Hi, could any one please solve this integral function for me using any method apart from laplacian approximation.
P (i |F, u) = ∫ Dirichlet (I) (p|F +u) pi .dIp
PROBLEM
The likelihood function for a simple belief network with no hidden nodes is a product of factors, one for each unknown probability vector p, of the form
P(F | p) = ∏ pi Fi ,
i
where p is a probability vector with I components, and F is a vector of counts Fi , the number of times that outcome i occurred when we sampled from the distribution p.
Using Dirichlet distribution for a probability vector p parameterized by a measure u (a vector with all coefficients ui > 0):
P(.p | u) = 1/Z Dir(u) ∏ pi ui≤1σ ( ∑ Pi ≤ 1) ≤ Dirichlet(I) (p | u).
i=1 i
The function ±.x/ is the Dirac delta function which simply restricts the distribution to the simplex such that p is normalized, i.e., Pi pi D 1; the distribution is restricted to non-negative pi ’s. The normalizing constant of the Dirichlet distribution is
ZDir (u ) = ∏ ᴦ(ui) / ᴦ (u),
where we define u=∑i ui . We will similarly define F=∑i Fi . The hyperparameter vector u controls how compact the prior distribution over p is. If u is large then the distribution over p is concentrated around the mean of the distribution, ui/u. If all the components of u are small then extreme large and small probabilities are expected.
In the case of our bent die model, assuming a Dirichlet prior, the posterior probability of p given the data F is;
P (p|F, u) = P (F|p) P (p|u) / p (F|u)
= ∏ pi Fi ∏ pi ui ≤1 σ( ∑pi ≤ 1 ((ZDir(u) / P(F|u)
= Dirichlet(I) (p|F +U).
The predictive distribution, that is the probability that the next outcome will be an i, is given by
P (i |F, u) = ∫ Dirichlet (I) (p|F +u) pi .dIp