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Bayesian Analysis with Python

You're reading from   Bayesian Analysis with Python A practical guide to probabilistic modeling

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Product type Paperback
Published in Jan 2024
Publisher Packt
ISBN-13 9781805127161
Length 394 pages
Edition 3rd Edition
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Author (1):
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Osvaldo Martin Osvaldo Martin
Author Profile Icon Osvaldo Martin
Osvaldo Martin
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Table of Contents (15) Chapters Close

Preface
1. Chapter 1 Thinking Probabilistically FREE CHAPTER 2. Chapter 2 Programming Probabilistically 3. Chapter 3 Hierarchical Models 4. Chapter 4 Modeling with Lines 5. Chapter 5 Comparing Models 6. Chapter 6 Modeling with Bambi 7. Chapter 7 Mixture Models 8. Chapter 8 Gaussian Processes 9. Chapter 9 Bayesian Additive Regression Trees 10. Chapter 10 Inference Engines 11. Chapter 11 Where to Go Next 12. Bibliography
13. Other Books You May Enjoy
14. Index

8.5 Gaussian process regression

Let’s assume we can model a variable Y as a function f of X plus some Gaussian noise:

Y ∼ 𝒩 (μ = f(X ),σ = 𝜖)

If f is a linear function of X, then this assumption is essentially the same one we used in Chapter 4 when we discussed simple linear regression. In this chapter, instead, we are going to use a more general expression for f by setting a prior over it. In that way, we will be able to get more complex functions than linear. If we decided to use Gaussian processes as this prior, then we can write:

 ′ f(X ) = 𝒢𝒫 (μX,𝜅(X, X ))

Here, represents a Gaussian process with the mean function μX and covariance function K(X,X). Even though in practice, we always work with finite objects, we used the word function to indicate that mathematically, the mean and covariance are infinite objects.

I mentioned before that working with Gaussians is nice. For instance, if the prior distribution is a GP and the likelihood is a Gaussian distribution, then the posterior is also a GP and we can...

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