Theory of probability distributions | Bayesian statistics

Posterior predictive distribution

In Bayesian statistics, the posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values. Given a set of N i.i.d. observations , a new value will be drawn from a distribution that depends on a parameter : It may seem tempting to plug in a single best estimate for , but this ignores uncertainty about , and because a source of uncertainty is ignored, the predictive distribution will be too narrow. Put another way, predictions of extreme values of will have a lower probability than if the uncertainty in the parameters as given by their posterior distribution is accounted for. A posterior predictive distribution accounts for uncertainty about . The posterior distribution of possible values depends on : And the posterior predictive distribution of given is calculated by marginalizing the distribution of given over the posterior distribution of given : Because it accounts for uncertainty about , the posterior predictive distribution will in general be wider than a predictive distribution which plugs in a single best estimate for . (Wikipedia).

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35 - Normal prior and likelihood - posterior predictive distribution

This video provides a derivation of the normal posterior predictive distribution for the case of a normal prior distribution and likelihood. If you are interested in seeing more of the material, arranged into a playlist, please visit: https://www.youtube.com/playlist?list=PLFDbGp5YzjqXQ4o

From playlist Bayesian statistics: a comprehensive course

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29 - Posterior predictive distribution: example Disease

This video provides an introduction to the concept of posterior predictive distributions, using the example of disease prevalence in a population. Here we consider the case of a beta prior and binomial likelihood; resulting in a beta-binomial posterior. If you are interested in seeing mo

From playlist Bayesian statistics: a comprehensive course

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(ML 7.9) Posterior distribution for univariate Gaussian (part 1)

Computing the posterior distribution for the mean of the univariate Gaussian, with a Gaussian prior (assuming known prior mean, and known variances). The posterior is Gaussian, showing that the Gaussian is a conjugate prior for the mean of a Gaussian.

From playlist Machine Learning

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(ML 7.10) Posterior distribution for univariate Gaussian (part 2)

Computing the posterior distribution for the mean of the univariate Gaussian, with a Gaussian prior (assuming known prior mean, and known variances). The posterior is Gaussian, showing that the Gaussian is a conjugate prior for the mean of a Gaussian.

From playlist Machine Learning

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(ML 10.6) Predictive distribution for linear regression (part 3)

How to compute the (posterior) predictive distribution for a new point, under a Bayesian model for linear regression.

From playlist Machine Learning

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(ML 10.7) Predictive distribution for linear regression (part 4)

How to compute the (posterior) predictive distribution for a new point, under a Bayesian model for linear regression.

From playlist Machine Learning

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(ML 10.5) Predictive distribution for linear regression (part 2)

How to compute the (posterior) predictive distribution for a new point, under a Bayesian model for linear regression.

From playlist Machine Learning

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33 - Normal prior conjugate to normal likelihood - intuition

This video provides some intuition for the properties of the posterior distribution for the case of a normal prior and likelihood. If you are interested in seeing more of the material, arranged into a playlist, please visit: https://www.youtube.com/playlist?list=PLFDbGp5YzjqXQ4oE4w9GVWdio

From playlist Bayesian statistics: a comprehensive course

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(ML 10.4) Predictive distribution for linear regression (part 1)

How to compute the (posterior) predictive distribution for a new point, under a Bayesian model for linear regression.

From playlist Machine Learning

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Stanford CS229: Machine Learning | Summer 2019 | Lecture 9 - Bayesian Methods - Parametric & Non

For more information about Stanford’s Artificial Intelligence professional and graduate programs, visit: https://stanford.io/3ptRUmB Anand Avati Computer Science, PhD To follow along with the course schedule and syllabus, visit: http://cs229.stanford.edu/syllabus-summer2019.html

From playlist Stanford CS229: Machine Learning Course | Summer 2019 (Anand Avati)

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Kerrie Mengersen: Bayesian Modelling

Abstract: This tutorial will be a beginner’s introduction to Bayesian statistical modelling and analysis. Simple models and computational tools will be described, followed by a discussion about implementing these approaches in practice. A range of case studies will be presented and possibl

From playlist Probability and Statistics

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Aki Vehtari: Model assessment, selection and averaging

Abstract: The tutorial covers cross-validation, and projection predictive approaches for model assessment, selection and inference after model selection and Bayesian stacking for model averaging. The talk is accompanied with R notebooks using rstanarm, bayesplot, loo, and projpred packages

From playlist Probability and Statistics

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Statistical Rethinking Fall 2017 - week02 lecture04

Week 02, lecture 04 for Statistical Rethinking: A Bayesian Course with Examples in R and Stan, taught at MPI-EVA in Fall 2017. This lecture covers Chapter 4. Slides are available here: https://speakerdeck.com/rmcelreath Additional information on textbook and R package here: http://xcel

From playlist Statistical Rethinking Fall 2017

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JASP 0.14 Tutorial: Learn Bayes Module PART 1 (Binomial Estimation) (Episode 24)

In this JASP tutorial, I learn how to understand Bayesian statistical methods using the "Learn Bayes" module, available in the latest version of JASP! I did not do Bayesian stats in graduate school, and most frequently use classical/frequentist statistical methods. In "part one" of my two-

From playlist JASP Tutorials

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Statistical Rethinking 2022 Lecture 02 - Bayesian Inference

Bayesian updating, sampling posterior distributions, computing posterior and prior predictive distributions Course materials: https://github.com/rmcelreath/stat_rethinking_2022 Intro music: https://www.youtube.com/watch?v=QH_VKWStK98 Chapters: 00:00 Introduction 04:53 Garden of forking

From playlist Statistical Rethinking 2022

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Statistical Rethinking 2022 Lecture 07 - Overfitting

Slides and other course materials: https://github.com/rmcelreath/stat_rethinking_2022 Music: Intro: https://www.youtube.com/watch?v=R9bwnY05GoU Pause: https://www.youtube.com/watch?v=wAPCSnAhhC8 Chapters: 00:00 Introduction 04:26 Problems of prediction 07:00 Cross-validation 22:00 Regula

From playlist Statistical Rethinking 2022

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Statistical Rethinking 2023 - 03 - Geocentric Models

Slides and other materials: https://github.com/rmcelreath/stat_rethinking_2023 Intro music: https://www.youtube.com/watch?v=ayARo_IGV7g Flow: https://www.youtube.com/watch?v=oriuG649ypM Pause: https://www.youtube.com/watch?v=lT5lFeaInl4 Outline 00:00 Introduction 13:56 Gaussian distribut

From playlist Statistical Rethinking 2023

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34 - Normal prior and likelihood - prior predictive distribution

This video provides a derivation of the normal prior predictive distribution for the case of a normal prior distribution and likelihood. If you are interested in seeing more of the material, arranged into a playlist, please visit: https://www.youtube.com/playlist?list=PLFDbGp5YzjqXQ4oE4w9

From playlist Bayesian statistics: a comprehensive course

Related pages

Sufficient statistic | Exponential family | Conjugate prior | Mean | Probability density function | Compound probability distribution | Prediction interval | Marginal distribution | Bayesian statistics | Dirichlet-multinomial distribution | Student's t-distribution | Variance | Likelihood function | Conditional probability | Multinomial distribution | Normal distribution | Marginal likelihood | Normalizing constant | Binomial distribution | Beta-binomial distribution