Laplace approximation in Bayesian inference

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SUMMARY

The discussion centers on implementing the Laplace approximation for Bayesian inference in a Python project involving MCMC to estimate four or more parameters. The formula for evidence, E, is provided as \( E = P(x_0)2\pi^{n/2}|C|^{1/2} \), where C represents the covariance matrix and \( P(x_0) \) is the maximum posterior value. The user seeks guidance on obtaining \( P(x_0) \) for more than two parameters, indicating a challenge in visualizing high-dimensional data. A suggestion is made to refer to the corner library for potential solutions.

PREREQUISITES
  • Understanding of Bayesian inference principles
  • Familiarity with MCMC (Markov Chain Monte Carlo) methods
  • Knowledge of covariance matrices in statistical analysis
  • Experience with Python programming and data visualization libraries, particularly Matplotlib
NEXT STEPS
  • Explore the corner library for visualizing high-dimensional distributions in Python
  • Research advanced techniques for calculating posterior distributions in Bayesian inference
  • Learn about the implementation of Laplace approximation in Python
  • Investigate alternative methods for visualizing n-dimensional data beyond histograms
USEFUL FOR

Data scientists, statisticians, and researchers involved in Bayesian modeling and inference, particularly those working with MCMC methods and high-dimensional parameter estimation.

BRN
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Hello everybody,
I am working on a Python project in which I have to make Bayesian inference to estimate 4 or more parameters using MCMC.
I also need to evaluate the evidence and I thought to do so through the Laplace approximation in n-dimensions:

$$ E = P(x_0)2\pi^{n/2}|C|^{1/2} $$

Where C is the parameter's covariance matrix and ##P(x_0)## is the maximum value that assumes the posterior.
Getting the covariance matrix is not a problem, but I don't know how get FX0. If they were only 2 parameters I could use matplotlib.hist2d, but being more than 4 parameters...
How could I do?
Some idea?

Thank you!
 
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BRN said:
FX0.
What is this ?
Are you looking for something like this

https://corner.readthedocs.io/en/latest/pages/quickstart.html
 

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