Is it appropriate to approximate an unknown probability distribution with a beta distribution? I am creating a model that utilize the output probability distribution of another model, as an input. The model outputs a probability of an event occurring per year, the distribution is a probability between (between 0 and 1) and right skewed, it is defined by Mean = 0.019, Mode = 0.009, 90% confidence interval: 5th percentile = 0.0002, 95th percentile = 0.07. I want to use a beta distribution so I can perform a Bayesian update using a binomial likelihood for the update.
However, I am uncertain of 1) if it is appropriate to use a beta distribution (as opposed to a lognormal) and how to prove if it is appropriate or not, and 2) how to determine the parameters of the beta distribution from the available information.
In relation to 2) I have looked at several articles on cross validate which seem related but I cannot quite work out how to implement:
Calculate the confidence interval for the mean of a beta distribution
Calculating the parameters of a Beta distribution using the mean and variance
This one seems the closest and easiest to implement: Selecting alpha and beta parameters for a beta distribution, based on a mode and a 95% credible interval
In the model I am creating it is possible to define a reparametrized beta distribution according to Mean and standard deviation, so I thought that I would be able to solve equations with the above mentioned values to obtain the standard deviation and define the beta distribution but now I am not sure if this is possible.

A beta distribution has support on $[0,1]$ while a log-normal distribution has support on $(0,\infty)$ so it is likely that at least one would be inappropriate for your problem, possibly both.  In particular the log-normal might look a strange choice for modelling a probability.
You have four pieces of information, which is in a sense too much information for finding two parameters of a beta distribution.  So you could only find an approximation, and optimising it would depend on how important the differences are.
For example $\alpha=1.5,\beta = 75$ would give

*

*a mean of about $0.0196$, close to your figure

*a mode of about $0.00671$, lower than your figure

*a 5th percentile of about $0.00234$, more than ten times your figure

*a 95th percentile of about $0.0506$, lower than your figure

and you would need to judge whether these (or those for some other choices of $\alpha$ and $\beta$) were good enough for your purposes.  If not, then using a beta distribution may not be what you want
