Terminology for Bayesian Posterior Mean of Probability with Uniform Prior If $p \sim$ Uniform$(0,1)$, and $X \sim$ Bin$(n, p)$, then the posterior mean of $p$ is given by $\frac{X+1}{n+2}$. 
Is there a common name for this estimator? I've found it solves lots of people's problems and I'd like to be able to point people to a reference, but haven't been able to find the right name for it. 
I vaguely recall this being called something like the "+1/+2 estimator" in a stats 101 book but that's not a very searchable term. 
 A: You could call it a shrinkage estimator. The estimator is closer to $.5$ than the more ubiquitous sample mean.
A: With prior $\mathsf{Unif}(0,1) \equiv \mathsf{Beta}(\alpha_0=1,\beta_0 =1)$ and likelihood
$\mathsf{Binom}(n, \theta)$ showing $x$ successes in $n$ trials, the posterior distribution is $\mathsf{Beta}(\alpha_n=1 + x,\; \beta_n = 1 + n - x).$ 
(This is easily seen by multiplying the kernels of the prior and likelihood to get the kernel of the posterior.)
Then the posterior mean
is  $$\mu_n = \frac{\alpha_n}{\alpha_n+\beta} = \frac{x+1}{n+2}.$$
In a Bayesian context, just using the terminology posterior mean may be best. (The median of the posterior distribution and the maximum of its PDF have also been used to summarize posterior information.)
Notes: (1) Here you are using $\mathsf{Beta}(1,1)$ as a noninformative prior distribution. On sound theoretical grounds, some Bayesian statisticians prefer to use
the Jeffreys prior $\mathsf{Beta}(\frac 1 2, \frac 1 2)$ as a noninformative prior. Then the posterior mean is $\mu_n = \frac{x+.5}{n+1}.$
(2) In making frequentist confidence intervals Agresti and Coull have suggested "adding two successes and two failures" to the sample in order to get a confidence interval based on the estimator $\hat p = \frac{x+2}{n+4},$ which has more accurate coverage probabilities (than the traditional Wald interval using $\hat p = \frac x n).$ David Moore has dubbed this a plus-four estimator in some of his widely-used elementary statistics texts, and the terminology has been used by others. I would not be surprised to see your estimator called 'plus two' and Jeffries' called 'plus one'.
(3) All of these estimators have the effect of 'shrinking the estimator towards 1/2' and so they have been called 'shrinkage estimators,' (a term that is much more widely used, particularly in James-Stein inference). See Answer (+1) by @Taylor.
A: This is called Laplace's smoothing, or Laplace's rule of succession, as Pierre-Simon Laplace used it for estimating the probability the sun rises again tomorrow: "We thus find that an event having occurred a number of times, the probability that it will happen again the next time is equal to this number increased by the unit, divided by the same number increased by two units."

Essai philosophique sur les probabilités par le marquis de Laplace
