Example #1
0
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)
Example #2
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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)
Example #3
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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))
Example #4
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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))
Example #5
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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))
Example #6
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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))
Example #7
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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))
Example #8
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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))
Example #9
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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))
Example #10
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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))
Example #11
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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))
Example #12
0
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))
Example #13
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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)
Example #14
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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)
Example #15
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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
Example #16
0
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
Example #17
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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))
Example #18
0
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))
Example #19
0
        :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)}
    """