650 Index Prior beliefs, 12, 60, 192, 225 Bayesian inference and, 68 beta distribution and, 81­82 rational inference and, 68 Prior distribution, 196, 197f, 198, 200, 395, 422­423 Bayesian analysis and, 275­277 from BUGS, 414 folded-t, 494 gamma, 402 in hierarchical model, 493­495 NHST and, 275 normal, 408 over bias, 192­193 overt/influence of, 275­277 regression coefficients and, 488 with strong dependency, 205f transformed data and, 402 with weak dependency, 204f Prior precision, 393 Prior predictive, 58 Prior probability, 303­304, 309 Probabilistic prediction, 377­379 Probability approximation of, 27 inferential statistics and, 24 Metropolis algorithm and, 124 outside/inside the head and, 25­30 long-run relative frequency as, 25­28 subjective belief and, 28­30 as possibilities number assignment, 29­30 properties of, 29­30 relative, 54 Probability density, 31­37, 108 definition of, 34 dx written next to interval and, 34 exercise for, 49 functions of, 33­37 as greater than 1, 33 integral from, 44 interval density and, 34 intervals v. exact values for, 31 as non-uniform, 33 as not exact, 32­33 probability mass and, 32 reparameterization of, 518 spinner examples of, 31­32, 32f Probability density function, 390, 391f beliefs models and, 158 mean of distribution and, 37 non-negative values/1 integration and, 35 normal distribution and, 36 properties of, 33­37 Probability distribution, 30­40, 361, 384 central tendency of, 39 continuous variables and, 44 defining of, 30 density and, 31­37 discrete/probability mass and, 30 intervals v. exact values for, 31 reparameterization of, 630­632 requirements for, 84 as simple/complex, 30 variance of, 38 Probability mass, 30 Bayes' rule and, 102 continuous variables and, 34 probability density and, 32 Probit function, 377 Probit regression, 578 Product space, 256 Proportionality, of linear function, 361­362 Proportions, exercise for differences of, 188­189 Proposal distribution, 124, 130, 150, 165 problems of, 131­132 Prospective power analysis, 332 Pseudoprior accuracy of, 251 BUGS example with, 246­253 definition for, 249 exercise for, 262­263 filtration-condensation experiment and, 246 results from use of, 251­252 transdimensional MCMC and, 253, 257 Pseudo-random number generators, 27, 27n1 Pythagorean theorem, 364 Python, 17 R R action example of, 16­17 aggregate function in, 544 arguments in functions of, 91­92 as-needed learning of, 17 basics of, 14­20 as Bayesian reasoner, 15 CamelBack notation and, 19 command syntax within, 20, 21 contour plot in, 162 cumulative normal function of, 578 dnorm in, v. BUGS, 260 download/installation of, 15 ease/use/accessibility of, 14 font size in, 17, 545 graph drawn by, 17f help for, 17 invoking/use of, 15 logistic regression in, 553 matrix command for, 69 multifactor ANOVA in, 520­522 new function defining and, 91­92 par function in, 17 perspective plot in, 160 plot function in, 16 programming in editing in, 18 running of, 19­20 variable naming in, 18­19 programming language of, 3 pseudo-random number generators and, 27 quadratic function in, 16 reference card for, 17 script and, 15 sequence function in, 16 Tinn-R editor for, 18, 20 Windows emulator and, 15 R code, 47­49, 228­235, 335­346, 409­414 for Bayes' rule, 69­71 for Bayesian oneway ANOVA, 495­497, 507­512 for Bayesian two-group ANOVA, 535­544 for beta distribution, 91­95 BRugs: robust linear regression, 439­443 BRugs: simple linear regression with repeated measures, 443­447 Q Quadratic trend, 16, 429­430 Quatrains, 2, 2n2