curv (Continuous Random Variables) is a python package for creating and manipulating continuous random variables in a bayesian network. You may find it useful for performing bayesian inference on "broad" distributions with high kurtosis or highly dimensional joint distributions where discrete sampling would require excessive memory.
Typical usage often looks like this:
#!/usr/bin/env python
import curv as cv
A = cv.normal(1,2)
B = cv.uniform(-5,4)
C = 4 + A - 2*B
# D = C * A**2 # Not implemented
cv.plotPDF(D,-10,10)
cv.plotNet()
cv.plotCHF(D,-10,10)Dependencies include:
- Numpy
- Scipy
- Matplotlib
Curv stores its random variables as continuous functions as opposed to storing discrete probabilities in sampled bins. This data structure reduces memory consumption (since distributions are not sampled and stored as individual points).
As a result, curv would be useful for handling large, highly dependent bayesian networks with high dimensional joint probability distribution functions without second-order approxiations (such as Chow-Liu trees).
Curv would also be useful for distributions with high kurtosis, where discrete sampling over a large number of bins would be needed to capture rare events at the edges of the distribution.
Consider the following: A and B are both primitive RVs. Then define: C = A + B Since A and B are primitives and therefore independent, C's pdf is the convolution of A and B's pdfs. Now consider the following: D = A + C This is not a straightforward convolution of the pdfs, since C is not independent with A. Instead, this is a projection of the joint pdf perpendicular to the y=x line. Since we are internally storing the characteristic functions, this projection is equivalent to slicing the joint cfun along the line y=x and scaling the result (by central-slice theorm).
Our problem is to find the joint CHF ϕX, Y(tx, ty) as a function of ϕX(tx) given Y = X + a where a is a constant.
In terms of PDFs, if Y = X + a then the joint PDF is given by math:: f{X,Y}(x,y) = f_X(x)delta(y-x-a) Taking the Fourier transform we have math:: phi{X,Y}(t_x,t_y) = iint f_X(x)delta(y-x-a)e^{-i(t_x x+t_y y)} ,dx,dy Solving the double integral, we obtain math:: phi{X,Y}(t_x,t_y) = e^{-i t_y a} phi_X(t_x+t_y)
Our problem is to find the joint CHF ϕX, Y, Z(tx, ty, tz) as a function of ϕX, Y(tx, ty) given Z = X + Y.
In terms of PDFs, if Z = X + Y then the joint PDF is given by math:: f{X,Y,Z}(x,y,z) = f{X,Y}(x,y)delta(z-x-y) Taking the Fourier transform we have math:: phi{X,Y,Z}(t_x,t_y,t_z) = iiint f{X,Y}(x,y)delta(z-x-y)e^{-i(t_x x+t_y y+t_z z)} ,dx,dy,dz Solving the triple integral, we obtain math:: phi{X,Y,Z}(t_x,t_y,t_z) = phi{X,Y}(t_x+t_z,t_y+t_z)
This is a direct result of the central slice theorm.
Note that if we start with a joint chf with non-1st generation parents, then we work as if the parents were mutually independent with the sum, since we are already conditioning on the 1st generation parents.
For the subtraction of two random variables Z = X − Y, we may similarly derive the expression math:: phi{X,Y,Z}(t_x,t_y,t_z) = phi{X,Y}(t_x+t_z,t_y-t_z)
Our problem is to find the joint CHF ϕX, Y(tx, ty) as a function of ϕX(tx) given Y = aX where a is a constant.
In terms of PDFs, if Y = aX then the joint PDF is given by math:: f{X,Y}(x,y) = f_X(x)delta(y-ax) Taking the Fourier transform we have math:: phi{X,Y}(t_x,t_y) = iint f_X(x)delta(y-ax)e^{-i(t_x x+t_y y)} ,dx,dy Solving the double integral, we obtain math:: phi{X,Y}(t_x,t_y) = phi_X(t_x + a t_y)