FindDistributionParameters fails with custom distribution?

Question

Context

I would like to find the MaximumLikelihood solution of a customized PDF

Let's start with a built in PDF. Following the documentation

dat = RandomVariate[LaplaceDistribution[2, 1], 1000];
param=FindDistributionParameters[dat, LaplaceDistribution[μ, σ],
 ParameterEstimator -> {"MaximumLikelihood", Method -> "NMaximize"}]

(* {μ->2.27258,σ->0.521354} *)

Show[Plot[
PDF[LaplaceDistribution[μ, σ] /. param, x], {x, -5, 5}], 
Histogram[dat, Automatic, "PDF"]]

works as expected. It finds a good estimator of $\mu$ and $\sigma$.

The problem

Now let me do the same with a customized PDF. Here I just impose that my custom PDF cannot be evaluated before it is given numerical values.

Clear[myLaplaceDistribution];
myLaplaceDistribution[μ_?NumberQ, σ_?NumberQ] := 
 LaplaceDistribution[μ, σ]

Then

dat = RandomVariate[LaplaceDistribution[2, 1], 10];
FindDistributionParameters[dat, myLaplaceDistribution[μ, σ],
 ParameterEstimator -> {"MaximumLikelihood", Method -> "NMaximize"}]

does not return a maximum likelihood estimate.

I am using 10.3.0 for Mac OS X x86 (64-bit) (October 9, 2015)

Question:

Any suggestions on how to make FindDistributionParameters work with unevaluated PDFs?

PS: I am aware of this https://mathematica.stackexchange.com/a/107914/1089 but here this question is a bit more general than simply a transformed distribution? And I have tried

dat = RandomVariate[LaplaceDistribution[2, 1], 10];
FindDistributionParameters[dat, 
 myLaplaceDistribution[μ, σ], {{μ, 
   Mean[dat]}, {σ, Mean[dat]}},
 ParameterEstimator -> {"MaximumLikelihood", Method -> "NMaximize"}]

it does not seems to help.

Update

This related answer https://mathematica.stackexchange.com/a/61426/1089 does not seem to help.

If I define explicitly the domain for the PDF

  Clear[myLaplaceDistribution2];
  myLaplaceDistribution2[μ_?NumberQ, σ_?NumberQ] := 
  ProbabilityDistribution[
  PDF[LaplaceDistribution[μ, σ], x], {x, -Infinity, 
   Infinity}, Assumptions -> (μ ∈ Reals && σ > 0)]

It still fails

dat = RandomVariate[LaplaceDistribution[2, 1], 10];
FindDistributionParameters[dat, 
 myLaplaceDistribution2[μ, σ], {{μ, 
   Mean[dat]}, {σ, Mean[dat]}},
 ParameterEstimator -> {"MaximumLikelihood", Method -> "NMaximize"}]

As @J.M. points out one can use the fact that Mathematica can cope with the fact the PDF need not be normalized. As follows

Clear[myLaplaceDistribution3];
myLaplaceDistribution3[μ_, σ_] = 
 ProbabilityDistribution[
  2 PDF[LaplaceDistribution[μ, σ], 
    x], {x, -∞, ∞}, 
  Assumptions -> (μ ∈ Reals && σ > 0), 
  Method -> "Normalize"]

(Note the factor of 2 in front of PDF to make the PDF not normalized.)

Then

dat = RandomVariate[LaplaceDistribution[2, 1], 10];
FindDistributionParameters[dat, myLaplaceDistribution3[μ, σ],
 ParameterEstimator -> {"MaximumLikelihood"}]

works.

I still think there must be situations where the PDF cannot be known before its arguments are known, and where Maximum likelihood analysis would make sense?

Note that I can always make my own:

MyFindDistributionParameters[data_, distrib_, var_] :=
 NMaximize[{Total[Log@ PDF[distrib, #] & /@ data], 
   DistributionParameterAssumptions[distrib]}, var][[2]];

MyFindDistributionParameters[dat,LaplaceDistribution[μ, σ], {μ, σ}]

but I was hoping Mathematica would provide me with a more efficient algorithm? (this seems to be 10 times slower than the built in function).

Answer 1

If you follow @J.M. 's advice removing ?NumberQ from the definition of the probability distribution makes everything work fine:

Clear[myLaplaceDistribution];
SeedRandom[12345];
myLaplaceDistribution[μ_, σ_] := LaplaceDistribution[μ, σ]
dat = RandomVariate[LaplaceDistribution[2, 1], 10];
FindDistributionParameters[dat, myLaplaceDistribution[μ, σ],
 ParameterEstimator -> {"MaximumLikelihood", Method -> "NMaximize"}]
(* {μ -> 1.8804870321227085,σ -> 0.7153183538699862} *)

I don't know what you mean by "Here I just impose that my custom PDF cannot be evaluated before it is given numerical values." Your first example doesn't have the two parameters evaluated as numbers and it works fine:

param=FindDistributionParameters[dat, LaplaceDistribution[μ, σ],
 ParameterEstimator -> {"MaximumLikelihood", Method -> "NMaximize"}]