def logpdf(x, a, b, loc=0, scale=1):
x, a, b, loc, scale = _promote_args_inexact("beta.logpdf", x, a, b, loc,
scale)
one = lax._const(x, 1)
shape_term = lax.neg(betaln(a, b))
y = lax.div(lax.sub(x, loc), scale)
log_linear_term = lax.add(xlogy(lax.sub(a, one), y),
xlog1py(lax.sub(b, one), lax.neg(y)))
log_probs = lax.sub(lax.add(shape_term, log_linear_term), lax.log(scale))
return where(logical_or(lax.gt(x, lax.add(loc, scale)), lax.lt(x, loc)),
-inf, log_probs)
def triangular_solve_jvp_rule_a(g_a, ans, a, b, left_side, lower, transpose_a,
conjugate_a, unit_diagonal):
m, n = b.shape[-2:]
k = 1 if unit_diagonal else 0
g_a = np.tril(g_a, k=-k) if lower else np.triu(g_a, k=k)
g_a = lax.neg(g_a)
g_a = np.swapaxes(g_a, -1, -2) if transpose_a else g_a
g_a = np.conj(g_a) if conjugate_a else g_a
dot = partial(lax.dot if g_a.ndim == 2 else lax.batch_matmul,
precision=lax.Precision.HIGHEST)
def a_inverse(rhs):
return triangular_solve(a, rhs, left_side, lower, transpose_a,
conjugate_a, unit_diagonal)
# triangular_solve is about the same cost as matrix multplication (~n^2 FLOPs
# for matrix/vector inputs). Order these operations in whichever order is
# cheaper.
if left_side:
assert g_a.shape[-2:] == a.shape[-2:] == (m, m) and ans.shape[-2:] == (
m, n)
if m > n:
return a_inverse(dot(g_a, ans)) # A^{-1} (∂A X)
else:
return dot(a_inverse(g_a), ans) # (A^{-1} ∂A) X
else:
assert g_a.shape[-2:] == a.shape[-2:] == (n, n) and ans.shape[-2:] == (
m, n)
if m < n:
return a_inverse(dot(ans, g_a)) # (X ∂A) A^{-1}
else:
return dot(ans, a_inverse(g_a)) # X (∂A A^{-1})
def logpdf(x, loc=0, scale=1):
x, loc, scale = _promote_args_inexact("norm.logpdf", x, loc, scale)
two = _constant_like(x, 2)
scale_sqrd = lax.pow(scale, two)
log_normalizer = lax.log(lax.mul(_constant_like(x, 2 * np.pi), scale_sqrd))
quadratic = lax.div(lax.pow(lax.sub(x, loc), two), scale_sqrd)
return lax.div(lax.neg(lax.add(log_normalizer, quadratic)), two)
def arccosh(x):
# Note: arccosh is multi-valued for complex input, and lax.acosh uses a different
# convention than np.arccosh.
out = lax.acosh(*_promote_args_inexact("arccosh", x))
if dtypes.issubdtype(out.dtype, np.complexfloating):
out = _where(real(out) < 0, lax.neg(out), out)
return out
def logpdf(x, loc=0, scale=1):
x, loc, scale = _promote_args_inexact("cauchy.logpdf", x, loc, scale)
pi = _constant_like(x, np.pi)
scaled_x = lax.div(lax.sub(x, loc), scale)
normalize_term = lax.log(lax.mul(pi, scale))
return lax.neg(
lax.add(normalize_term, lax.log1p(lax.mul(scaled_x, scaled_x))))
def cdf(x, loc=0, scale=1):
x, loc, scale = _promote_args_inexact("laplace.cdf", x, loc, scale)
half = _constant_like(x, 0.5)
one = _constant_like(x, 1)
zero = _constant_like(x, 0)
diff = lax.div(lax.sub(x, loc), scale)
return lax.select(lax.le(diff, zero), lax.mul(half, lax.exp(diff)),
lax.sub(one, lax.mul(half, lax.exp(lax.neg(diff)))))
def logpdf(x, b, loc=0, scale=1):
x, b, loc, scale = _promote_args_inexact("pareto.logpdf", x, b, loc, scale)
one = _constant_like(x, 1)
scaled_x = lax.div(lax.sub(x, loc), scale)
normalize_term = lax.log(lax.div(scale, b))
log_probs = lax.neg(
lax.add(normalize_term, lax.mul(lax.add(b, one), lax.log(scaled_x))))
return where(lax.lt(x, lax.add(loc, scale)), -inf, log_probs)
def logpmf(k, n, a, b, loc=0):
"""JAX implementation of scipy.stats.betabinom.logpmf."""
k, n, a, b, loc = _promote_args_inexact("betabinom.logpmf", k, n, a, b,
loc)
y = lax.sub(lax.floor(k), loc)
one = _lax_const(y, 1)
zero = _lax_const(y, 0)
combiln = lax.neg(
lax.add(lax.log1p(n),
betaln(lax.add(lax.sub(n, y), one), lax.add(y, one))))
beta_lns = lax.sub(betaln(lax.add(y, a), lax.add(lax.sub(n, y), b)),
betaln(a, b))
log_probs = lax.add(combiln, beta_lns)
y_cond = logical_or(lax.lt(y, lax.neg(loc)), lax.gt(y, lax.sub(n, loc)))
log_probs = where(y_cond, -inf, log_probs)
n_a_b_cond = logical_or(logical_or(lax.lt(n, one), lax.lt(a, zero)),
lax.lt(b, zero))
return where(n_a_b_cond, nan, log_probs)
def triangular_solve_jvp_rule_a(g_a, ans, a, b, left_side, lower, transpose_a,
conjugate_a):
g_a = lax.neg(g_a)
g_a = np.swapaxes(g_a, -1, -2) if transpose_a else g_a
tmp = triangular_solve(a, g_a, left_side, lower, transpose_a, conjugate_a)
dot = lax.dot if g_a.ndim == 2 else lax.batch_matmul
if left_side:
return dot(tmp, ans)
else:
return dot(ans, tmp)
def logaddexp(x1, x2):
x1, x2 = _promote_args_inexact("logaddexp", x1, x2)
amax = lax.max(x1, x2)
if dtypes.issubdtype(x1.dtype, np.floating):
delta = lax.sub(x1, x2)
return lax.select(lax_internal._isnan(delta),
lax.add(x1, x2), # NaNs or infinities of the same sign.
lax.add(amax, lax.log1p(lax.exp(lax.neg(lax.abs(delta))))))
else:
delta = lax.sub(lax.add(x1, x2), lax.mul(amax, _constant_like(amax, 2)))
out = lax.add(amax, lax.log1p(lax.exp(delta)))
return lax.complex(lax.real(out), _wrap_between(lax.imag(out), np.pi))
def logpdf(x, df, loc=0, scale=1):
x, df, loc, scale = _promote_args_inexact("t.logpdf", x, df, loc, scale)
two = _lax_const(x, 2)
scaled_x = lax.div(lax.sub(x, loc), scale)
df_over_two = lax.div(df, two)
df_plus_one_over_two = lax.add(df_over_two, _lax_const(x, 0.5))
normalize_term_const = lax.mul(lax.mul(scale, scale), _lax_const(x, np.pi))
normalize_term_tmp = lax.div(lax.log(lax.mul(normalize_term_const, df)), two)
normalize_term = lax.sub(lax.add(lax.lgamma(df_over_two), normalize_term_tmp),
lax.lgamma(df_plus_one_over_two))
quadratic = lax.div(lax.mul(scaled_x, scaled_x), df)
return lax.neg(lax.add(normalize_term, lax.mul(df_plus_one_over_two, lax.log1p(quadratic))))
def logpdf(x, df, loc=0, scale=1):
x, df, loc, scale = _promote_args_inexact("chi2.logpdf", x, df, loc, scale)
one = _constant_like(x, 1)
two = _constant_like(x, 2)
y = lax.div(lax.sub(x, loc), scale)
df_on_two = lax.div(df, two)
kernel = lax.sub(lax.mul(lax.sub(df_on_two, one), lax.log(y)), lax.div(y,two))
nrml_cnst = lax.neg(lax.add(lax.lgamma(df_on_two),lax.div(lax.mul(lax.log(two), df),two)))
log_probs = lax.add(lax.sub(nrml_cnst, lax.log(scale)), kernel)
return where(lax.lt(x, loc), -inf, log_probs)
def triangular_solve_jvp_rule_a(g_a, ans, a, b, left_side, lower, transpose_a,
conjugate_a, unit_diagonal):
k = 1 if unit_diagonal else 0
g_a = np.tril(g_a, k=-k) if lower else np.triu(g_a, k=k)
g_a = lax.neg(g_a)
g_a = np.swapaxes(g_a, -1, -2) if transpose_a else g_a
g_a = np.conj(g_a) if conjugate_a else g_a
tmp = triangular_solve(a, g_a, left_side, lower, transpose_a, conjugate_a,
unit_diagonal)
dot = lax.dot if g_a.ndim == 2 else lax.batch_matmul
if left_side:
return dot(tmp, ans)
else:
return dot(ans, tmp)
def _random_poisson(rng_key, lmbda, shape):
"""
References
----------
.. [1] Knuth, Donald E. Art of computer programming, volume 2:
Seminumerical algorithms. Addison-Wesley Professional, 2014 (p 137).
"""
L = lax.exp(lax.neg(lmbda))
k = np.zeros(shape=shape)
p = np.ones(shape=shape)
is_done = p < L
while not is_done.all():
_, rng_key = random.split(rng_key)
u = random.uniform(rng_key, shape=shape)
p = np.where(is_done, p, u * p)
k = np.where(is_done, k, k + 1)
is_done = p < L
return k
series, = series_in
primal_out = prim.bind(x)
c0, cs = jet(deriv, primals_in, series_in)
c = [c0] + cs
u = [x] + series
v = [primal_out] + [None] * len(series)
for k in range(1, len(v)):
v[k] = fact(k - 1) * sum(
_scale(k, j) * c[k - j] * u[j] for j in range(1, k + 1))
primal_out, *series_out = v
return primal_out, series_out
def_deriv(
lax.erf_p, lambda x: lax.mul(lax._const(x, 2. / np.sqrt(np.pi)),
lax.exp(lax.neg(lax.square(x)))))
def def_comp(prim, comp):
"""
Define the jet rule for a primitive in terms of a composition of simpler primitives.
"""
jet_rules[prim] = partial(jet, comp)
def_comp(lax.expm1_p, lambda x: lax.exp(x) - 1)
def_comp(lax.log1p_p, lambda x: lax.log(1 + x))
def_comp(lax.sqrt_p, lambda x: x**0.5)
def_comp(lax.rsqrt_p, lambda x: x**-0.5)
def_comp(lax.asinh_p, lambda x: lax.log(x + lax.sqrt(lax.square(x) + 1)))
def_comp(lax.acosh_p, lambda x: lax.log(x + lax.sqrt(lax.square(x) - 1)))
def eager_unary(state):
a = jax.device_put(1)
lax.neg(a).block_until_ready()
while state:
lax.neg(a).block_until_ready()
def eager_unary_dispatch(state):
a = jax.device_put(1)
lax.neg(a)
while state:
lax.neg(a)
def expit(x):
x = asarray(x)
one = lax._const(x, 1)
return lax.div(one, lax.add(one, lax.exp(lax.neg(x))))
def expn_jvp(n, primals, tangents):
(x, ), (x_dot, ) = primals, tangents
return expn(n, x), lax.mul(lax.neg(x_dot),
expn(lax.sub(n, _lax_const(n, 1)), x))
def logpdf(x):
x, = _promote_args_inexact("logistic.logpdf", x)
two = _lax_const(x, 2)
half_x = lax.div(x, two)
return lax.mul(lax.neg(two), jnp.logaddexp(half_x, lax.neg(half_x)))
def logpdf(x, loc=0, scale=1):
x, loc, scale = _promote_args_inexact("expon.logpdf", x, loc, scale)
log_scale = lax.log(scale)
linear_term = lax.div(lax.sub(x, loc), scale)
log_probs = lax.neg(lax.add(linear_term, log_scale))
return where(lax.lt(x, loc), -inf, log_probs)
def sf(x):
return expit(lax.neg(x))
def sinh(x):
x, = _promote_to_result_dtype(onp.sinh, x)
return lax.div(lax.sub(lax.exp(x), lax.exp(lax.neg(x))),
_constant_like(x, 2))
def logpdf(x, loc=0, scale=1):
x, loc, scale = _promote_args_inexact("laplace.logpdf", x, loc, scale)
two = _constant_like(x, 2)
linear_term = lax.div(lax.abs(lax.sub(x, loc)), scale)
return lax.neg(lax.add(linear_term, lax.log(lax.mul(two, scale))))
def fun(x, y, z): pred = lax.lt(x, 3) true_fun = lambda y: y false_fun = lambda z: lax.neg(z) return lax.cond(pred, y, true_fun, z, false_fun)
def cosh(x):
x, = _promote_to_result_dtype(onp.cosh, x)
return lax.div(lax.add(lax.exp(x), lax.exp(lax.neg(x))),
_constant_like(x, 2))
def entr(x):
x, = _promote_args_inexact("entr", x)
return lax.select(lax.lt(x, _constant_like(x, 0)),
lax.full_like(x, -np.inf), lax.neg(xlogy(x, x)))
def expit(x):
x, = _promote_args_inexact("expit", x)
one = _lax_const(x, 1)
return lax.div(one, lax.add(one, lax.exp(lax.neg(x))))
def deriv_prop(prim, deriv, primals_in, series_in):
x, = primals_in
series, = series_in
primal_out = prim.bind(x)
c0, cs = jet(deriv, primals_in, series_in)
c = [c0] + cs
u = [x] + series
v = [primal_out] + [None] * len(series)
for k in range(1, len(v)):
v[k] = fact(k-1) * sum(_scale(k, j) * c[k-j] * u[j] for j in range(1, k + 1))
primal_out, *series_out = v
return primal_out, series_out
def_deriv(lax.erf_p, lambda x: lax.mul(lax._const(x, 2. / np.sqrt(np.pi)), lax.exp(lax.neg(lax.square(x)))))
def def_comp(prim, comp):
"""
Define the jet rule for a primitive in terms of a composition of simpler primitives.
"""
jet_rules[prim] = partial(jet, comp)
def_comp(lax.expm1_p, lambda x: lax.exp(x) - 1)
def_comp(lax.log1p_p, lambda x: lax.log(1 + x))
def_comp(lax.sqrt_p, lambda x: x ** 0.5)
def_comp(lax.rsqrt_p, lambda x: x ** -0.5)
def_comp(lax.asinh_p, lambda x: lax.log(x + lax.sqrt(lax.square(x) + 1)))
def_comp(lax.acosh_p, lambda x: lax.log(x + lax.sqrt(lax.square(x) - 1)))
def logpdf(self, x):
x = x * 1.0
return lax.add(
xlogy(x, lax.log(self.lmbda)),
lax.add(lax.neg(self.lmbda), lax.neg(lax.lgamma(x))),
)