I am using R and `MCMCpack`

to do a Bayesian analysis of some data that I have. I have generated posterior distributions (`postDist`

) on the means of some parameters (`y1,y2,y3`

) using `MCMCregress`

(`postDist <- MCMCRegress( x ~ y + z ,...)`

).

Now, I would like to take those posterior distributions on the means and generate a posterior distribution on the difference between the means. Is that a reasonable thing to do in a Bayesian analysis, and if so, how do you do it (either in theory or in R)?

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#### Best Answer

First, the method and theory, in brief: The goal is to approximate the target distribution $p(theta|D)$ where $theta$ is a vector parameter and $D$ is observed data, given some prior distribution $p(theta)$. At each stage of the MCMC chain, the sampling algorithm proposes a new parameter vector $theta$. (This process varies depending on the flavor of algorithm, and the proposal distribution.) Given a proposed $theta$, it then computes the product $p(D|theta_{proposed})p(theta_{proposed})$, which by Bayes rule is proportional to the posterior distribution $p(theta|D)$. It accepts the proposal with probability $max(frac{p(theta_{proposed})}{p(theta_{current})},1)$. *If* a number of requirements are met, this chain will produce a representative sample from the posterior distribution. (In brief, it requires a proposal process that adequately covers the posterior distribution, proper burn-in, and convergence.)

If those requirements are met, one can view the MCMC sample as an approximation to the posterior. Each individual sample value is one sampled vector of values for $theta$; likewise, differencing two sampled parameters over the entire sample produces an approximated distribution of the difference between the two parameters. (I'm not familiar with MCMCPack, but I gather from your code and comment that `postDist[,"y2"]`

and `postDist[,"y2"]`

are vectors of samples from the posterior, so this should work.) This is one benefit of MCMC methods: If the parameters covary, then solving for their sum or difference analytically depends on knowing their joint distribution.

By the by, I began learning Bayesian methods with Kruschke's *Doing Bayesian Data Analysis*, and I highly recommend his chapters explaining MCMC algorithms. It's a *very* approachable, intuitive treatment.

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