# Reconstructing probability density function from marginal distributions

#### federico.carminati

##### New Member
Good afternoon everybody,
all my apologies if this is a stupid question and sorry in advance because I am sure I will not do a good job in explaining my problem, since it is not entirely clear to me either. Here it is. Suppose I have an unknown multidimensional probability density distribution

f(x_{1},\dots,x_{n})

but I know all the univariate marginal distributions

f_{i}(x) = \int{f(x_{1},\dots,x_{n}) dx_{j,j \neq i}}

and I have a constraint

g(x_{1},\dots,x_{n})=0

I know I cannot reconstruct f, since there is probably a family of f's that satisfy these conditions. However I would like to know whether it is possible to characterize this family of functions and, under specific conditions, find the functional form of at least one candidate that would satisfy the above constraints.

Thanks in advance for any help or pointer to literature or brilliant idea or simply interested comment.

Best regard,

Federico

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#### Dason

##### Ambassador to the humans
Without saying more about either the univariate distributions or the constraint I don't think there is much one could say in general.

#### federico.carminati

##### New Member
I know, but there is very little I can say more. The scattering data for neutron induced reactions come as marginal distributions, either as discrete functions or as different continuous functions. The balance of energy-momentum is not conserved, so the problem is to create a joined distribution that respects kinematic constraints. There are hundreds of distributions. So the problem is really generic.

#### Dason

##### Ambassador to the humans
What is the constraint you have and why do you need to create a joint distribution?

#### federico.carminati

##### New Member
For each energy of the incoming neutron there is a distribution of the kinematic variables of the outgoing particles, which could be from 2 to several. Each particle is characterized by an energy and an angle theta (distributions are symmetric in phi). So I have 2 x n variables. The constraint is that the energy-momentum is balanced, that is P-\sum{p_{i}}=0, where P and p_{i} are lorentz four vectors momentum-energy. I need a constrained joint distribution because I want to sample the variables being sure that the physical constrains are respected. I could sample from the marginal and impose the kinematic constraint, and if it does not work, resample. This would be slow but more importantly, this would be wrong, since I would alter the distribution.