Same or different? The Bayesian way Say I have the following model:
$$\text{Poisson}(\lambda) \sim \begin{cases} 
      \lambda_1 & \text{if  } t \lt \tau \\
      \lambda_2  & \text{if } t \geq \tau 
\end{cases} 
$$
And I infer the posteriors for $\lambda_1$ and $\lambda_2$ shown below from my data. Is there a Bayesian way of telling (or quantifying) if $\lambda_1$ and $\lambda_2$ are the same or different? 
Perhaps measuring the probability that $\lambda_1$ is different from $\lambda_2$? Or perhaps using KL divergences? 
For example, how can I measure $p(\lambda_2 \neq \lambda_1)$, or at least, $p(\lambda_2 \gt \lambda_1)$?
In general, once you have the posteriors shown below (assume non-zero PDF values everywhere for both), what is a good way of answering this question?

Update
It seems that this question can be answered in two ways:


*

*If we have samples of the posteriors, we could look at the fraction of the samples where $\lambda_1 \neq \lambda_2$ (or equivalently  $\lambda_2 > \lambda_1$). @Cam.Davidson.Pilon included an answer that would address this problem using such samples.

*Integrating some sort of difference of the posteriors. And that's an important part of my question. What would that integration look like? Presumably the sampling approach would approximate this integral, but I would like to know the formulation of this integral.
Note: The plots above come from this material.
 A: I think a better question is, are they significantly different?
To answer this, we need to compute $P(\lambda_2 > \lambda_1)$. Call this quantity $p$. If $p \approx 0.50$, then there's equal chance one is larger than the other.  On the other hand, If $p$ is really close to 1, then we can be confident that yes $\lambda_2$ is larger (read: different) than $\lambda_1$. 
How do we compute $p$? It's trivial in a Bayesian MCMC framework. We have samples from the posterior, so lets just compute the chace that samples from $\lambda_2$ are larger than $\lambda_1$:
 p = np.mean( lambda_2_samples > lambda_1_samples )
 print p

I apologize for not including this in the book, I'll definetly add it as I think it's one of the most useful ideas in Bayesian inference
A: As stated, this question is trivial. Assuming $\lambda_1$ and $\lambda_2$ are continuous random variables, $\Pr(\lambda_1=\lambda_2)=0$. 
I suspect you are interested in the probability that $\lambda_1$ and $\lambda_2$ are within some $\epsilon$ of each other. In that case, the area of the the difference in the two posterior densities on the interval $[-\epsilon/2, \epsilon/2]$ is your answer. Larger values of overlap indicate that the two posteriors are more similar.
If you would prefer to work with simulated results (and for most problems, we don't have the luxury of choice), simply take the proportion of the results where $\lambda_2>\lambda_1$ as an approximation.
