Beta Bernoulli - distribution of p conditional on successes I have a Beta-Bernoulli process $\textbf{X}=\{X_1, X_2...X_n\}$, $\textbf{p}=\{p_1, p_2...p_n\}$ where:
$$X_i \sim Bernoulli(p_i)$$
$$p_i \sim Beta(\alpha,\beta)$$
where $\alpha$ and $\beta$ are known.
Is there an analytical solution for distribution of $p'$ for the subset of trials that were successful:
$f_{p'|X_i=1}$
It's easier to illustrate in a simulation setting.
That is, given sequence $\textbf{X}$, let $\textbf{X'}$ be the subset of $\textbf{X'}$ containing successes only: $$\textbf{X}'=\{X_i|X_i = 1\} = \{1,1,1...1\}$$
What is the distribution of $p$ that generated $\textbf{X'}$?
For example, below is the result from a simulation. The blue line is the distribution of $p$ from all trials ($p\sim Beta(2,2)$). Orange line is the conditional distribution of $p$ from successes only. Green line is the distribution of $p$ from failures only. So, the overall distribution of $p$ is a mixture of the two.

 A: I suppose this is supposed to be a Bayesian question
If the prior for $p$ is $\operatorname{Beta}(\alpha, \beta)$ with density proportional to $p^{\alpha-1}(1-p)^{\beta-1}$ and you then observe $k$ successes and $n-k$ failures in $\mathbf{X}$ with likelihood proportional to $p^k(1-p)^{n-k}$
then the posterior density for $p$ given $\mathbf{X}$  is proportional to $p^{\alpha-1+ k}(1-p)^{\beta-1+ n-k}$ so the posterior distribution for $p$ is  $\operatorname{Beta}(\alpha+k, \beta+n-k)$
If $\mathbf{X}$ is a single observation and is a success then this mean the posterior distribution for $p$ is  $\operatorname{Beta}(\alpha+1, \beta)$

I now believe you think you are asking a different question, where different values of $p$ are drawn from $\operatorname{Beta}(\alpha, \beta)$ giving $p_1, p_2, \ldots, p_n$, and then each $X_i$ is drawn from  $\operatorname{Bernoulli}(p_i)$; you want to know the distribution of those $p_i$ corresponding to $X_i=1$.
This is equivalent to the final line of my initial answer on a single successful observation: the distribution in question is $\operatorname{Beta}(\alpha+1, \beta)$
To try and convince you, here are a couple of simulations in R of $10^6$ initial draws, first with $\alpha=2, \beta=2$ and then with $\alpha=2, \beta=8$, drawing the simulated densities for the successful cases (black) and comparing this with the theoretical densities for $\operatorname{Beta}(\alpha+1, \beta)$.  They are a very close match
set.seed(2021)

positives <- function(samplesize, alpha, beta){
  p <- rbeta(samplesize, alpha, beta)
  X <- rbinom(samplesize, 1, p)
  success <- (X == 1)
  psuccess <- p[success]
  plot(density(psuccess, from=0, to=1))
  curve(dbeta(x, alpha+1, beta), col="red", from=0, to=1, add=TRUE)
  c(mean(success))
  }

positives(10^6, alpha=1, beta=2)

and the chart above, starting with $p\sim \operatorname{Beta}(1,2)$, shows a $\operatorname{Beta}(2,2)$ distribution for the $p_i$ corresponding to $X_i=1$.

and the chart above, starting with $p\sim \operatorname{Beta}(2,2)$, shows a $\operatorname{Beta}(3,2)$ distribution for the $p_i$ corresponding to $X_i=1$. Then
positives(10^6, alpha=2, beta=8)


and the chart above, starting with $p\sim \operatorname{Beta}(2,8)$, shows a $\operatorname{Beta}(3,8)$ distribution for the $p_i$ corresponding to $X_i=1$.
