It is often useful to construct a distribution \(d^\prime\) which is consistent with some marginal aspects of \(d\), but otherwise optimizes some information measure. For example, perhaps we are interested in constructing a distribution which matches pairwise marginals with another, but otherwise has maximum entropy:
In : from dit.algorithms.scipy_optimizers import MaxEntOptimizer ImportErrorTraceback (most recent call last) <ipython-input-1-5fd9c8e847a4> in <module>() ----> 1 from dit.algorithms.scipy_optimizers import MaxEntOptimizer ImportError: No module named scipy_optimizers In : xor = dit.example_dists.Xor() In : meo = MaxEntOptimizer(xor, [[0,1], [0,2], [1,2]]) NameErrorTraceback (most recent call last) <ipython-input-3-f55de6eaf234> in <module>() ----> 1 meo = MaxEntOptimizer(xor, [[0,1], [0,2], [1,2]]) NameError: name 'MaxEntOptimizer' is not defined In : meo.optimize() NameErrorTraceback (most recent call last) <ipython-input-4-ffaf8eaa6f1b> in <module>() ----> 1 meo.optimize() NameError: name 'meo' is not defined In : dp = meo.construct_dist() NameErrorTraceback (most recent call last) <ipython-input-5-6f4e9c4137b2> in <module>() ----> 1 dp = meo.construct_dist() NameError: name 'meo' is not defined In : print(dp) NameErrorTraceback (most recent call last) <ipython-input-6-2b98a304971b> in <module>() ----> 1 print(dp) NameError: name 'dp' is not defined
There are three special functions to handle common optimization problems:
In : from dit.algorithms import maxent_dist, marginal_maxent_dists, pid_broja
The first is maximum entropy distributions with specific fixed marginals. It encapsulates the steps run above:
In : print(maxent_dist(xor, [[0,1], [0,2], [1,2]])) Class: Distribution Alphabet: ('0', '1') for all rvs Base: linear Outcome Class: str Outcome Length: 3 RV Names: None x p(x) 000 2444573/19556583 001 2048462/16387697 010 1/8 011 3274257/26194055 100 3976016/31808129 101 1795124/14360991 110 1927555/15420441 111 1/8
The second constructs several maximum entropy distributions, each with all subsets of variables of a particular size fixed:
In : k0, k1, k2, k3 = marginal_maxent_dists(xor)
k0 is the maxent dist corresponding the same alphabets as
k1 fixes \(p(x_0)\), \(p(x_1)\), and \(p(x_2)\);
k2 fixes \(p(x_0, x_1)\), \(p(x_0, x_2)\), and \(p(x_1, x_2)\) (as in the
maxent_dist example above), and finally
k3 fixes \(p(x_0, x_1, x_2)\) (e.g. is the distribution we started with).
Partial Information Decomposition¶
Finally, we have
pid_broja(). This computes the 2 input, 1 output partial information decomposition as defined [BRO+14]. We can compute the partial information decomposition where \(X_0\) and \(X_1\) are interpreted as inputs, and \(X_2\) as the output, with the following code:
In : sources = [, ] In : target =  In : pid_broja(xor, sources, target) TypeErrorTraceback (most recent call last) <ipython-input-12-91b5087e14d1> in <module>() ----> 1 pid_broja(xor, sources, target) TypeError: 'module' object is not callable
indicating that the redundancy (R) is zero, neither input provides unique informaiton (U0, U1), and there is 1 bit of synergy (S).
Creating Your Own Optimizer¶
dit.algorithms.scipy_optimizers provides two optimization classes for optimizing some quantity while matching arbitrary margins from a reference distribution. The first,
dit.algorithms.scipy_optimizers.BaseConvexOptimizer, is for use when the objective is convex, while the second,
dit.algorithms.scipy_optimizers.BaseNonConvexOptimizer is for use when the objective is non-convex. Simply subclass one of these two and impliment the
objective method and it is good to go.