using Distributions
p = 0.2
x = rand(Bernoulli(0.2),50)
fit_mle(Bernoulli(p), x)
MethodError: no method matching fit_mle(::Bernoulli{Float64}, ::Vector{Bool})

I know that

fit_mle(Bernoulli, x)

works.
Adding the following seems to do what I want

function fit_mle(g::Distribution{Univariate,Discrete}, args...)
fit_mle(typeof(g), args...)
end
function fit_mle(g::Distribution{Univariate,Continuous}, args...)
fit_mle(typeof(g), args...)
end

However, I don’t understand why fit_mle(Bernoulli(p), x) is not authorized (or coded this way)?
Sure, the parameter p does not matter for the usual function where the MLE is explicit.
In my applications, I write extensions of fit_mle to other distributions where the actual parameter p is useful. It can be an initialization for an optimization algorithm.
Or when the distributions are Products or Mixture, the type is not enough and the whole distribution structure is needed.

I get your point, the first p = p_ini is just an initial point here not what you look for.
In optimization, it would not work because rosenbrock([1.5, 2.5]) is a scalar and not a function.
For distribution fitting Bernoulli(p) carries the same info as just the type Bernoulli.
In fact, it carries more info because of the p_ini, in some situation it might be useless I agree (like for Benoulli).
Imagine a Product distribution of an Exponential and Normal distribution.
If one were to write a fit_mle(::Type{<:Product}, x) how do you extract the informations over the component of the Product if you just accept the type ? Whereas fit_mle(g::Product, x) has everything you need.
Same thing for a Mixture distribution.

I am still uncomfortable with the types, so there might be a way to do that I do not understand.