There are several operations possible on joint random variables. Let's consider the standard xor distribution:
.. ipython::
In [1]: d = dit.Distribution(['000', '011', '101', '110'], [1/4]*4)
In [2]: d.set_rv_names('XYZ')
.. py:module:: dit.npdist
:mod:`dit` supports two ways of selecting only a subset of random variables. :meth:`~Distribution.marginal` returns a distribution containing only the random variables specified, whereas :meth:`~Distribution.marginalize` return a distribution containing all random variables except the ones specified:
.. ipython::
:doctest:
In [3]: print(d.marginal('XY'))
Class: Distribution
Alphabet: ('0', '1') for all rvs
Base: linear
Outcome Class: str
Outcome Length: 2
RV Names: ('X', 'Y')
x p(x)
00 1/4
01 1/4
10 1/4
11 1/4
In [4]: print(d.marginalize('XY'))
Class: Distribution
Alphabet: ('0', '1') for all rvs
Base: linear
Outcome Class: str
Outcome Length: 1
RV Names: ('Z',)
x p(x)
0 1/2
1 1/2
.. automethod:: Distribution.marginal
.. automethod:: Distribution.marginalize
We can also condition on a subset of random variables:
.. ipython::
In [5]: marginal, cdists = d.condition_on('XY')
@doctest
In [6]: print(marginal)
Class: Distribution
Alphabet: ('0', '1') for all rvs
Base: linear
Outcome Class: str
Outcome Length: 2
RV Names: ('X', 'Y')
x p(x)
00 1/4
01 1/4
10 1/4
11 1/4
@doctest
In [7]: print(cdists[0]) # XY = 00
Class: Distribution
Alphabet: ('0', '1') for all rvs
Base: linear
Outcome Class: str
Outcome Length: 1
RV Names: ('Z',)
x p(x)
0 1
@doctest
In [8]: print(cdists[1]) # XY = 01
Class: Distribution
Alphabet: ('0', '1') for all rvs
Base: linear
Outcome Class: str
Outcome Length: 1
RV Names: ('Z',)
x p(x)
1 1
@doctest
In [9]: print(cdists[2]) # XY = 10
Class: Distribution
Alphabet: ('0', '1') for all rvs
Base: linear
Outcome Class: str
Outcome Length: 1
RV Names: ('Z',)
x p(x)
1 1
@doctest
In [10]: print(cdists[3]) # XY = 11
Class: Distribution
Alphabet: ('0', '1') for all rvs
Base: linear
Outcome Class: str
Outcome Length: 1
RV Names: ('Z',)
x p(x)
0 1
.. automethod:: Distribution.condition_on
.. py:module:: dit.algorithms.lattice
We can construct the join of two random variables:
X \join Y = \min \{ V | V \imore X \land V \imore Y \}
Where \min is understood to be minimizing with respect to the entropy.
.. ipython:: In [11]: from dit.algorithms.lattice import join @doctest In [12]: print(join(d, ['XY'])) Class: ScalarDistribution Alphabet: (0, 1, 2, 3) Base: linear x p(x) 0 1/4 1 1/4 2 1/4 3 1/4
.. autofunction:: join
.. autofunction:: insert_join
We can construct the meet of two random variabls:
X \meet Y = \max \{ V | V \iless X \land V \iless Y \}
Where \max is understood to be maximizing with respect to the entropy.
.. ipython::
In [13]: from dit.algorithms.lattice import meet
In [14]: outcomes = ['00', '01', '10', '11', '22', '33']
In [15]: d2 = dit.Distribution(outcomes, [1/8]*4 + [1/4]*2, sample_space=outcomes)
In [16]: d2.set_rv_names('XY')
@doctest
In [17]: print(meet(d2, ['X', 'Y']))
Class: ScalarDistribution
Alphabet: (0, 1, 2)
Base: linear
x p(x)
0 1/4
1 1/4
2 1/2
.. autofunction:: meet
.. autofunction:: insert_meet
.. py:module:: dit.algorithms.minimal_sufficient_statistic
This method constructs the minimal sufficient statistic of X about Y: X \mss Y:
X \mss Y = \min \{ V | V \iless X \land \I[X:Y] = \I[V:Y] \}
.. ipython::
In [18]: from dit.algorithms import insert_mss
In [19]: d2 = dit.Distribution(['00', '01', '10', '11', '22', '33'], [1/8]*4 + [1/4]*2)
@doctest
In [20]: print(insert_mss(d2, -1, [0], [1]))
Class: Distribution
Alphabet: (('0', '1', '2', '3'), ('0', '1', '2', '3'), ('0', '1', '2'))
Base: linear
Outcome Class: str
Outcome Length: 3
RV Names: None
x p(x)
002 1/8
012 1/8
102 1/8
112 1/8
220 1/4
331 1/4
Again, \min is understood to be over entropies.
.. autofunction:: mss
.. autofunction:: insert_mss