How would you do Bayesian ANOVA and regression in R? I have a fairly simple dataset consisting of one independent variable, one dependent variable, and a categorical variable. 
I have plenty of experience running frequentist tests like aov() and lm(), but I cannot figure out how to perform their bayesian equivalents in R. 
I would like to run a bayesian linear regression on the first two variables and a bayesian analysis of variance using the categorical variable as the groupings, but I cannot find any simple examples on how to do this with R. Can someone provide a basic example for both? Additionally, what exactly are the output statistics created by bayesian analysis and what do they express?
I am not very well-versed in stats, but the consensus seems to be that using basic tests with p-values is now thought to be somewhat misguided, and I am trying to keep up.
Regards.
 A: The BayesFactor package (demonstrated here: http://bayesfactorpcl.r-forge.r-project.org/ and available on CRAN) allows Bayesian ANOVA and regression. It uses Bayes factors for model comparison and allows posterior sampling for estimation.
A: This is quite convenient with the LearnBayes package. 
fit <- lm(Sepal.Length ~ Species, data=iris, x=TRUE, y=TRUE)
library(LearnBayes)
posterior_sims <- blinreg(fit$y, fit$x, 50000)

The blinreg function uses a noninformative prior by default, and this yields an inference very close to the frequentist one.
Estimates:
> # frequentist 
> fit$coefficients
      (Intercept) Speciesversicolor  Speciesvirginica 
            5.006             0.930             1.582 
> # Bayesian
> colMeans(posterior_sims$beta)
      X(Intercept) XSpeciesversicolor  XSpeciesvirginica 
         5.0066682          0.9291718          1.5807763 

Confidence intervals:
> # frequentist
> confint(fit)
                      2.5 %   97.5 %
(Intercept)       4.8621258 5.149874
Speciesversicolor 0.7265312 1.133469
Speciesvirginica  1.3785312 1.785469
> # Bayesian
> apply(posterior_sims$beta, 2, function(x) quantile(x, c(0.025, 0.975)))
      X(Intercept) XSpeciesversicolor XSpeciesvirginica
2.5%      4.862444          0.7249691          1.376319
97.5%     5.149735          1.1343101          1.783060

