.. _mcmc_pdfs-label:

Probability Density Functions for MCMC Sampling
-----------------------------------------------

  - 'normal' : normal distribution, with probability density function(PDF) parameterized as follows:

    .. math::
       p(y|\mu,\sigma)=\frac{1}{\sqrt{2\pi}\sigma}\exp(-{1\over 2}({{y-\mu} \over {\sigma}})^2)

  - 'skew_normal' : skew normal distribution, with probability density function parameterized as follows:

    .. math::
       p(y|\xi,\omega,\alpha)={1\over{\omega\sqrt{2\xi}}}\exp(-{1\over 2}({{y-\xi}\over{\omega}})^2)(1+erf(\alpha({{y-\xi}\over{\omega\sqrt{2}}})))

  - 'student_t' : student-t distribution, with density function parameterized as follows:

    .. math::
       p(y|\nu,\mu,\sigma)={{\Gamma((\nu+1)/2))}\over{\Gamma(\nu/2)}}{1\over{\sqrt{\nu\pi}\sigma}}(1+{1\over\nu}({{y-\mu}\over \sigma})^2)^{-(\nu+1)/2}

  - 'cauchy' : Cauchy distribution, with probability density function parameterized as follows:

    .. math:: p(y|\mu,\sigma)={1\over{\pi\sigma(1+(y-\mu)^2/\sigma^2}}

  - 'laplace' : Laplace distribution, with probability density function parameterized as follows:

    .. math:: p(y|\mu,\sigma) = {1\over{2\sigma}}\exp(-{|y-\mu|\over\sigma})

  - 'logistic' : Logistic distribution, with probability density function parameterized as follows:

    .. math:: p(y|\mu,\sigma) = {1\over\sigma}\exp(-{{y-\mu}\over\sigma})(1+\exp(-{{y-\mu}\over\sigma}))^2

  - 'gumbel' : Gumbel distribution, with probability density function parameterized as follows:

    .. math:: p(y|\mu,\beta) = {1\over\beta}\exp(-{{y-\mu}\over\beta} - \exp(-{{y-\mu}\over\beta}))

  - 'exponential' : Exponential distribution, with probability density function parameterized as follows:

    .. math:: p(y|\beta) = \beta\exp(-\beta y), y\geq 0

  - 'chi_square' : Chi-square distribution, with probability density function parameterized as follows:

    .. math:: p(y|\nu) = {2^{-\nu/2}\over\Gamma(\nu/2)}y^{\nu/2-1}exp(-{1\over 2}y), y\geq 0

  - 'invchi_square' : Inverse Chi-square distribution, with probability density function parameterized as follows:

    .. math:: p(y|\nu)={2^{-\nu/2}\over\Gamma(\nu/2)}y^{-\nu/2-1}exp(-{1\over {2y}})

  - 'gamma' : Gamma distribution, with probability density function parameterized as follows:

    .. math:: p(y|\alpha,\beta)={{\beta^\alpha}\over\Gamma(\alpha)}y^{\alpha-1}\exp(-\beta y)

  - 'weibull' : Weibull distribution, with probability density function parameterized as follows:

    .. math:: p(y|\alpha,\sigma)={\alpha\over\sigma}({y\over\sigma})^{\alpha-1}\exp(-({\alpha\over\sigma})^\alpha)

  - 'frechet' : Frechet distribution, with probability density function parameterized as follows:

    .. math:: p(y|\alpha,\sigma)={\alpha\over\sigma}({y\over\sigma})^{-\alpha-1}\exp(-({y\over\sigma})^{-\alpha})

  - 'rayleigh' : Rayleigh distribution, with probability density function parameterized as follows:

    .. math:: p(y|\sigma)={y\over\sigma^2}\exp(-{y^2\over{2\sigma^2}})

  - 'multinormal' : Multivariate normal distribution, with probability density function parameterized as follows:

    .. math:: p(y|\mu, \Sigma)={1\over{(2\pi)^{K/2}\sqrt{|\Sigma|}}}\exp(-{1\over 2}(y-\mu)^T\Sigma^{-1}(y-\mu))

  - 'multinormalprec' : Multivariate normal distribution with precision parameterization, whose probability density
    function is parameterized as follows:

    .. math:: p(y|\mu, \Omega)={\sqrt{|\Omega|}\over{(2\pi)^{K/2}}}\exp(-{1\over 2}(y-\mu)^T\Omega(y-\mu))

  - 'multinormalcholesky' : Multivariate normal distribution with Cholesky parameterization, whose probability density
    function is parameterized as follows:

    .. math:: p(y|\mu,L)={1\over{(2\pi)^{K/2}|L|}}\exp(-{1\over 2}(y-\mu)^T(LL^T)^{-1}(y-\mu))

  - 'multistudent_t' : Multivariate student-t distribution, with probability density function parameterized as follows:

    .. math::
       p(y|\nu,\mu,\Sigma)={1\over{(\pi\nu)^{K/2}\sqrt{|\Sigma|}}}{{\Gamma((\nu+K)/2)}\over{\Gamma(\nu/2)}}(1+{1\over\nu}(y-\mu)^T\Sigma^{-1}(y-\mu))^{-(\nu+K)/2}

  - 'dirichlet' : Dirichlet distribution, with probability density function parameterized as follows:

    .. math:: p(y|\alpha)={{\Gamma(\sum_{k=1}^K\alpha_k)}\over{\prod_{k=1}^K\Gamma(\alpha_k)}}\prod_{k=1}^K y_k^{\alpha_k-1}

  - 'lognormal' : Lognormal distribution, with probability density function specified as follows:

    .. math:: p(y|\mu,\sigma)=\frac{1}{\sqrt{2\pi}\sigma y}\exp(-\frac{1}{2}(\frac{\log{y}-\mu}{\sigma})^2)

  - 'invgamma' : Inverse Gamma distribution, with probability density function specified as follows:

    .. math:: p(y|\alpha, \beta)=\frac{\beta^\alpha}{\Gamma(\alpha)}y^{-\alpha-1}\exp(-\beta\frac{1}{y})

  - 'beta' : Beta distribution, with probability density function specified as follows:

    .. math:: p(y|\alpha,\beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}y^{\alpha-1}(1-y)^{\beta-1}

  - 'pareto' : Pareto distribution, with probability density function specified as follows:

    .. math:: p(y|y_{min},\alpha)=\frac{\alpha y^\alpha_{min}}{y^{\alpha + 1}}

  - 'lomax' : Lomax distribution, with probability density function specified as follows:

    .. math:: p(y|\lambda, \alpha) = \frac{\alpha}{\lambda}(1+\frac{y}{\lambda})^{-\alpha-1}