Optimization

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 [1]: from dit.algorithms.distribution_optimizers import MaxEntOptimizer

In [2]: xor = dit.example_dists.Xor()

In [3]: meo = MaxEntOptimizer(xor, [[0,1], [0,2], [1,2]])

In [4]: meo.optimize()
Out[4]: 
     fun: -3.0000017320700905
     jac: array([-3.00000018, -3.00000015, -3.00000006, -2.9999997 , -2.99999976,
       -2.99999982, -2.99999994, -2.99999964])
 message: 'Optimization terminated successfully.'
    nfev: 892
     nit: 81
    njev: 81
  status: 0
 success: True
       x: array([0.1250001 , 0.12500008, 0.12500006, 0.12500008, 0.12500008,
       0.12500005, 0.12500006, 0.12500007])

In [5]: dp = meo.construct_dist()

In [6]: print(dp)
Class:          Distribution
Alphabet:       ('0', '1') for all rvs
Base:           linear
Outcome Class:  str
Outcome Length: 3
RV Names:       None

x     p(x)
000   1793883/14351063
001   2538379/20307033
010   1569035/12552281
011   6389891/51119127
100   7040815/56326519
101   2856306/22850449
110   1/8
111   1845800/14766399

Helper Functions

There are three special functions to handle common optimization problems:

In [7]: from dit.algorithms import maxent_dist, marginal_maxent_dists

The first is maximum entropy distributions with specific fixed marginals. It encapsulates the steps run above:

In [8]: 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   1257477/10059817
001   1093061/8744487
010   1576401/12611207
011   1184638/9477105
100   1254660/10037279
101   994733/7957865
110   983321/7866569
111   760230/6081839

The second constructs several maximum entropy distributions, each with all subsets of variables of a particular size fixed:

In [9]: k0, k1, k2, k3 = marginal_maxent_dists(xor)

where k0 is the maxent dist corresponding the same alphabets as xor; 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).