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):
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):
# 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]
