[SciPy-user] computing Bayesian credible intervals : help on constrained optimisation schemes?

[Johann Cohen-Tanugi]

Hello,
I am attaching my script. I followed Neil's suggestion but it fails to 
converge, due seemingly to issues with the fact that a and b must be 
positive, which I cannot enforce with fsolve, AFAIK.
I am ready to dive into constrained schemes, especially in openOpt, but 
I was hoping for some suggestions beforehand as to which path to follow 
to solve this problem now.
I remind that I am trying to find a and b so that :
integral from a to b of p(x) = Q
and p(a)=p(b)
where Q is given (0.95 in my script) and p is a Poisson posterior pdf 
for ON/OFF source experiments. a,b, and x are source rates, and as such 
are positive.
People will have recognized the computation of a Bayesian credible 
interval here!!

thanks a lot in advance,
Johann

Neil Martinsen-Burrell wrote:
> Johann Cohen-Tanugi <cohen <at> slac.stanford.edu> writes:
>
>   
>> hi Neil, thanks for your answer and sorry I was not clear enough. Of 
>> course I require the 2 conditions. 1) defines *a* credible interval if p 
>> is a posterior pdf; and 2) sets a constraint that for common situation 
>> yield *the* standard Bayesian credible interval. I will have a look at 
>> brentq, I do not know what it refers to.
>>     
>
> scipy.optimize.brentq is Brent's method for finding a root of a given scalar
> equation.  Since you are looking for two values, a and b, with two conditions,
> then Brent's method is not appropriate (barring some symmetry-based reduction to
> one variable).  I like to use scipy.optimize.fsolve to find roots of
> multivariable equations, such as
>
> def solve_me(x): # x is an array of the values you are solving for
>     a,b = x
>     integral_error = quad(density, a , b) - q
>     prob_difference = density(b) - density(a)
>     return np.array([integral_error, prob_difference])
>
> fsolve(solve_me, [0.0, 1.0])  # initial guess is a = 0, b = 1
>
>
>
>
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