Conditional probability with sympy

Question

Since it is not a math related question but about using a library to do symbolic computation, SO is better suited to answer this than {math|stats}.stackexchange.com

I want to use Sympy to calculate the following:

This is the code I'm using in Sympy

from sympy.stats import Normal, density, DiscreteUniform, P, given
from sympy import Symbol, pprint, symbols, Symbol, Eq
sigma = Symbol("sigma", positive=True)
mu = DiscreteUniform('mu', [1,2])
N = Normal('normal', mu, sigma)

sampling_dist = given(N, Eq(mu, 1))
prior = P(Eq(mu, 1))
marginal = P(Eq(mu, 1))*given(N, Eq(mu, 1))+P(Eq(mu, 2))*given(N, Eq(mu,2))
post = prior * sampling_dist / marginal

Now I want to

1. be able to print the equation for the posterior distribution (I expect sigma and x to be the only unknown)
2. Plot the posterior by fixing sigma with a known value

I tried to print the equation by asking the density with

density(post)(Symbol('x'))

And I get the following error

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-6-9b0e1c9ada72> in <module>()
----> 1 density(post)(Symbol('x'))

/Users/alexis/anaconda3/lib/python3.4/site-packages/sympy/stats/rv.py in density(expr, condition, evaluate, numsamples, **kwargs)
    692                 **kwargs)
    693 
--> 694     return Density(expr, condition).doit(evaluate=evaluate, **kwargs)
    695 
    696 

/Users/alexis/anaconda3/lib/python3.4/site-packages/sympy/stats/rv.py in doit(self, evaluate, **kwargs)
    643             isinstance(pspace(expr), SinglePSpace)):
    644             return expr.pspace.distribution
--> 645         result = pspace(expr).compute_density(expr, **kwargs)
    646 
    647         if evaluate and hasattr(result, 'doit'):

/Users/alexis/anaconda3/lib/python3.4/site-packages/sympy/stats/rv.py in pspace(expr)
    430         return rvs[0].pspace
    431     # Otherwise make a product space
--> 432     return ProductPSpace(*[rv.pspace for rv in rvs])
    433 
    434 

/Users/alexis/anaconda3/lib/python3.4/site-packages/sympy/stats/rv.py in __new__(cls, *spaces)
    278         # Overlapping symbols
    279         if len(symbols) < sum(len(space.symbols) for space in spaces):
--> 280             raise ValueError("Overlapping Random Variables")
    281 
    282         if all(space.is_Finite for space in spaces):

ValueError: Overlapping Random Variables

Even if I subs sigma with a constant I get the same error. I (obviouslu) don't understand what I'm doing wrong.

Answer 1

I don't fully understand what you're trying to accomplish, but I ran your code in a Jupyter notebook and examined the output of post with sympy.init_printing() enabled, which displays the following:

The exception that gets thrown, ValueError: Overlapping Random Variables, suggests that SymPy simply doesn't know what to do with the expression. I suggest you double-check the construction of your expression, testing the output in a more piecemeal way.