Corey Chivers on P(A|B) ∝P(B|A)P(A)

If you can write the likelihood function for your model, MHadaptive will take care of the rest (ie. all that MCMC business). I wrote this R package to simplify the estimation of posterior distributions of arbitrary models. Here’s how it works:

1) Define your model (ie the likelihood * prior). In this example, lets build a simple linear regression model.

(log)Likelihood:

li_reg<-function(pars,data)
{
a<-pars[1]      #intercept
b<-pars[2]      #slope
sd_e<-pars[3]   #error (residuals)
if(sd_e<=0){return(NaN)}
pred <- a + b * data[,1]
log_likelihood<-sum( dnorm(data[,2],pred,sd_e, log=TRUE) )
prior<- prior_reg(pars)
return(log_likelihood + prior)
}

Prior:

prior_reg<-function(pars)
{
a<-pars[1]          #intercept
b<-pars[2]          #slope
epsilon<-pars[3]    #error

prior_a<-dnorm(a,0,100,log=TRUE)     ## non-informative (flat) priors on all
prior_b<-dnorm(b,0,100,log=TRUE)     ## parameters.
prior_epsilon<-dgamma(epsilon,1,1/100,log=TRUE)

return(prior_a + prior_b + prior_epsilon)
}

Now lets just simulate some data to give it a test run:

x<-runif(30,5,15)
y<-x+rnorm(30,0,5) ##Slope=1, intercept=0, epsilon=5
d<-cbind(x,y)
plot(x,y)

2) The function Metro_Hastings() does all the work. Just pass your model to this function, sit back and let MC Emmcee take it away.

mcmc_r<-Metro_Hastings(li_func=li_reg,pars=c(0,1,1),
par_names=c('a','b','epsilon'),data=d)

3) You can view the posteriors of all model parameters using plotMH()

plotMH(mcmc_r)

By default, this will also plot the pair-wise correlation between all parameters.

4) Print posterior Credible Intervals.

BCI(mcmc_r)
#              0.025    0.975
# a       -5.3345970 6.841016
# b        0.4216079 1.690075
# epsilon  3.8863393 6.660037

Tadda! Easy-Peasy.

For a more introductory look at Bayesian modeling, check out the slides and script file from an R workshop I gave at McGill on the subject. There are some additional optional arguments you can pass to Metro_Hastings() which you can read all about in the package manual. You can get the package source from cran, or from my github page.

*notes: MHadaptive is only recommended for relatively low dimensional models. Mixing can be quite slow for higher order models.