OpenBUGS中的常用分布与函数.doc

Appendix I DistributionsContents topDiscrete Univariate Bernoulli Binomial Categorical Negative Binomial Poisson Non-central HypergeometricContinuous Univariate Beta Chi-squared Double Exponential Exponential Flat Gamma Generalized Extreme Value Generalized F Generalized Gamma Generalized Pareto Generic LogLikelihood Distribution Log-normal Logistic Normal Pareto Student-t Uniform WeibullDiscrete Multivariate MultinomialContinuous Multivariate Dirichlet Multivariate Normal Multivariate Student-t WishartDiscrete Univariatetoptop appendix iBernoullir dbern(p) Binomialr dbin(p, n) Categoricalr dcat(p) Negative Binomialx dnegbin(p, r) Poissonr dpois(lambda) Non-central Hypergeometricx dhyper(n, m, N, psi)Continuous Univariatetoptop appendix iBetap dbeta(a, b) Chi-squaredx dchisqr(k) Double Exponentialx ddexp(mu, tau) Exponentialx dexp(lambda) Flatx dflat()constant value for all x; not a proper distributionGammax dgamma(r, mu) Generalized Extreme Value x dgev(mu,sigma,eta) Generalized Fx df(n,m,mu,tau)Reduces to the standard F for mu=0, tau=1.Generalized Gammax dggamma(r, mu, beta) Generalized Paretox dgpar(mu,sigma,eta) GenericLogLikelihood distributionx dloglik(lambda) exp(lambda); NB does not depend on x. See Generic sampling distributions.Log-normalx dlnorm(mu, tau) Logisticx dlogis(mu, tau) Normalx dnorm(mu, tau) Paretox dpar(alpha, c) Student-tx dt(mu, tau, k) Uniformx dunif(a, b) Weibullx dweib(v, lambda) Discrete Multivariatetoptop appendix iMultinomialx dmulti(p, N) Continuous Multivariatetoptop appendix iDirichletp ddirich(alpha) May also be speltddirchas in WinBUGS.Multivariate Normalx dmnorm(mu, T,) Multivariate Student-tx dmt(mu, T, k) Wishartx, dwish(R, k) Appendix II Functions and FunctionalsFunction arguments represented by e can be expressions, those by s must be scalar-valued nodes in the graph and those represented by v must be vector-valued nodes in a graph. Some function arguments must be stochastic nodes. Functionals are described using a similar notation to functions, the special notation F(x) is used to describe the function on which the functional acts. See example Functionals for details. Systems of ordinary differential equations and their solution can be described in the BUGS language by using the special D(x1:n, t) notation. See example ode for details. Scalar functions top abs(e) absolute value of e, |e| arccos(e) inverse cosine of e arccosh(e) inverse hyperbolic cosine of e arcsin(e) inverse sine of e arcsinh(e) inverse hyperbolic sine of e arctan(e) inverse tangent of e arctanh(e) inverse hyperbolic tangent of e cloglog(e) complementary log log of e, ln(-ln(1 - e) cos(e) cosine of e cosh(e) hyperbolic cosine of e cumulative(s1, s2) tail area of distribution of s1 up to the value of s2, s1 must be stochastic, s1 and s2 can be the same ? cut(e) cuts edges in the graph - see Use of the cut functiondensity(s1, s2) density of distribution of s1 at value of s2, s1 must be a stochastic node supplied as data, s1 and s2 can be the same.deviance(s1, s2) deviance of distribution of s1 at value of s2, s1 must be a stochastic node supplied as data, s1 and s2 can be the same. equals(e1, e2) 1 if value of e1 equals value of e2; 0 otherwise exp(e) exp(e) gammap(s1, s2) partial (incomplete) gamma function, value of standard gamma density with parameter s1 integrated up to s2 ilogit(e) exp(e)/ (1 + exp(e) icloglog(e) 1 - exp( - exp(e)integral(F(s), s1, s2, s3) definite integral of function F(s) between s = s1 and s = s2 to accuracy s3 log(e) natural logarithm of e logfact(e) ln(e!) loggam(e) logarithm of gamma function of e logit(e) ln(e/ (1 - e) max(e1, e2) e1 if e1 e2; e2 otherwise min(e1, e2) e1 if e1 = 0; 0 otherwisetan(e) tangent of e tanh(e) hyperbolic tangent of e trunc(e) greatest integer less than or equal to eVector functions top inprod(v1, v2)inner product of v1 and v2,Siv1iv2i interp.lin(e, v1, v2) v2p + (v2p+1 - v2p) * (e - v1p) / (v1p+1 - v1p) where the elements of v1 are in ascending order and p is such that v1p e v1p+1.Given function values in the vector v2 evaluated at the points in v1, this estimates the function value at a new point e by simple linear interpolation using the closest bounding pair of points. For example, given the population in 1991,2001 and 2011, we might want to estimate the population in 2004. inverse(v) inverse of symmetric positive-definite matrix v logdet(v) log of determinant of v for symmetric positive-definite v mean(v) Sivi / nn = dim(v) eigen.vals(v) eigenvalues of matrix vode(v1, v2, D(v3, s1), s2, s3)solution of system of ordinary differential equations at grid of points v2 given initial values v1 at time s2 solved to accuracy s3. v3 is a vector of components of the system of ode and s1 is the time variable. See the PDF files in the Diff/Docudirectory of the OpenBUGS installation for further details. prod(v) Pivi p.valueM(v)v must be a multivariate stochastic node, returns a vector of ones and zeros depending on if a sample from the prior is less than value of the corresponding component of v rank(v, s) number of components of vless than or equal to s