estimation of the model evidence integral and the robust prediction integral is discussed using samples. |
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| Lecture 13 - Markov Chain Monte Carlo algorithms. Metropolis-Hasting algorithm for probability distribution sampling. Uncertainty propagation using posterior samples. Introduction to Markov Chain Monte Carlo algorithms and their applications. Markov Chains and the Metropolis-Hastings algorithm for drawing samples from any complex distribution. The Metropolis-Hastings algorithm is explained thoroughly, and illustrative videos of its use are shown for different configurations of parameters. Finally, the propagation of uncertainty is discussed on the basis of the posterior samples from the Metropolis-Hastings algorithm. |
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| Theory #14 | |
| Theory #15 |
