def apply(*given):
x_equality, inequality = given
assert inequality.is_Unequality
xy_probability, zero = inequality.args
assert xy_probability.is_Probability
assert zero.is_zero
xy_joint = xy_probability.arg
assert x_equality.is_Equality
x_t_given, x_t = x_equality.args
assert x_t_given.is_Conditioned
_x_t, x_t_historic = x_t_given.args
assert x_t == _x_t
x, t = x_t.args
assert x_t_historic == x[:t].as_boolean()
x_boolean, y_boolean = xy_joint._argset
if x_boolean != x.as_boolean():
x_boolean, y_boolean = y_boolean, x_boolean
y = y_boolean.lhs
assert t.is_positive
i = Symbol.i(integer=True)
j = Symbol.j(integer=True)
return Equality(ArgMax[i](P(y[i] | x)),
ArgMax[i](P(y[i]) * Product[j](P(x[j] | y[i]))),
given=given)
def apply(*given):
x, y = process_assumptions(*given)
n, d = x.shape
t = Symbol.t(integer=True, domain=[0, n - 1])
i = Symbol.i(integer=True)
joint_probability = P(x[:t + 1], y[:t + 1])
emission_probability = P(x[i] | y[i])
transition_probability = P(y[i] | y[i - 1])
y_given_x_probability = P(y | x)
y = pspace(y).symbol
G = Symbol.G(shape=(d, d),
definition=LAMBDA[y[i - 1],
y[i]](-sympy.log(transition_probability)))
s = Symbol.s(shape=(n, ), definition=LAMBDA[t](-log(joint_probability)))
x = Symbol.x(shape=(n, d),
definition=LAMBDA[y[i], i](-sympy.log(emission_probability)))
z = Symbol.z(shape=(n, d),
definition=LAMBDA[y[t], t](Piecewise(
(Sum[y[0:t]](sympy.E**-s[t]), t > 0),
(sympy.E**-s[t], True))))
# assert z.is_extended_positive
x_quote = Symbol.x_quote(shape=(n, d),
definition=-LAMBDA[t](sympy.log(z[t])))
# assert x_quote.is_real
return Equality(x_quote[t + 1], -log(Sum(exp(-x_quote[t] - G))) + x[t + 1], given=given), \
Equality(-log(y_given_x_probability), log(Sum(exp(-x_quote[n - 1]))) + s[n - 1], given=given)
def apply(*given):
given_x, given_y, unequality = given
assert given_x.is_Equality
assert given_y.is_Equality
assert unequality.is_Unequality
assert unequality.rhs.is_zero
x = given_x.rhs
y = given_y.rhs
assert unequality.lhs.arg.lhs in {x, y}
eq = given_x.lhs
assert eq.is_Conditioned
_x, z = eq.args
eq = given_y.lhs
assert eq.is_Conditioned
_y, _z = eq.args
assert x == _x
assert y == _y
assert z == _z
assert x.is_random and y.is_random and z.is_random
return Equality(P(x, y, given=z), P(x, y), given=given)
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
z = Symbol.z(real=True, random=True)
Eq << apply(x.is_independent_of(z), y.is_independent_of(z), Unequal(P(y), 0))
Eq << bayes.equality.equality.product.apply(Eq[0])
Eq.bayes_yz = bayes.equality.equality.product.apply(Eq[1])
Eq.bayes_xyz = bayes.theorem.apply(Eq[0].domain_definition(), var=P(x, y))
Eq << Eq[2].subs(Eq[1].reversed)
Eq.given_addition = bayes.equality.equality.given_addition.condition_probability.apply(Eq[0], Eq[-1])
Eq << bayes.inequality.inequality.joint.apply(Eq[-1])
Eq << bayes.theorem.apply(Eq[-1], var=x)
Eq << Eq.bayes_xyz.subs(Eq[-1])
Eq << Eq[-1].subs(Eq.bayes_yz)
Eq << algebre.scalar.inequality.equality.apply(Eq[-1], Eq[0].domain_definition())
Eq << Eq[-1].subs(Eq.given_addition)
Eq << bayes.theorem.apply(Eq[2], var=x)
Eq << Eq[-2].subs(Eq[-1].reversed)
Eq << Eq[-1].reversed
def apply(*given):
equality, unequal = given
if equality.is_Unequality:
equality, unequal = unequal, equality
assert unequal.is_Unequality
assert unequal.lhs.is_Probability
assert unequal.rhs.is_zero
_y, z = unequal.lhs.arg.args
assert equality.is_Equality
lhs, rhs = equality.args
if lhs.is_Probability:
assert rhs.is_Probability
lhs = lhs.arg
rhs = rhs.arg
assert lhs.is_Conditioned
x, y = lhs.args
assert x == rhs
if _y != y:
_y, z = z, _y
assert _y == y
if x.is_symbol:
return Equality(x | y & z, x | z, given=given)
else:
return Equality(P(x, given=y & z), P(x, given=z), given=given)
def apply(*given):
x, y = process_assumptions(*given)
n, d = x.shape
t = Symbol.t(integer=True, domain=[0, n - 1])
i = Symbol.i(integer=True)
joint_probability_t = P(x[:t + 1], y[:t + 1])
joint_probability = P(x, y)
emission_probability = P(x[i] | y[i])
transition_probability = P(y[i] | y[i - 1])
y = pspace(y).symbol
G = Symbol.G(shape=(d, d),
definition=LAMBDA[y[i - 1],
y[i]](-sympy.log(transition_probability)))
s = Symbol.s(shape=(n, ), definition=LAMBDA[t](-log(joint_probability_t)))
x = Symbol.x(shape=(n, d),
definition=LAMBDA[y[i], i](-sympy.log(emission_probability)))
x_quote = Symbol.x_quote(shape=(n, d),
definition=LAMBDA[y[t], t](Piecewise(
(MIN[y[0:t]](s[t]), t > 0), (s[0], True))))
assert x_quote.is_real
return Equality(x_quote[t + 1], x[t + 1] + MIN(x_quote[t] + G), given=given), \
Equality(MAX[y](joint_probability), exp(-MIN(x_quote[n - 1])), given=given)
def prove(Eq):
x = Symbol.x(real=True, random=True)
Eq << apply(Unequal(P(x), 0))
Eq << GreaterThan(P(x), 0, plausible=True)
Eq <<= Eq[-1] & Eq[0]
def apply(*given):
x, y = process_assumptions(*given)
n, _ = x.shape
t = Symbol.t(integer=True, domain=[0, n - 1])
i = Symbol.i(integer=True)
return Equality(P(x[:t + 1], y[:t + 1]),
P(x[0] | y[0]) * P(y[0]) *
Product[i:1:t](P(y[i] | y[i - 1]) * P(x[i] | y[i])),
given=given)
def prove(Eq):
# d is the number of output labels
# oo is the length of the sequence
d = Symbol.d(integer=True, positive=True)
n = Symbol.n(integer=True, positive=True)
x = Symbol.x(shape=(n, d), real=True, random=True, given=True)
y = Symbol.y(shape=(n, ),
integer=True,
domain=[0, d - 1],
random=True,
given=True)
i = Symbol.i(integer=True)
t = Symbol.t(integer=True, domain=[0, n])
joint_probability_t = P(x[:t + 1], y[:t + 1])
emission_probability = P(x[i] | y[i])
transition_probability = P(y[i] | y[i - 1])
given = Equality(
joint_probability_t,
P(x[0] | y[0]) * P(y[0]) *
Product[i:1:t](transition_probability * emission_probability))
y = pspace(y).symbol
G = Symbol.G(shape=(d, d),
definition=LAMBDA[y[i - 1],
y[i]](-sympy.log(transition_probability)))
s = Symbol.s(shape=(n, ), definition=LAMBDA[t](-log(joint_probability_t)))
x = Symbol.x(shape=(n, d),
definition=LAMBDA[y[i], i](-sympy.log(emission_probability)))
Eq.s_definition, Eq.G_definition, Eq.x_definition, Eq.given, Eq.logits_recursion = apply(
G, x, s, given=given)
Eq << Eq.s_definition.this.rhs.subs(Eq.given)
Eq << Eq[-1].this.rhs.args[1].as_Add()
Eq << Eq[-1].subs(Eq.x_definition.subs(i, 0).reversed)
Eq << Eq[-1].this.rhs.args[-1].args[1].as_Add()
Eq << Eq[-1].this.rhs.args[-1].args[1].function.as_Add()
Eq << Eq[-1].this.rhs.args[-1].args[1].as_two_terms()
Eq << Eq[-1].subs(Eq.x_definition.reversed).subs(Eq.G_definition.reversed)
Eq << Eq[-1].this.rhs.args[-1].bisect({0})
Eq << Eq[-1].subs(t, t + 1) - Eq[-1]
s = Eq.s_definition.lhs.base
Eq << Eq[-1].this.rhs.simplify() + s[t]
def apply(given):
assert given.is_Equality
lhs, rhs = given.args
assert lhs.is_Conditioned
x, y = lhs.args
assert x == rhs
return Equality(P(x, y), P(x) * P(y), given=given)
def prove(Eq):
t = Symbol.t(integer=True, positive=True)
n = Symbol.n(integer=True, positive=True)
m = Symbol.m(integer=True, positive=True)
x = Symbol.x(real=True, shape=(n, ), random=True)
y = Symbol.y(real=True, shape=(m, ), random=True)
Eq << apply(Equal(x[t] | x[:t], x[t]), Unequal(P(x, y), 0))
i = Eq[-1].lhs.variable
j = Eq[-1].rhs.function.args[-1].variable
Eq << bayes.inequality.et.apply(Eq[1]).split()
Eq << bayes.theorem.apply(Eq[-2], var=y[i])
Eq.y_given_x = algebre.scalar.inequality.equality.apply(Eq[-1],
Eq[-3]).reversed
Eq << bayes.inequality.inequality.joint_slice.apply(Eq[1], [j, i])
Eq << bayes.inequality.et.apply(Eq[-1]).split()
Eq << bayes.theorem.apply(Eq[-1], var=x)
Eq.y_given_x = Eq.y_given_x.subs(Eq[-1])
Eq << Eq.y_given_x.argmax((i, ))
Eq << bayes.inequality.inequality.joint_slice.apply(Eq[1], Slice[:t, i])
Eq.xt_given_x_historic = bayes.equality.equality.given_addition.joint_probability.apply(
Eq[0], Eq[-1])
Eq.xt_given_yi_nonzero = bayes.inequality.inequality.conditioned.apply(
Eq[-1], wrt=y[i])
Eq << P(x[:t + 1] | y[i]).bayes_theorem(x[:t]).as_Or()
Eq << (Eq[-1] & Eq.xt_given_yi_nonzero).split()
Eq << Eq[-1].subs(Eq.xt_given_x_historic)
Eq << algebre.scalar.inequality.equality.apply(Eq[-1],
Eq.xt_given_yi_nonzero)
Eq << Eq[-1].product((t, 1, n - 1))
t = Eq[-1].rhs.variable
Eq << Eq[-1] / Eq[-1].lhs.args[-1]
Eq << Eq[-1].this.rhs.limits_subs(t, j)
Eq << Eq[2].subs(Eq[-1].reversed)
def apply(given):
assert given.is_Unequality
assert given.lhs.is_Probability
assert given.rhs.is_zero
eq = given.lhs.arg
assert eq.is_Conditioned
return Unequal(P(eq.lhs), 0, given=given)
def apply(given, wrt):
assert given.is_Unequality
assert given.lhs.is_Probability
assert given.rhs.is_zero
probability = given.lhs
p = probability.marginalize(wrt)
return Unequal(P(p.arg | wrt), 0, given=given)
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
Eq << apply(Unequal(P(x | y), 0))
Eq << bayes.inequality.inequality.joint.apply(Eq[0])
Eq << bayes.inequality.et.apply(Eq[-1]).split()
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
Eq << apply(Unequal(P(x, y), 0))
Eq.x_marginal_probability, Eq.y_marginal_probability = P(x, y).total_probability_theorem(y), P(x, y).total_probability_theorem(x)
_y = Eq.x_marginal_probability.lhs.variable
_x = Eq.y_marginal_probability.lhs.variable
Eq << bayes.inequality.strict_greater_than.apply(Eq[0])
Eq <<= Eq[-1].integrate((_y,)), Eq[-1].integrate((_x,))
Eq <<= Eq[-2].subs(Eq.x_marginal_probability), Eq[-1].subs(Eq.y_marginal_probability)
Eq <<= algebre.scalar.strict_greater_than.inequality.apply(Eq[-1]) & algebre.scalar.strict_greater_than.inequality.apply(Eq[-2])
def apply(*given):
equality, inequality = given
assert equality.is_Equality
assert inequality.is_Unequality
assert inequality.rhs.is_zero
inequality.lhs.is_Probability
x = inequality.lhs.arg
lhs, rhs = equality.args
assert lhs.is_Probability
_x, y = lhs.arg.args
if x != _x:
_x, y = y, _x
assert x == _x
assert rhs == P(x) * P(y)
return Equality(P(y, given=x), P(y), given=given)
def apply(given, wrt=None):
assert given.is_Unequality
assert given.lhs.is_Probability
assert given.rhs.is_zero
eqs = given.lhs.arg
assert eqs.is_And
if wrt is not None:
lhs, rhs = [], []
for eq in eqs.args:
if eq.lhs in wrt:
rhs.append(eq)
else:
lhs.append(eq)
lhs = And(*lhs)
rhs = And(*rhs)
return And(Unequal(P(lhs), 0), Unequal(P(rhs), 0), given=given)
else:
return And(*[Unequal(P(eq), 0) for eq in eqs.args], given=given)
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
z = Symbol.z(real=True, random=True)
Eq << apply(Equality(x | y, x), Unequal(P(y, z), 0))
Eq << bayes.inequality.inequality.conditioned.apply(Eq[1], y)
Eq << bayes.equality.equality.given_addition.condition_probability.apply(Eq[0], Eq[-1])
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
given = Equality(x | y, x)
Eq << apply(given, Unequal(P(x), 0))
Eq << bayes.equality.equality.product.apply(given)
Eq << bayes.equality.equality.independence.apply(Eq[-1], Eq[1])
def prove(Eq):
n = Symbol.n(integer=True, domain=[2, oo])
x = Symbol.x(real=True, shape=(n,), random=True)
Eq << apply(Unequal(P(x), 0))
t = Symbol.t(integer=True, domain=[1, n - 1])
Eq << Eq[0].this.lhs.arg.bisect(Slice[:t])
Eq << bayes.inequality.et.apply(Eq[-1]).split()
def apply(given):
assert given.is_Unequality
assert given.lhs.is_Probability
assert given.rhs.is_zero
eq = given.lhs.arg
x, _x = eq.args
assert _x == pspace(x).symbol
n = x.shape[0]
t = Symbol.t(integer=True, domain=[1, n - 1])
return Unequal(P(x[:t]), 0, given=given)
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
given = Equality(x | y, x)
Eq << apply(given, Unequal(P(x, y), 0))
Eq << bayes.inequality.et.apply(Eq[1]).split()
Eq << bayes.equality.equality.symmetry.apply(Eq[0], Eq[-2])
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
z = Symbol.z(real=True, random=True)
Eq << apply(x.is_independent_of(z), y.is_independent_of(z),
Unequal(P(x, y), 0))
Eq << bayes.inequality.et.apply(Eq[2]).split()
Eq << bayes.equality.equality.conditional_joint_probability.nonzero.apply(
Eq[0], Eq[1], Eq[-1])
def process_assumptions(*given):
x_independence_assumption, y_independence_assumption, xy_independence_assumption, xy_nonzero_assumption = given
assert xy_nonzero_assumption.is_Unequality
assert xy_nonzero_assumption.rhs.is_zero
x = x_independence_assumption.rhs.base
y = y_independence_assumption.lhs.lhs.base
assert y_independence_assumption.lhs.lhs == y_independence_assumption.rhs.lhs
assert xy_nonzero_assumption.lhs == P(x, y)
assert xy_independence_assumption.rhs.base == y
return x, y
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
Eq << apply(Unequal(P(x, y), 0), y)
Eq << bayes.inequality.et.apply(Eq[0]).split()
Eq << bayes.theorem.apply(Eq[-1], var=Eq[0].lhs)
Eq << Eq[0].subs(Eq[-1])
Eq << algebre.scalar.inequality.inequality.apply(Eq[-3], Eq[-1])
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
given = Unequal(P(x), 0)
Eq << apply(given, y)
Eq << Eq[-1].lhs.bayes_theorem(x)
Eq << Eq[-1].as_Or()
Eq << (Eq[-1] & Eq[0]).split()
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
Eq << apply(Unequal(P(x | y), 0))
Eq << Eq[0].domain_definition()
Eq << bayes.theorem.apply(Eq[-1], var=x)
Eq << Eq[0] * Eq[2]
Eq << Eq[-2].subs(Eq[-1])
def prove(Eq):
n = Symbol.n(integer=True, domain=[2, oo])
x = Symbol.x(real=True, shape=(n, ), random=True)
y = Symbol.y(real=True, shape=(n, ), random=True)
t = Symbol.t(integer=True, domain=[1, n - 1])
Eq << apply(Unequal(P(x, y), 0), Slice[:t, :t])
Eq << Eq[0].this.lhs.arg.args[-1].bisect(Slice[:t])
Eq << Eq[-1].this.lhs.arg.args[0].bisect(Slice[:t])
Eq << bayes.inequality.et.apply(Eq[-1], wrt={x[:t], y[:t]}).split()
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
z = Symbol.z(real=True, random=True)
Eq << apply(Equality(x | y, x), Unequal(P(z | y), 0))
Eq << P(x | y, z).bayes_theorem(z)
Eq << Eq[-1].as_Or()
Eq << (Eq[-1] & Eq[1]).split()
Eq << Eq[-1].subs(Eq[0])
Eq << bayes.inequality.inequality.marginal.apply(Eq[1])
Eq << bayes.theorem.apply(Eq[-1], var=x)
Eq << Eq[-3].subs(Eq[-1])
Eq << algebre.scalar.inequality.equality.apply(Eq[-1], Eq[-3])
Eq << Eq[-1].reversed
def prove(Eq):
x = Symbol.x(real=True, random=True)
y = Symbol.y(real=True, random=True)
given = Equality(P(x, y), P(x) * P(y))
Eq << apply(given, Unequal(P(x), 0))
Eq << Eq[-1].simplify()
Eq << bayes.theorem.apply(Eq[1], var=y)
Eq << Eq[-1].subs(Eq[0])
Eq << Eq[-1] - Eq[-1].rhs
Eq << Eq[-1].this.lhs.collect(P(x))
Eq << Eq[-1].as_Or()
Eq << (Eq[-1] & Eq[1]).split()
Eq << Eq[-1].reversed