What is sigma2 in output of MCMC regression by MCMCregress of MCMCpack package of R

Question

I am using following data and code:

> myiris = iris
> myiris$Species = as.numeric(iris$Species)
> head(myiris)

> MCMCregress(formula = Species ~ ., data = myiris)
> summary(MCMCregress(Species~., data=myiris))

Iterations = 1001:11000
Thinning interval = 1 
Number of chains = 1 
Sample size per chain = 10000 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                 Mean       SD  Naive SE Time-series SE
(Intercept)   1.18538 0.206918 2.069e-03      1.998e-03
Sepal.Length -0.11138 0.058294 5.829e-04      5.829e-04
Sepal.Width  -0.04027 0.060478 6.048e-04      6.397e-04
Petal.Length  0.22816 0.057331 5.733e-04      5.733e-04
Petal.Width   0.60974 0.095370 9.537e-04      9.537e-04
sigma2        0.04875 0.005833 5.833e-05      5.978e-05

2. Quantiles for each variable:

                 2.5%      25%      50%        75%    97.5%
(Intercept)   0.77743  1.04672  1.18547  1.3244075 1.586649
Sepal.Length -0.22727 -0.15082 -0.11067 -0.0721836 0.001095
Sepal.Width  -0.15863 -0.08112 -0.04035 -0.0001155 0.078274
Petal.Length  0.11611  0.18907  0.22845  0.2672629 0.339776
Petal.Width   0.42368  0.54458  0.61028  0.6739717 0.794600
sigma2        0.03853  0.04460  0.04834  0.0522567 0.061303

What is sigma2 in the output and what is its signficance?

Answer 1

Linear regression model is

$$ y_i = X_i \beta + \varepsilon_i , \ \ \varepsilon_i \sim \mathcal{N}(0, \sigma^2)$$

so as you can see, $\sigma^2$ is error variance. On another hand, you can look at the model as

$$ y_i \sim \mathcal{N}(X_i \beta,\ \sigma^2) $$

in this case, $\sigma^2$ is a variance of distribution of $y_i$'s. So if you assume Bayesian point of view, where both you variables and parameters are random variables with a certain distributions, then $\sigma^2$ becomes an additional parameter in your model and not just "error variance" like in traditional OLS regression. In Bayesian approach you get a whole distribution of $y_i$'s, that is described by its mean and variance. So replying to your comment, I don't know what you mean by "ignoring" this value, but what can be said is that it is a parameter of model defined like this that describes distribution of $y_i$'s. You can read more on linear regression in Bayesian approach e.g. on Kruschke's page.

See also the JSS paper on MCMCpack (Martin, Quinn, & Park, 2011) for learning more on this library.