Why should we use 16-50-84 percentile instead of $\mu$ and $\sigma$ to characterize sampling results such as MCMC Can someone explain why we should use the 16-50-84 percentile rather than mean and standard deviation to characterize the average and uncertainties of the sampling results, such as MCMC (the Markov Chain Monte Carlo sampling)?
I look up the source code of a python package corner and I find they are using the 16-50-84 percentile. Can someone explain the difference between the 16-50-84 percentile and mean/standard deviation and when we should use which?
 A: Who told you that? First, there's no particular reason why those percentiles are more useful than other percentiles. As noticed by other answers, mean and standard deviation easily translate to percentiles for normal distribution, but not necessarily for other distributions. Depending on what characteristics of the distribution are important for your particular use case, you can use any descriptive statistics that are useful for that case. In some cases, percentiles may be easier to interpret (e.g. think of exponential distribution that has a mean close to zero and is skewed).
A: When you have a normal distribution, the following are true.

*

*The mean $\mu$ is at the $50^{\text{th}}$ percentile.


*Within one standard deviation $\sigma$ of the mean, $68\%$ of the data are contained.


*The distribution is symmetric about the mean.
This means that $34\%$ of the density is above $\mu$ but below $\mu+\sigma$, and $34\%$ of the density is below $\mu$ but above $\mu-\sigma$.
$$
50+34 = 84\\
50-34 = 16
$$
Those numbers$-$$16$, $50$, and $84$$-$look familiar, don't they?
In the normal distribution, $\mu-\sigma$ corresponds to the $16^{\text{th}}$ percentile, and $\mu+\sigma$ corresponds to the $84^{\text{th}}$ percentile.
In other words, when you have a normal distribution from your MCMC, you get $\mu-\sigma$, $\mu$, and $\mu+\sigma$ by looking at those those three percentiles, $16$, $50$, and $84$, respectively. When you don't have a normal distribution, you still get the same quantiles with which you are familiar from the normal distribution.
If you do not have a normal distribution, then $\mu-\sigma$, $\mu$, and $\mu+\sigma$ might correspond to percentiles other than $16$, $50$, and $84$, respectively.
A: You can use either, and in fact I suggest you use both. The mean, median, standard deviations, and other percentiles all provide different information as summary statistics of the posterior. I'd also definitely recommend a histogram (or other density estimate) of posterior draws.
