513 lines
16 KiB
Python
513 lines
16 KiB
Python
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"""
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Finite Discrete Random Variables Module
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See Also
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========
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sympy.stats.frv_types
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sympy.stats.rv
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sympy.stats.crv
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"""
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from itertools import product
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from sympy.concrete.summations import Sum
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from sympy.core.basic import Basic
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from sympy.core.cache import cacheit
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from sympy.core.function import Lambda
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from sympy.core.mul import Mul
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from sympy.core.numbers import (I, nan)
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from sympy.core.relational import Eq
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from sympy.core.singleton import S
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from sympy.core.symbol import (Dummy, Symbol)
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from sympy.core.sympify import sympify
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.piecewise import Piecewise
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from sympy.logic.boolalg import (And, Or)
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from sympy.sets.sets import Intersection
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from sympy.core.containers import Dict
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from sympy.core.logic import Logic
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from sympy.core.relational import Relational
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from sympy.core.sympify import _sympify
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from sympy.sets.sets import FiniteSet
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from sympy.stats.rv import (RandomDomain, ProductDomain, ConditionalDomain,
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PSpace, IndependentProductPSpace, SinglePSpace, random_symbols,
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sumsets, rv_subs, NamedArgsMixin, Density, Distribution)
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class FiniteDensity(dict):
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"""
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A domain with Finite Density.
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"""
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def __call__(self, item):
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"""
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Make instance of a class callable.
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If item belongs to current instance of a class, return it.
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Otherwise, return 0.
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"""
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item = sympify(item)
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if item in self:
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return self[item]
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else:
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return 0
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@property
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def dict(self):
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"""
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Return item as dictionary.
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"""
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return dict(self)
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class FiniteDomain(RandomDomain):
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"""
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A domain with discrete finite support
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Represented using a FiniteSet.
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"""
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is_Finite = True
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@property
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def symbols(self):
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return FiniteSet(sym for sym, val in self.elements)
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@property
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def elements(self):
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return self.args[0]
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@property
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def dict(self):
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return FiniteSet(*[Dict(dict(el)) for el in self.elements])
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def __contains__(self, other):
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return other in self.elements
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def __iter__(self):
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return self.elements.__iter__()
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def as_boolean(self):
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return Or(*[And(*[Eq(sym, val) for sym, val in item]) for item in self])
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class SingleFiniteDomain(FiniteDomain):
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"""
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A FiniteDomain over a single symbol/set
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Example: The possibilities of a *single* die roll.
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"""
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def __new__(cls, symbol, set):
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if not isinstance(set, FiniteSet) and \
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not isinstance(set, Intersection):
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set = FiniteSet(*set)
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return Basic.__new__(cls, symbol, set)
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@property
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def symbol(self):
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return self.args[0]
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@property
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def symbols(self):
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return FiniteSet(self.symbol)
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@property
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def set(self):
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return self.args[1]
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@property
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def elements(self):
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return FiniteSet(*[frozenset(((self.symbol, elem), )) for elem in self.set])
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def __iter__(self):
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return (frozenset(((self.symbol, elem),)) for elem in self.set)
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def __contains__(self, other):
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sym, val = tuple(other)[0]
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return sym == self.symbol and val in self.set
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class ProductFiniteDomain(ProductDomain, FiniteDomain):
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"""
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A Finite domain consisting of several other FiniteDomains
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Example: The possibilities of the rolls of three independent dice
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"""
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def __iter__(self):
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proditer = product(*self.domains)
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return (sumsets(items) for items in proditer)
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@property
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def elements(self):
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return FiniteSet(*self)
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class ConditionalFiniteDomain(ConditionalDomain, ProductFiniteDomain):
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"""
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A FiniteDomain that has been restricted by a condition
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Example: The possibilities of a die roll under the condition that the
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roll is even.
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"""
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def __new__(cls, domain, condition):
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"""
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Create a new instance of ConditionalFiniteDomain class
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"""
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if condition is True:
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return domain
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cond = rv_subs(condition)
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return Basic.__new__(cls, domain, cond)
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def _test(self, elem):
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"""
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Test the value. If value is boolean, return it. If value is equality
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relational (two objects are equal), return it with left-hand side
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being equal to right-hand side. Otherwise, raise ValueError exception.
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"""
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val = self.condition.xreplace(dict(elem))
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if val in [True, False]:
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return val
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elif val.is_Equality:
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return val.lhs == val.rhs
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raise ValueError("Undecidable if %s" % str(val))
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def __contains__(self, other):
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return other in self.fulldomain and self._test(other)
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def __iter__(self):
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return (elem for elem in self.fulldomain if self._test(elem))
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@property
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def set(self):
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if isinstance(self.fulldomain, SingleFiniteDomain):
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return FiniteSet(*[elem for elem in self.fulldomain.set
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if frozenset(((self.fulldomain.symbol, elem),)) in self])
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else:
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raise NotImplementedError(
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"Not implemented on multi-dimensional conditional domain")
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def as_boolean(self):
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return FiniteDomain.as_boolean(self)
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class SingleFiniteDistribution(Distribution, NamedArgsMixin):
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def __new__(cls, *args):
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args = list(map(sympify, args))
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return Basic.__new__(cls, *args)
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@staticmethod
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def check(*args):
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pass
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@property # type: ignore
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@cacheit
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def dict(self):
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if self.is_symbolic:
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return Density(self)
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return {k: self.pmf(k) for k in self.set}
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def pmf(self, *args): # to be overridden by specific distribution
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raise NotImplementedError()
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@property
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def set(self): # to be overridden by specific distribution
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raise NotImplementedError()
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values = property(lambda self: self.dict.values)
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items = property(lambda self: self.dict.items)
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is_symbolic = property(lambda self: False)
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__iter__ = property(lambda self: self.dict.__iter__)
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__getitem__ = property(lambda self: self.dict.__getitem__)
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def __call__(self, *args):
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return self.pmf(*args)
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def __contains__(self, other):
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return other in self.set
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#=============================================
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#========= Probability Space ===============
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#=============================================
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class FinitePSpace(PSpace):
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"""
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A Finite Probability Space
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Represents the probabilities of a finite number of events.
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"""
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is_Finite = True
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def __new__(cls, domain, density):
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density = {sympify(key): sympify(val)
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for key, val in density.items()}
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public_density = Dict(density)
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obj = PSpace.__new__(cls, domain, public_density)
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obj._density = density
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return obj
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def prob_of(self, elem):
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elem = sympify(elem)
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density = self._density
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if isinstance(list(density.keys())[0], FiniteSet):
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return density.get(elem, S.Zero)
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return density.get(tuple(elem)[0][1], S.Zero)
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def where(self, condition):
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assert all(r.symbol in self.symbols for r in random_symbols(condition))
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return ConditionalFiniteDomain(self.domain, condition)
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def compute_density(self, expr):
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expr = rv_subs(expr, self.values)
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d = FiniteDensity()
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for elem in self.domain:
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val = expr.xreplace(dict(elem))
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prob = self.prob_of(elem)
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d[val] = d.get(val, S.Zero) + prob
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return d
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@cacheit
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def compute_cdf(self, expr):
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d = self.compute_density(expr)
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cum_prob = S.Zero
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cdf = []
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for key in sorted(d):
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prob = d[key]
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cum_prob += prob
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cdf.append((key, cum_prob))
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return dict(cdf)
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@cacheit
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def sorted_cdf(self, expr, python_float=False):
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cdf = self.compute_cdf(expr)
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items = list(cdf.items())
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sorted_items = sorted(items, key=lambda val_cumprob: val_cumprob[1])
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if python_float:
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sorted_items = [(v, float(cum_prob))
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for v, cum_prob in sorted_items]
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return sorted_items
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@cacheit
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def compute_characteristic_function(self, expr):
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d = self.compute_density(expr)
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t = Dummy('t', real=True)
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return Lambda(t, sum(exp(I*k*t)*v for k,v in d.items()))
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@cacheit
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def compute_moment_generating_function(self, expr):
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d = self.compute_density(expr)
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t = Dummy('t', real=True)
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return Lambda(t, sum(exp(k*t)*v for k,v in d.items()))
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def compute_expectation(self, expr, rvs=None, **kwargs):
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rvs = rvs or self.values
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expr = rv_subs(expr, rvs)
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probs = [self.prob_of(elem) for elem in self.domain]
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if isinstance(expr, (Logic, Relational)):
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parse_domain = [tuple(elem)[0][1] for elem in self.domain]
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bools = [expr.xreplace(dict(elem)) for elem in self.domain]
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else:
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parse_domain = [expr.xreplace(dict(elem)) for elem in self.domain]
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bools = [True for elem in self.domain]
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return sum([Piecewise((prob * elem, blv), (S.Zero, True))
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for prob, elem, blv in zip(probs, parse_domain, bools)])
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def compute_quantile(self, expr):
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cdf = self.compute_cdf(expr)
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p = Dummy('p', real=True)
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set = ((nan, (p < 0) | (p > 1)),)
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for key, value in cdf.items():
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set = set + ((key, p <= value), )
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return Lambda(p, Piecewise(*set))
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def probability(self, condition):
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cond_symbols = frozenset(rs.symbol for rs in random_symbols(condition))
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cond = rv_subs(condition)
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if not cond_symbols.issubset(self.symbols):
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raise ValueError("Cannot compare foreign random symbols, %s"
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%(str(cond_symbols - self.symbols)))
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if isinstance(condition, Relational) and \
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(not cond.free_symbols.issubset(self.domain.free_symbols)):
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rv = condition.lhs if isinstance(condition.rhs, Symbol) else condition.rhs
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return sum(Piecewise(
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(self.prob_of(elem), condition.subs(rv, list(elem)[0][1])),
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(S.Zero, True)) for elem in self.domain)
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return sympify(sum(self.prob_of(elem) for elem in self.where(condition)))
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def conditional_space(self, condition):
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domain = self.where(condition)
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prob = self.probability(condition)
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density = {key: val / prob
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for key, val in self._density.items() if domain._test(key)}
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return FinitePSpace(domain, density)
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def sample(self, size=(), library='scipy', seed=None):
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"""
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Internal sample method
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Returns dictionary mapping RandomSymbol to realization value.
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"""
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return {self.value: self.distribution.sample(size, library, seed)}
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class SingleFinitePSpace(SinglePSpace, FinitePSpace):
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"""
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A single finite probability space
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Represents the probabilities of a set of random events that can be
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attributed to a single variable/symbol.
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This class is implemented by many of the standard FiniteRV types such as
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Die, Bernoulli, Coin, etc....
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"""
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@property
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def domain(self):
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return SingleFiniteDomain(self.symbol, self.distribution.set)
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@property
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def _is_symbolic(self):
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"""
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Helper property to check if the distribution
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of the random variable is having symbolic
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dimension.
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"""
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return self.distribution.is_symbolic
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@property
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def distribution(self):
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return self.args[1]
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def pmf(self, expr):
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return self.distribution.pmf(expr)
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@property # type: ignore
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@cacheit
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def _density(self):
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return {FiniteSet((self.symbol, val)): prob
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for val, prob in self.distribution.dict.items()}
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||
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|
@cacheit
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|
def compute_characteristic_function(self, expr):
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|
|
if self._is_symbolic:
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|
|
d = self.compute_density(expr)
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t = Dummy('t', real=True)
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|
ki = Dummy('ki')
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|
return Lambda(t, Sum(d(ki)*exp(I*ki*t), (ki, self.args[1].low, self.args[1].high)))
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|
|
expr = rv_subs(expr, self.values)
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|
|
return FinitePSpace(self.domain, self.distribution).compute_characteristic_function(expr)
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|
|
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||
|
|
@cacheit
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|
|
def compute_moment_generating_function(self, expr):
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|
|
if self._is_symbolic:
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|
d = self.compute_density(expr)
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||
|
|
t = Dummy('t', real=True)
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||
|
|
ki = Dummy('ki')
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|
return Lambda(t, Sum(d(ki)*exp(ki*t), (ki, self.args[1].low, self.args[1].high)))
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|
|
expr = rv_subs(expr, self.values)
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||
|
|
return FinitePSpace(self.domain, self.distribution).compute_moment_generating_function(expr)
|
||
|
|
|
||
|
|
def compute_quantile(self, expr):
|
||
|
|
if self._is_symbolic:
|
||
|
|
raise NotImplementedError("Computing quantile for random variables "
|
||
|
|
"with symbolic dimension because the bounds of searching the required "
|
||
|
|
"value is undetermined.")
|
||
|
|
expr = rv_subs(expr, self.values)
|
||
|
|
return FinitePSpace(self.domain, self.distribution).compute_quantile(expr)
|
||
|
|
|
||
|
|
def compute_density(self, expr):
|
||
|
|
if self._is_symbolic:
|
||
|
|
rv = list(random_symbols(expr))[0]
|
||
|
|
k = Dummy('k', integer=True)
|
||
|
|
cond = True if not isinstance(expr, (Relational, Logic)) \
|
||
|
|
else expr.subs(rv, k)
|
||
|
|
return Lambda(k,
|
||
|
|
Piecewise((self.pmf(k), And(k >= self.args[1].low,
|
||
|
|
k <= self.args[1].high, cond)), (S.Zero, True)))
|
||
|
|
expr = rv_subs(expr, self.values)
|
||
|
|
return FinitePSpace(self.domain, self.distribution).compute_density(expr)
|
||
|
|
|
||
|
|
def compute_cdf(self, expr):
|
||
|
|
if self._is_symbolic:
|
||
|
|
d = self.compute_density(expr)
|
||
|
|
k = Dummy('k')
|
||
|
|
ki = Dummy('ki')
|
||
|
|
return Lambda(k, Sum(d(ki), (ki, self.args[1].low, k)))
|
||
|
|
expr = rv_subs(expr, self.values)
|
||
|
|
return FinitePSpace(self.domain, self.distribution).compute_cdf(expr)
|
||
|
|
|
||
|
|
def compute_expectation(self, expr, rvs=None, **kwargs):
|
||
|
|
if self._is_symbolic:
|
||
|
|
rv = random_symbols(expr)[0]
|
||
|
|
k = Dummy('k', integer=True)
|
||
|
|
expr = expr.subs(rv, k)
|
||
|
|
cond = True if not isinstance(expr, (Relational, Logic)) \
|
||
|
|
else expr
|
||
|
|
func = self.pmf(k) * k if cond != True else self.pmf(k) * expr
|
||
|
|
return Sum(Piecewise((func, cond), (S.Zero, True)),
|
||
|
|
(k, self.distribution.low, self.distribution.high)).doit()
|
||
|
|
|
||
|
|
expr = _sympify(expr)
|
||
|
|
expr = rv_subs(expr, rvs)
|
||
|
|
return FinitePSpace(self.domain, self.distribution).compute_expectation(expr, rvs, **kwargs)
|
||
|
|
|
||
|
|
def probability(self, condition):
|
||
|
|
if self._is_symbolic:
|
||
|
|
#TODO: Implement the mechanism for handling queries for symbolic sized distributions.
|
||
|
|
raise NotImplementedError("Currently, probability queries are not "
|
||
|
|
"supported for random variables with symbolic sized distributions.")
|
||
|
|
condition = rv_subs(condition)
|
||
|
|
return FinitePSpace(self.domain, self.distribution).probability(condition)
|
||
|
|
|
||
|
|
def conditional_space(self, condition):
|
||
|
|
"""
|
||
|
|
This method is used for transferring the
|
||
|
|
computation to probability method because
|
||
|
|
conditional space of random variables with
|
||
|
|
symbolic dimensions is currently not possible.
|
||
|
|
"""
|
||
|
|
if self._is_symbolic:
|
||
|
|
self
|
||
|
|
domain = self.where(condition)
|
||
|
|
prob = self.probability(condition)
|
||
|
|
density = {key: val / prob
|
||
|
|
for key, val in self._density.items() if domain._test(key)}
|
||
|
|
return FinitePSpace(domain, density)
|
||
|
|
|
||
|
|
|
||
|
|
class ProductFinitePSpace(IndependentProductPSpace, FinitePSpace):
|
||
|
|
"""
|
||
|
|
A collection of several independent finite probability spaces
|
||
|
|
"""
|
||
|
|
@property
|
||
|
|
def domain(self):
|
||
|
|
return ProductFiniteDomain(*[space.domain for space in self.spaces])
|
||
|
|
|
||
|
|
@property # type: ignore
|
||
|
|
@cacheit
|
||
|
|
def _density(self):
|
||
|
|
proditer = product(*[iter(space._density.items())
|
||
|
|
for space in self.spaces])
|
||
|
|
d = {}
|
||
|
|
for items in proditer:
|
||
|
|
elems, probs = list(zip(*items))
|
||
|
|
elem = sumsets(elems)
|
||
|
|
prob = Mul(*probs)
|
||
|
|
d[elem] = d.get(elem, S.Zero) + prob
|
||
|
|
return Dict(d)
|
||
|
|
|
||
|
|
@property # type: ignore
|
||
|
|
@cacheit
|
||
|
|
def density(self):
|
||
|
|
return Dict(self._density)
|
||
|
|
|
||
|
|
def probability(self, condition):
|
||
|
|
return FinitePSpace.probability(self, condition)
|
||
|
|
|
||
|
|
def compute_density(self, expr):
|
||
|
|
return FinitePSpace.compute_density(self, expr)
|