def weibull(x, l, k):
""" Weibull distribution log-likelihood.
:param x: *int, float, np.array.* :math:`x > 0`
:param l: *float.* Scale parameter. :math:`\\lambda > 0`
:param k: *float.* Shape parameter. :math:`k > 0`
"""
if outofbounds(l > 0, k > 0, x > 0):
return -np.inf
return np.sum(np.log(k/l) + (k-1)*np.log(x/l) - (x/l)**k)
def weibull(x, l, k):
""" Weibull distribution log-likelihood.
:param x: *int, float, np.array.* :math:`x > 0`
:param l: *float.* Scale parameter. :math:`\\lambda > 0`
:param k: *float.* Shape parameter. :math:`k > 0`
"""
if fails_constraints(l > 0, k > 0, x > 0):
return -np.inf
return np.sum(np.log(k / l) + (k - 1) * np.log(x / l) - (x / l)**k)
def student_t(x, nu=1):
""" Student's t log-likelihood
:param x: *int, float, np.array.*
:param nu: (optional) *int.* Degress of freedom.
.. math ::
\log{P(x; \\nu)} \propto \log{\Gamma \\left(\\frac{\\nu+1}{2} \\right)} - \
\log{\Gamma \left( \\frac{\\nu}{2} \\right) } - \
\\frac{1}{2}\log{\\nu} - \
\\frac{\\nu+1}{2}\log{\left(1 + \\frac{x^2}{\\nu} \\right)}
"""
return np.sum(np.log(gamma(0.5*(nu + 1))) - np.log(gamma(nu/2.)) - \
0.5*np.log(nu) - (nu+1)/2*np.log(1+x**2/nu))
def student_t(x, nu=1):
""" Student's t log-likelihood
:param x: *int, float, np.array.*
:param nu: (optional) *int.* Degress of freedom.
.. math ::
\log{P(x; \\nu)} \propto \log{\Gamma \\left(\\frac{\\nu+1}{2} \\right)} - \
\log{\Gamma \left( \\frac{\\nu}{2} \\right) } - \
\\frac{1}{2}\log{\\nu} - \
\\frac{\\nu+1}{2}\log{\left(1 + \\frac{x^2}{\\nu} \\right)}
"""
return np.sum(np.log(gamma(0.5*(nu + 1))) - np.log(gamma(nu/2.)) - \
0.5*np.log(nu) - (nu+1)/2*np.log(1+x**2/nu))
def laplace(x, mu, tau):
""" Laplace distribution log-likelihood
:param x: *int, float, np.array.* :math:`-\infty < \mu < \infty`
:param mu: *int, float, np.array.* Location parameter. :math:`-\infty < \mu < \infty`
:param tau: *int, float.* Scale parameter, :math:`\\tau > 0`
.. math ::
\log{P(x; \\mu, \\tau)} \propto \log{\\tau/2} - \\tau \\left|x - \mu \\right|
"""
if fails_constraints(tau > 0):
return -np.inf
return np.sum(np.log(tau) - tau * np.abs(x - mu))
def binomial(k, n, p):
""" Binomial distribution log-likelihood.
:param k: *int, np.array.* Number of successes. :math:`k <= n`
:param n: *int, np.array.* Number of trials. :math:`n > 0`
:param p: *int, float, np.array.* Success probability. :math:`0<= p <= 1`
.. math::
\log{P(k; n, p)} \propto k \log{p} + (n-k)\log{(1-p)}
"""
if k > n:
raise ValueError("k must be less than or equal to n")
if fails_constraints(0 < p, p < 1):
return -np.inf
return np.sum(k * np.log(p) + (n - k) * np.log(1 - p))
def beta(x, alpha=1, beta=1):
""" Beta distribution log-likelihood.
:param x: *float, np.array.* :math:`0 < x < 1`
:param alpha: (optional) *int, float.* Shape parameter, :math:`\\alpha > 0`
:param beta: (optional) *int, float.* Shape parameter, :math:`\\beta > 0`
.. math ::
\log{P(x; \\alpha, \\beta)} \propto (\\alpha - 1)\log{x} + \
(\\beta - 1) \log{(1 - x)}
"""
if fails_constraints(0 < x, x < 1, alpha > 0, beta > 0):
return -np.inf
return np.sum((alpha - 1) * np.log(x) + (beta - 1) * np.log(1 - x))
def laplace(x, mu, tau):
""" Laplace distribution log-likelihood
:param x: *int, float, np.array.* :math:`-\infty < \mu < \infty`
:param mu: *int, float, np.array.* Location parameter. :math:`-\infty < \mu < \infty`
:param tau: *int, float.* Scale parameter, :math:`\\tau > 0`
.. math ::
\log{P(x; \\mu, \\tau)} \propto \log{\\tau/2} - \\tau \\left|x - \mu \\right|
"""
if outofbounds(tau > 0):
return -np.inf
return np.sum(np.log(tau) - tau*np.abs(x - mu))
def beta(x, alpha=1, beta=1):
""" Beta distribution log-likelihood.
:param x: *float, np.array.* :math:`0 < x < 1`
:param alpha: (optional) *int, float.* Shape parameter, :math:`\\alpha > 0`
:param beta: (optional) *int, float.* Shape parameter, :math:`\\beta > 0`
.. math ::
\log{P(x; \\alpha, \\beta)} \propto (\\alpha - 1)\log{x} + \
(\\beta - 1) \log{(1 - x)}
"""
if outofbounds(0 < x, x < 1, alpha > 0, beta > 0):
return -np.inf
return np.sum((alpha - 1)*np.log(x) + (beta - 1)*np.log(1-x))
def binomial(k, n, p):
""" Binomial distribution log-likelihood.
:param k: *int, np.array.* Number of successes. :math:`k <= n`
:param n: *int, np.array.* Number of trials. :math:`n > 0`
:param p: *int, float, np.array.* Success probability. :math:`0<= p <= 1`
.. math::
\log{P(k; n, p)} \propto k \log{p} + (n-k)\log{(1-p)}
"""
if k > n:
raise ValueError("k must be less than or equal to n")
if outofbounds(0 < p, p < 1):
return -np.inf
return np.sum(k*np.log(p) + (n-k)*np.log(1-p))
def cauchy(x, alpha=0, beta=1):
""" Cauchy distribution log-likelihood.
:param x: *int, float, np.array.* :math:`-\infty < x < \infty`
:param alpha: *int, float, nparray.* Location parameter, :math:`-\infty < \\alpha < \infty`
:param beta: *int, float.* Scale parameter, :math:`\\beta > 0`
.. math::
\log{P(x; \\alpha, \\beta)} \propto -\log{\\beta} - \
\log{\left[1 + \left(\\frac{x - \\alpha}{\\beta}\\right)^2\\right]}
"""
if fails_constraints(beta > 0):
return -np.inf
return np.sum(-np.log(beta) - np.log(1 + ((x - alpha) / beta)**2))
def cauchy(x, alpha=0, beta=1):
""" Cauchy distribution log-likelihood.
:param x: *int, float, np.array.* :math:`-\infty < x < \infty`
:param alpha: *int, float, nparray.* Location parameter, :math:`-\infty < \\alpha < \infty`
:param beta: *int, float.* Scale parameter, :math:`\\beta > 0`
.. math::
\log{P(x; \\alpha, \\beta)} \propto -\log{\\beta} - \
\log{\left[1 + \left(\\frac{x - \\alpha}{\\beta}\\right)^2\\right]}
"""
if outofbounds(beta > 0):
return -np.inf
return np.sum(-np.log(beta) - np.log(1 + ((x - alpha)/beta)**2))
def exponential(x, rate=1):
""" Log likelihood of the exponential distribution.
:param x: *int, float, np.array.*
:param rate: (optional) *int, float, np.array.* Rate parameter, :math:`\lambda > 0`. Defaults to 1.
.. math ::
\log{P(x; \lambda)} \propto \log{\lambda} - \lambda x
"""
if outofbounds(rate > 0):
return -np.inf
if np.size(rate) != 1 and len(x) != len(rate):
raise ValueError('If rate is a vector, x must be the same size as rate.'
' We got x={}, rate={}'.format(x, rate))
return np.sum(np.log(rate) - rate*x)
def exponential(x, rate=1):
""" Log likelihood of the exponential distribution.
:param x: *int, float, np.array.*
:param rate: (optional) *int, float, np.array.* Rate parameter, :math:`\lambda > 0`. Defaults to 1.
.. math ::
\log{P(x; \lambda)} \propto \log{\lambda} - \lambda x
"""
if fails_constraints(x > 0, rate > 0):
return -np.inf
if np.size(rate) != 1 and len(x) != len(rate):
raise ValueError('If rate is a vector, x must be the same size as rate.'
' We got x={}, rate={}'.format(x, rate))
return np.sum(np.log(rate) - rate*x)
def poisson(x, rate=1):
""" Poisson distribution log-likelihood.
:param x: *int, float, np.array.* Event count.
:param rate: (optional) *int, float, np.array.* Rate parameter, :math:`\lambda > 0`. Defaults to 1.
.. math ::
\log{P(x; \lambda)} \propto x \log{\lambda} - \lambda
"""
if outofbounds(rate > 0):
return -np.inf
if np.size(rate) != 1 and len(x) != len(rate):
raise ValueError('If rate is a vector, x must be the same size as rate.'
' We got x={}, rate={}'.format(x, rate))
return np.sum(x*np.log(rate)) - np.size(x)*rate
def poisson(x, rate=1):
""" Poisson distribution log-likelihood.
:param x: *int, float, np.array.* Event count.
:param rate: (optional) *int, float, np.array.* Rate parameter, :math:`\lambda > 0`. Defaults to 1.
.. math ::
\log{P(x; \lambda)} \propto x \log{\lambda} - \lambda
"""
if fails_constraints(rate > 0):
return -np.inf
if np.size(rate) != 1 and len(x) != len(rate):
raise ValueError('If rate is a vector, x must be the same size as rate.'
' We got x={}, rate={}'.format(x, rate))
return np.sum(x*np.log(rate)) - np.size(x)*rate
def normal(x, mu=0, sig=1):
""" Normal distribution log-likelihood.
:param x: *int, float, np.array.*
:param mu: (optional) *int, float, np.array.*
Location parameter of the normal distribution. Defaults to 0.
:param sig: (optional) *int, float.*
Standard deviation of the normal distribution, :math:`\sigma > 0`.
Defaults to 1.
.. math::
\log{P(x; \mu, \sigma)} \propto -\log{\sigma} \
- \\frac{(x - \mu)^2}{2 \sigma^2}
"""
if np.size(mu) != 1 and len(x) != len(mu):
raise ValueError('If mu is a vector, x must be the same size as mu.'
' We got x={}, mu={}'.format(x, mu))
return np.sum(-np.log(sig) - (x - mu)**2/(2*sig**2))
def normal(x, mu=0, sig=1):
""" Normal distribution log-likelihood.
:param x: *int, float, np.array.*
:param mu: (optional) *int, float, np.array.*
Location parameter of the normal distribution. Defaults to 0.
:param sig: (optional) *int, float.*
Standard deviation of the normal distribution, :math:`\sigma > 0`.
Defaults to 1.
.. math::
\log{P(x; \mu, \sigma)} \propto -\log{\sigma} \
- \\frac{(x - \mu)^2}{2 \sigma^2}
"""
if np.size(mu) != 1 and len(x) != len(mu):
raise ValueError('If mu is a vector, x must be the same size as mu.'
' We got x={}, mu={}'.format(x, mu))
return np.sum(-np.log(sig) - (x - mu)**2 / (2 * sig**2))
:param k: *int, np.array.* Number of successes.
:param p: *int, float, np.array.* Success probability.
.. math ::
\log{P(x; r)} \propto x\log{p} + \
r\log{(1 - p)}
"""
if fails_constraints(r > 0):
return -np.inf
p = mu / (mu + r)
return np.sum(r * np.log(1 - p) + y * np.log(p) )
def beta(x, alpha=1, beta=1):
""" Beta distribution log-likelihood.
:param x: *float, np.array.* :math:`0 < x < 1`
:param alpha: (optional) *int, float.* Shape parameter, :math:`\\alpha > 0`
:param beta: (optional) *int, float.* Shape parameter, :math:`\\beta > 0`
.. math ::
\log{P(x; \\alpha, \\beta)} \propto (\\alpha - 1)\log{x} + \
(\\beta - 1) \log{(1 - x)}
"""