Original link:tecdat.cn/?p=3429

 

BART is a Bayesian nonparametric model that can be fitted using Backfitting MCMC.

I do not use any software package…… MCMC was implemented from scratch.

Consider covariable dataAnd resultsforThe theme,. In this example, the data looks like this:

We might consider the following probability model

 


Basically we use cubic polynomials to model the conditional mean. Note that this is a special case of the more general model

In this caseand. The modelHas a flat prior on each element of the parameter vector and has shape and rate on the variance parameterThe antigamma prior of.

Each condition is a posterioriIt’s all Gaussian. We can sample from this using conjugate Gibbs or Metropolis. We can also take the whole parameter vectorSampling as a block, but in this article we’ll stick to reverse fitting – which is itself a Gibbs sampler. We are still from each of the other conditionsConditional sampling in conditional posteriori. However, we use the key insight of each condition posterioriDepending on the other betaIs only represented by residuals

Intuitively,Is in the minusOther terms (no) after the left hand average. It’s also normally distributed,

Before normal, a posteriori can be calculated by conjugate

Backfitting MCMC proceeds as follows. First, except for initializing all beta builds. This is completely arbitrary – you can start with any parameter. Then, in each Gibbs iteration,

  1. To calculateWith the valueIn the current iteration. A posteriori sampleWith current extraction as the condition.
  2. To calculateWith the valueIn the current iteration. Pay attention to,Use the values from Step 1. Samples from the rear.
  3. Continue this process for all beta parameters.
  4. After all parameters are drawn, samples are taken. This posterior is another antigamma.

The term reverse fit seems appropriate because at each iteration we “exit”We want to use other beta versions for sample distribution.

To obtain a fitting regression line, we need to sample from a posteriori prediction distribution. We draw after step 4 in each Gibbs iterationValue to perform this operation

superscriptRepresents the use ofGibbs iterates the value of the parameter.


 

Thank you very much for reading this article, please leave a comment below if you have any questions!