R : Copyright 2005, The R Foundation for Statistical Computing Version 2.1.1 (2005-06-20), ISBN 3-900051-07-0 R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for a HTML browser interface to help. Type 'q()' to quit R. > ### *
> ### > attach(NULL, name = "CheckExEnv") > assign(".CheckExEnv", as.environment(2), pos = length(search())) # base > ## add some hooks to label plot pages for base and grid graphics > setHook("plot.new", ".newplot.hook") > setHook("persp", ".newplot.hook") > setHook("grid.newpage", ".gridplot.hook") > > assign("cleanEx", + function(env = .GlobalEnv) { + rm(list = ls(envir = env, all.names = TRUE), envir = env) + RNGkind("default", "default") + set.seed(1) + options(warn = 1) + delayedAssign("T", stop("T used instead of TRUE"), + assign.env = .CheckExEnv) + delayedAssign("F", stop("F used instead of FALSE"), + assign.env = .CheckExEnv) + sch <- search() + newitems <- sch[! sch %in% .oldSearch] + for(item in rev(newitems)) + eval(substitute(detach(item), list(item=item))) + missitems <- .oldSearch[! .oldSearch %in% sch] + if(length(missitems)) + warning("items ", paste(missitems, collapse=", "), + " have been removed from the search path") + }, + env = .CheckExEnv) > assign("..nameEx", "__{must remake R-ex/*.R}__", env = .CheckExEnv) # for now > assign("ptime", proc.time(), env = .CheckExEnv) > grDevices::postscript("Bolstad-Examples.ps") > assign("par.postscript", graphics::par(no.readonly = TRUE), env = .CheckExEnv) > options(contrasts = c(unordered = "contr.treatment", ordered = "contr.poly")) > options(warn = 1) > library('Bolstad') > > assign(".oldSearch", search(), env = .CheckExEnv) > assign(".oldNS", loadedNamespaces(), env = .CheckExEnv) > cleanEx(); ..nameEx <- "binobp" > > ### * binobp > > flush(stderr()); flush(stdout()) > > ### Name: binobp > ### Title: Binomial sampling with a beta prior > ### Aliases: binobp > ### Keywords: misc > > ### ** Examples > > ## simplest call with 6 successes observed in 8 trials and a Beta(1,1) uniform > ## prior > binobp(6,8) Posterior Mean : 0.7 Posterior Variance : 0.0190909 Posterior Std. Deviation : 0.1381699 Prob. Quantile ------ --------- 0.005 0.3073936 0.01 0.3436855 0.025 0.3999064 0.05 0.4503584 0.5 0.7137633 0.95 0.9022532 0.975 0.9251454 0.99 0.9466518 0.995 0.9584153 > > ## 6 successes observed in 8 trials and a non-uniform Beta(0.5,6) prior > binobp(6,8,0.5,6) Posterior Mean : 0.4482759 Posterior Variance : 0.0159564 Posterior Std. Deviation : 0.1263188 Prob. Quantile ------ --------- 0.005 0.1554803 0.01 0.1769862 0.025 0.2116211 0.05 0.2441832 0.5 0.4458341 0.95 0.6607604 0.975 0.698439 0.99 0.7397328 0.995 0.7661081 > > ## 4 successes observed in 12 trials with a non uniform Beta(3,3) prior > ## plot the stored prior, likelihood and posterior > results<-binobp(4,12,3,3,ret=TRUE) Posterior Mean : 0.3888889 Posterior Variance : 0.0125081 Posterior Std. Deviation : 0.1118397 Prob. Quantile ------ --------- 0.005 0.1370832 0.01 0.1552348 0.025 0.184437 0.05 0.2119082 0.5 0.3846872 0.95 0.5802946 0.975 0.6167163 0.99 0.6577095 0.995 0.6845936 > > par(mfrow=c(3,1)) > y.lims<-c(0,1.1*max(results$posterior,results$prior)) > > plot(results$theta,results$prior,ylim=y.lims,type="l" + ,xlab=expression(theta),ylab="Density",main="Prior") > polygon(results$theta,results$prior,col="red") > > plot(results$theta,results$likelihood,ylim=c(0,0.25),type="l" + ,xlab=expression(theta),ylab="Density",main="Likelihood") > polygon(results$theta,results$likelihood,col="green") > > plot(results$theta,results$posterior,ylim=y.lims,type="l" + ,xlab=expression(theta),ylab="Density",main="Posterior") > polygon(results$theta,results$posterior,col="blue") > > > > > > graphics::par(get("par.postscript", env = .CheckExEnv)) > cleanEx(); ..nameEx <- "binodp" > > ### * binodp > > flush(stderr()); flush(stdout()) > > ### Name: binodp > ### Title: Binomial sampling with a discrete prior > ### Aliases: binodp > ### Keywords: misc > > ### ** Examples > > ## simplest call with 6 successes observed in 8 trials and a uniform prior > binodp(6,8) Conditional distribution of x given theta and n: 0 1 2 3 4 5 6 7 8 0 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.111 0.3897 0.3897 0.1705 0.0426 0.0067 0.0007 0.0000 0.0000 0.0000 0.222 0.1339 0.3061 0.3061 0.1749 0.0625 0.0143 0.0020 0.0002 0.0000 0.333 0.0390 0.1561 0.2731 0.2731 0.1707 0.0683 0.0171 0.0024 0.0002 0.444 0.0091 0.0581 0.1626 0.2602 0.2602 0.1665 0.0666 0.0152 0.0015 0.556 0.0015 0.0152 0.0666 0.1665 0.2602 0.2602 0.1626 0.0581 0.0091 0.667 0.0002 0.0024 0.0171 0.0683 0.1707 0.2731 0.2731 0.1561 0.0390 0.778 0.0000 0.0002 0.0020 0.0143 0.0625 0.1749 0.3061 0.3061 0.1339 0.889 0.0000 0.0000 0.0000 0.0007 0.0067 0.0426 0.1705 0.3897 0.3897 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 Joint distribution: 0 1 2 3 4 5 6 7 8 [1,] 0.1000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [2,] 0.0390 0.0390 0.0171 0.0043 0.0007 0.0001 0.0000 0.0000 0.0000 [3,] 0.0134 0.0306 0.0306 0.0175 0.0062 0.0014 0.0002 0.0000 0.0000 [4,] 0.0039 0.0156 0.0273 0.0273 0.0171 0.0068 0.0017 0.0002 0.0000 [5,] 0.0009 0.0058 0.0163 0.0260 0.0260 0.0167 0.0067 0.0015 0.0002 [6,] 0.0002 0.0015 0.0067 0.0167 0.0260 0.0260 0.0163 0.0058 0.0009 [7,] 0.0000 0.0002 0.0017 0.0068 0.0171 0.0273 0.0273 0.0156 0.0039 [8,] 0.0000 0.0000 0.0002 0.0014 0.0062 0.0175 0.0306 0.0306 0.0134 [9,] 0.0000 0.0000 0.0000 0.0001 0.0007 0.0043 0.0171 0.0390 0.0390 [10,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1000 Marginal distribution of x: 0 1 2 3 4 5 6 7 8 [1,] 0.1573 0.0928 0.0998 0.1001 0.1 0.1001 0.0998 0.0928 0.1573 Prior Likelihood Posterior 0 0.1 0.000000e+00 0.000000e+00 0.111 0.1 4.162919e-06 4.170772e-05 0.222 0.1 2.039830e-04 2.043678e-03 0.333 0.1 1.707057e-03 1.710277e-02 0.444 0.1 6.660670e-03 6.673236e-02 0.556 0.1 1.626140e-02 1.629208e-01 0.667 0.1 2.731291e-02 2.736444e-01 0.778 0.1 3.061020e-02 3.066795e-01 0.889 0.1 1.705132e-02 1.708348e-01 1 0.1 0.000000e+00 0.000000e+00 > > ## same as previous example but with more possibilities for theta > binodp(6,8,n.theta=100) Conditional distribution of x given theta and n: 0 1 2 3 4 5 6 7 8 0 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.01 0.9220 0.0753 0.0027 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.02 0.8494 0.1401 0.0101 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.03 0.7818 0.1954 0.0214 0.0013 0.0001 0.0000 0.0000 0.0000 0.0000 0.04 0.7190 0.2422 0.0357 0.0030 0.0002 0.0000 0.0000 0.0000 0.0000 0.051 0.6606 0.2811 0.0523 0.0056 0.0004 0.0000 0.0000 0.0000 0.0000 0.061 0.6064 0.3130 0.0707 0.0091 0.0007 0.0000 0.0000 0.0000 0.0000 0.071 0.5562 0.3385 0.0902 0.0137 0.0013 0.0001 0.0000 0.0000 0.0000 0.081 0.5096 0.3584 0.1103 0.0194 0.0021 0.0001 0.0000 0.0000 0.0000 0.091 0.4665 0.3732 0.1306 0.0261 0.0033 0.0003 0.0000 0.0000 0.0000 0.101 0.4266 0.3835 0.1508 0.0339 0.0048 0.0004 0.0000 0.0000 0.0000 0.111 0.3897 0.3897 0.1705 0.0426 0.0067 0.0007 0.0000 0.0000 0.0000 0.121 0.3557 0.3925 0.1895 0.0523 0.0090 0.0010 0.0001 0.0000 0.0000 0.131 0.3243 0.3921 0.2075 0.0627 0.0119 0.0014 0.0001 0.0000 0.0000 0.141 0.2953 0.3891 0.2243 0.0739 0.0152 0.0020 0.0002 0.0000 0.0000 0.152 0.2686 0.3838 0.2398 0.0857 0.0191 0.0027 0.0002 0.0000 0.0000 0.162 0.2441 0.3764 0.2540 0.0979 0.0236 0.0036 0.0004 0.0000 0.0000 0.172 0.2215 0.3674 0.2666 0.1105 0.0286 0.0048 0.0005 0.0000 0.0000 0.182 0.2008 0.3570 0.2777 0.1234 0.0343 0.0061 0.0007 0.0000 0.0000 0.192 0.1818 0.3455 0.2872 0.1364 0.0405 0.0077 0.0009 0.0001 0.0000 0.202 0.1644 0.3330 0.2951 0.1494 0.0473 0.0096 0.0012 0.0001 0.0000 0.212 0.1485 0.3198 0.3014 0.1623 0.0546 0.0118 0.0016 0.0001 0.0000 0.222 0.1339 0.3061 0.3061 0.1749 0.0625 0.0143 0.0020 0.0002 0.0000 0.232 0.1206 0.2920 0.3093 0.1872 0.0708 0.0171 0.0026 0.0002 0.0000 0.242 0.1085 0.2777 0.3111 0.1991 0.0796 0.0204 0.0033 0.0003 0.0000 0.253 0.0974 0.2634 0.3114 0.2104 0.0889 0.0240 0.0041 0.0004 0.0000 0.263 0.0874 0.2490 0.3104 0.2211 0.0984 0.0281 0.0050 0.0005 0.0000 0.273 0.0783 0.2348 0.3082 0.2311 0.1083 0.0325 0.0061 0.0007 0.0000 0.283 0.0700 0.2208 0.3047 0.2404 0.1185 0.0374 0.0074 0.0008 0.0000 0.293 0.0625 0.2071 0.3002 0.2488 0.1288 0.0427 0.0088 0.0010 0.0001 0.303 0.0557 0.1937 0.2947 0.2563 0.1393 0.0484 0.0105 0.0013 0.0001 0.313 0.0495 0.1807 0.2883 0.2629 0.1498 0.0546 0.0125 0.0016 0.0001 0.323 0.0440 0.1681 0.2811 0.2685 0.1603 0.0612 0.0146 0.0020 0.0001 0.333 0.0390 0.1561 0.2731 0.2731 0.1707 0.0683 0.0171 0.0024 0.0002 0.343 0.0345 0.1445 0.2646 0.2768 0.1810 0.0757 0.0198 0.0030 0.0002 0.354 0.0305 0.1335 0.2554 0.2794 0.1910 0.0836 0.0228 0.0036 0.0002 0.364 0.0269 0.1229 0.2459 0.2810 0.2007 0.0918 0.0262 0.0043 0.0003 0.374 0.0237 0.1130 0.2360 0.2816 0.2101 0.1003 0.0299 0.0051 0.0004 0.384 0.0208 0.1035 0.2257 0.2813 0.2190 0.1091 0.0340 0.0061 0.0005 0.394 0.0182 0.0947 0.2153 0.2799 0.2274 0.1183 0.0384 0.0071 0.0006 0.404 0.0159 0.0863 0.2048 0.2777 0.2353 0.1276 0.0433 0.0084 0.0007 0.414 0.0139 0.0785 0.1942 0.2745 0.2426 0.1372 0.0485 0.0098 0.0009 0.424 0.0121 0.0712 0.1836 0.2705 0.2492 0.1469 0.0541 0.0114 0.0010 0.434 0.0105 0.0644 0.1730 0.2657 0.2551 0.1567 0.0602 0.0132 0.0013 0.444 0.0091 0.0581 0.1626 0.2602 0.2602 0.1665 0.0666 0.0152 0.0015 0.455 0.0078 0.0522 0.1524 0.2539 0.2645 0.1763 0.0735 0.0175 0.0018 0.465 0.0067 0.0468 0.1423 0.2470 0.2680 0.1861 0.0808 0.0200 0.0022 0.475 0.0058 0.0419 0.1325 0.2396 0.2707 0.1957 0.0884 0.0228 0.0026 0.485 0.0050 0.0373 0.1230 0.2316 0.2724 0.2051 0.0965 0.0260 0.0031 0.495 0.0042 0.0332 0.1138 0.2231 0.2733 0.2143 0.1050 0.0294 0.0036 0.505 0.0036 0.0294 0.1050 0.2143 0.2733 0.2231 0.1138 0.0332 0.0042 0.515 0.0031 0.0260 0.0965 0.2051 0.2724 0.2316 0.1230 0.0373 0.0050 0.525 0.0026 0.0228 0.0884 0.1957 0.2707 0.2396 0.1325 0.0419 0.0058 0.535 0.0022 0.0200 0.0808 0.1861 0.2680 0.2470 0.1423 0.0468 0.0067 0.545 0.0018 0.0175 0.0735 0.1763 0.2645 0.2539 0.1524 0.0522 0.0078 0.556 0.0015 0.0152 0.0666 0.1665 0.2602 0.2602 0.1626 0.0581 0.0091 0.566 0.0013 0.0132 0.0602 0.1567 0.2551 0.2657 0.1730 0.0644 0.0105 0.576 0.0010 0.0114 0.0541 0.1469 0.2492 0.2705 0.1836 0.0712 0.0121 0.586 0.0009 0.0098 0.0485 0.1372 0.2426 0.2745 0.1942 0.0785 0.0139 0.596 0.0007 0.0084 0.0433 0.1276 0.2353 0.2777 0.2048 0.0863 0.0159 0.606 0.0006 0.0071 0.0384 0.1183 0.2274 0.2799 0.2153 0.0947 0.0182 0.616 0.0005 0.0061 0.0340 0.1091 0.2190 0.2813 0.2257 0.1035 0.0208 0.626 0.0004 0.0051 0.0299 0.1003 0.2101 0.2816 0.2360 0.1130 0.0237 0.636 0.0003 0.0043 0.0262 0.0918 0.2007 0.2810 0.2459 0.1229 0.0269 0.646 0.0002 0.0036 0.0228 0.0836 0.1910 0.2794 0.2554 0.1335 0.0305 0.657 0.0002 0.0030 0.0198 0.0757 0.1810 0.2768 0.2646 0.1445 0.0345 0.667 0.0002 0.0024 0.0171 0.0683 0.1707 0.2731 0.2731 0.1561 0.0390 0.677 0.0001 0.0020 0.0146 0.0612 0.1603 0.2685 0.2811 0.1681 0.0440 0.687 0.0001 0.0016 0.0125 0.0546 0.1498 0.2629 0.2883 0.1807 0.0495 0.697 0.0001 0.0013 0.0105 0.0484 0.1393 0.2563 0.2947 0.1937 0.0557 0.707 0.0001 0.0010 0.0088 0.0427 0.1288 0.2488 0.3002 0.2071 0.0625 0.717 0.0000 0.0008 0.0074 0.0374 0.1185 0.2404 0.3047 0.2208 0.0700 0.727 0.0000 0.0007 0.0061 0.0325 0.1083 0.2311 0.3082 0.2348 0.0783 0.737 0.0000 0.0005 0.0050 0.0281 0.0984 0.2211 0.3104 0.2490 0.0874 0.747 0.0000 0.0004 0.0041 0.0240 0.0889 0.2104 0.3114 0.2634 0.0974 0.758 0.0000 0.0003 0.0033 0.0204 0.0796 0.1991 0.3111 0.2777 0.1085 0.768 0.0000 0.0002 0.0026 0.0171 0.0708 0.1872 0.3093 0.2920 0.1206 0.778 0.0000 0.0002 0.0020 0.0143 0.0625 0.1749 0.3061 0.3061 0.1339 0.788 0.0000 0.0001 0.0016 0.0118 0.0546 0.1623 0.3014 0.3198 0.1485 0.798 0.0000 0.0001 0.0012 0.0096 0.0473 0.1494 0.2951 0.3330 0.1644 0.808 0.0000 0.0001 0.0009 0.0077 0.0405 0.1364 0.2872 0.3455 0.1818 0.818 0.0000 0.0000 0.0007 0.0061 0.0343 0.1234 0.2777 0.3570 0.2008 0.828 0.0000 0.0000 0.0005 0.0048 0.0286 0.1105 0.2666 0.3674 0.2215 0.838 0.0000 0.0000 0.0004 0.0036 0.0236 0.0979 0.2540 0.3764 0.2441 0.848 0.0000 0.0000 0.0002 0.0027 0.0191 0.0857 0.2398 0.3838 0.2686 0.859 0.0000 0.0000 0.0002 0.0020 0.0152 0.0739 0.2243 0.3891 0.2953 0.869 0.0000 0.0000 0.0001 0.0014 0.0119 0.0627 0.2075 0.3921 0.3243 0.879 0.0000 0.0000 0.0001 0.0010 0.0090 0.0523 0.1895 0.3925 0.3557 0.889 0.0000 0.0000 0.0000 0.0007 0.0067 0.0426 0.1705 0.3897 0.3897 0.899 0.0000 0.0000 0.0000 0.0004 0.0048 0.0339 0.1508 0.3835 0.4266 0.909 0.0000 0.0000 0.0000 0.0003 0.0033 0.0261 0.1306 0.3732 0.4665 0.919 0.0000 0.0000 0.0000 0.0001 0.0021 0.0194 0.1103 0.3584 0.5096 0.929 0.0000 0.0000 0.0000 0.0001 0.0013 0.0137 0.0902 0.3385 0.5562 0.939 0.0000 0.0000 0.0000 0.0000 0.0007 0.0091 0.0707 0.3130 0.6064 0.949 0.0000 0.0000 0.0000 0.0000 0.0004 0.0056 0.0523 0.2811 0.6606 0.96 0.0000 0.0000 0.0000 0.0000 0.0002 0.0030 0.0357 0.2422 0.7190 0.97 0.0000 0.0000 0.0000 0.0000 0.0001 0.0013 0.0214 0.1954 0.7818 0.98 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0101 0.1401 0.8494 0.99 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0027 0.0753 0.9220 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 Joint distribution: 0 1 2 3 4 5 6 7 8 [1,] 0.0100 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [2,] 0.0092 0.0008 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [3,] 0.0085 0.0014 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [4,] 0.0078 0.0020 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [5,] 0.0072 0.0024 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [6,] 0.0066 0.0028 0.0005 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 [7,] 0.0061 0.0031 0.0007 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 [8,] 0.0056 0.0034 0.0009 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 [9,] 0.0051 0.0036 0.0011 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 [10,] 0.0047 0.0037 0.0013 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 [11,] 0.0043 0.0038 0.0015 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 [12,] 0.0039 0.0039 0.0017 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 [13,] 0.0036 0.0039 0.0019 0.0005 0.0001 0.0000 0.0000 0.0000 0.0000 [14,] 0.0032 0.0039 0.0021 0.0006 0.0001 0.0000 0.0000 0.0000 0.0000 [15,] 0.0030 0.0039 0.0022 0.0007 0.0002 0.0000 0.0000 0.0000 0.0000 [16,] 0.0027 0.0038 0.0024 0.0009 0.0002 0.0000 0.0000 0.0000 0.0000 [17,] 0.0024 0.0038 0.0025 0.0010 0.0002 0.0000 0.0000 0.0000 0.0000 [18,] 0.0022 0.0037 0.0027 0.0011 0.0003 0.0000 0.0000 0.0000 0.0000 [19,] 0.0020 0.0036 0.0028 0.0012 0.0003 0.0001 0.0000 0.0000 0.0000 [20,] 0.0018 0.0035 0.0029 0.0014 0.0004 0.0001 0.0000 0.0000 0.0000 [21,] 0.0016 0.0033 0.0030 0.0015 0.0005 0.0001 0.0000 0.0000 0.0000 [22,] 0.0015 0.0032 0.0030 0.0016 0.0005 0.0001 0.0000 0.0000 0.0000 [23,] 0.0013 0.0031 0.0031 0.0017 0.0006 0.0001 0.0000 0.0000 0.0000 [24,] 0.0012 0.0029 0.0031 0.0019 0.0007 0.0002 0.0000 0.0000 0.0000 [25,] 0.0011 0.0028 0.0031 0.0020 0.0008 0.0002 0.0000 0.0000 0.0000 [26,] 0.0010 0.0026 0.0031 0.0021 0.0009 0.0002 0.0000 0.0000 0.0000 [27,] 0.0009 0.0025 0.0031 0.0022 0.0010 0.0003 0.0000 0.0000 0.0000 [28,] 0.0008 0.0023 0.0031 0.0023 0.0011 0.0003 0.0001 0.0000 0.0000 [29,] 0.0007 0.0022 0.0030 0.0024 0.0012 0.0004 0.0001 0.0000 0.0000 [30,] 0.0006 0.0021 0.0030 0.0025 0.0013 0.0004 0.0001 0.0000 0.0000 [31,] 0.0006 0.0019 0.0029 0.0026 0.0014 0.0005 0.0001 0.0000 0.0000 [32,] 0.0005 0.0018 0.0029 0.0026 0.0015 0.0005 0.0001 0.0000 0.0000 [33,] 0.0004 0.0017 0.0028 0.0027 0.0016 0.0006 0.0001 0.0000 0.0000 [34,] 0.0004 0.0016 0.0027 0.0027 0.0017 0.0007 0.0002 0.0000 0.0000 [35,] 0.0003 0.0014 0.0026 0.0028 0.0018 0.0008 0.0002 0.0000 0.0000 [36,] 0.0003 0.0013 0.0026 0.0028 0.0019 0.0008 0.0002 0.0000 0.0000 [37,] 0.0003 0.0012 0.0025 0.0028 0.0020 0.0009 0.0003 0.0000 0.0000 [38,] 0.0002 0.0011 0.0024 0.0028 0.0021 0.0010 0.0003 0.0001 0.0000 [39,] 0.0002 0.0010 0.0023 0.0028 0.0022 0.0011 0.0003 0.0001 0.0000 [40,] 0.0002 0.0009 0.0022 0.0028 0.0023 0.0012 0.0004 0.0001 0.0000 [41,] 0.0002 0.0009 0.0020 0.0028 0.0024 0.0013 0.0004 0.0001 0.0000 [42,] 0.0001 0.0008 0.0019 0.0027 0.0024 0.0014 0.0005 0.0001 0.0000 [43,] 0.0001 0.0007 0.0018 0.0027 0.0025 0.0015 0.0005 0.0001 0.0000 [44,] 0.0001 0.0006 0.0017 0.0027 0.0026 0.0016 0.0006 0.0001 0.0000 [45,] 0.0001 0.0006 0.0016 0.0026 0.0026 0.0017 0.0007 0.0002 0.0000 [46,] 0.0001 0.0005 0.0015 0.0025 0.0026 0.0018 0.0007 0.0002 0.0000 [47,] 0.0001 0.0005 0.0014 0.0025 0.0027 0.0019 0.0008 0.0002 0.0000 [48,] 0.0001 0.0004 0.0013 0.0024 0.0027 0.0020 0.0009 0.0002 0.0000 [49,] 0.0000 0.0004 0.0012 0.0023 0.0027 0.0021 0.0010 0.0003 0.0000 [50,] 0.0000 0.0003 0.0011 0.0022 0.0027 0.0021 0.0011 0.0003 0.0000 [51,] 0.0000 0.0003 0.0011 0.0021 0.0027 0.0022 0.0011 0.0003 0.0000 [52,] 0.0000 0.0003 0.0010 0.0021 0.0027 0.0023 0.0012 0.0004 0.0000 [53,] 0.0000 0.0002 0.0009 0.0020 0.0027 0.0024 0.0013 0.0004 0.0001 [54,] 0.0000 0.0002 0.0008 0.0019 0.0027 0.0025 0.0014 0.0005 0.0001 [55,] 0.0000 0.0002 0.0007 0.0018 0.0026 0.0025 0.0015 0.0005 0.0001 [56,] 0.0000 0.0002 0.0007 0.0017 0.0026 0.0026 0.0016 0.0006 0.0001 [57,] 0.0000 0.0001 0.0006 0.0016 0.0026 0.0027 0.0017 0.0006 0.0001 [58,] 0.0000 0.0001 0.0005 0.0015 0.0025 0.0027 0.0018 0.0007 0.0001 [59,] 0.0000 0.0001 0.0005 0.0014 0.0024 0.0027 0.0019 0.0008 0.0001 [60,] 0.0000 0.0001 0.0004 0.0013 0.0024 0.0028 0.0020 0.0009 0.0002 [61,] 0.0000 0.0001 0.0004 0.0012 0.0023 0.0028 0.0022 0.0009 0.0002 [62,] 0.0000 0.0001 0.0003 0.0011 0.0022 0.0028 0.0023 0.0010 0.0002 [63,] 0.0000 0.0001 0.0003 0.0010 0.0021 0.0028 0.0024 0.0011 0.0002 [64,] 0.0000 0.0000 0.0003 0.0009 0.0020 0.0028 0.0025 0.0012 0.0003 [65,] 0.0000 0.0000 0.0002 0.0008 0.0019 0.0028 0.0026 0.0013 0.0003 [66,] 0.0000 0.0000 0.0002 0.0008 0.0018 0.0028 0.0026 0.0014 0.0003 [67,] 0.0000 0.0000 0.0002 0.0007 0.0017 0.0027 0.0027 0.0016 0.0004 [68,] 0.0000 0.0000 0.0001 0.0006 0.0016 0.0027 0.0028 0.0017 0.0004 [69,] 0.0000 0.0000 0.0001 0.0005 0.0015 0.0026 0.0029 0.0018 0.0005 [70,] 0.0000 0.0000 0.0001 0.0005 0.0014 0.0026 0.0029 0.0019 0.0006 [71,] 0.0000 0.0000 0.0001 0.0004 0.0013 0.0025 0.0030 0.0021 0.0006 [72,] 0.0000 0.0000 0.0001 0.0004 0.0012 0.0024 0.0030 0.0022 0.0007 [73,] 0.0000 0.0000 0.0001 0.0003 0.0011 0.0023 0.0031 0.0023 0.0008 [74,] 0.0000 0.0000 0.0000 0.0003 0.0010 0.0022 0.0031 0.0025 0.0009 [75,] 0.0000 0.0000 0.0000 0.0002 0.0009 0.0021 0.0031 0.0026 0.0010 [76,] 0.0000 0.0000 0.0000 0.0002 0.0008 0.0020 0.0031 0.0028 0.0011 [77,] 0.0000 0.0000 0.0000 0.0002 0.0007 0.0019 0.0031 0.0029 0.0012 [78,] 0.0000 0.0000 0.0000 0.0001 0.0006 0.0017 0.0031 0.0031 0.0013 [79,] 0.0000 0.0000 0.0000 0.0001 0.0005 0.0016 0.0030 0.0032 0.0015 [80,] 0.0000 0.0000 0.0000 0.0001 0.0005 0.0015 0.0030 0.0033 0.0016 [81,] 0.0000 0.0000 0.0000 0.0001 0.0004 0.0014 0.0029 0.0035 0.0018 [82,] 0.0000 0.0000 0.0000 0.0001 0.0003 0.0012 0.0028 0.0036 0.0020 [83,] 0.0000 0.0000 0.0000 0.0000 0.0003 0.0011 0.0027 0.0037 0.0022 [84,] 0.0000 0.0000 0.0000 0.0000 0.0002 0.0010 0.0025 0.0038 0.0024 [85,] 0.0000 0.0000 0.0000 0.0000 0.0002 0.0009 0.0024 0.0038 0.0027 [86,] 0.0000 0.0000 0.0000 0.0000 0.0002 0.0007 0.0022 0.0039 0.0030 [87,] 0.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0021 0.0039 0.0032 [88,] 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0019 0.0039 0.0036 [89,] 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0017 0.0039 0.0039 [90,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0015 0.0038 0.0043 [91,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0013 0.0037 0.0047 [92,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0011 0.0036 0.0051 [93,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0009 0.0034 0.0056 [94,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0007 0.0031 0.0061 [95,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0028 0.0066 [96,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0024 0.0072 [97,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0020 0.0078 [98,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0014 0.0085 [99,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0008 0.0092 [100,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0100 Marginal distribution of x: 0 1 2 3 4 5 6 7 8 [1,] 0.1151 0.1099 0.11 0.11 0.11 0.11 0.11 0.1099 0.1151 Prior Likelihood Posterior 0 0.01 0.000000e+00 0.000000e+00 0.01 0.01 2.914262e-13 2.649330e-12 0.02 0.01 1.827258e-11 1.661144e-10 0.03 0.01 2.038668e-10 1.853335e-09 0.04 0.01 1.121718e-09 1.019744e-08 0.051 0.01 4.189404e-09 3.808549e-08 0.061 0.01 1.224475e-08 1.113159e-07 0.071 0.01 3.021624e-08 2.746931e-07 0.081 0.01 6.587175e-08 5.988341e-07 0.091 0.01 1.306221e-07 1.187473e-06 0.101 0.01 2.403568e-07 2.185062e-06 0.111 0.01 4.162919e-07 3.784472e-06 0.121 0.01 6.858079e-07 6.234618e-06 0.131 0.01 1.083262e-06 9.847837e-06 0.141 0.01 1.650755e-06 1.500686e-05 0.152 0.01 2.438836e-06 2.217124e-05 0.162 0.01 3.507136e-06 3.188305e-05 0.172 0.01 4.924903e-06 4.477185e-05 0.182 0.01 6.771448e-06 6.155863e-05 0.192 0.01 9.136462e-06 8.305876e-05 0.202 0.01 1.212022e-05 1.101839e-04 0.212 0.01 1.583366e-05 1.439424e-04 0.222 0.01 2.039830e-05 1.854391e-04 0.232 0.01 2.594602e-05 2.358729e-04 0.242 0.01 3.261869e-05 2.965335e-04 0.253 0.01 4.056766e-05 3.687970e-04 0.263 0.01 4.995308e-05 4.541190e-04 0.273 0.01 6.094303e-05 5.540276e-04 0.283 0.01 7.371254e-05 6.701141e-04 0.293 0.01 8.844240e-05 8.040220e-04 0.303 0.01 1.053179e-04 9.574356e-04 0.313 0.01 1.245273e-04 1.132066e-03 0.323 0.01 1.462601e-04 1.329638e-03 0.333 0.01 1.707057e-04 1.551870e-03 0.343 0.01 1.980508e-04 1.800462e-03 0.354 0.01 2.284782e-04 2.077074e-03 0.364 0.01 2.621637e-04 2.383307e-03 0.374 0.01 2.992748e-04 2.720681e-03 0.384 0.01 3.399678e-04 3.090616e-03 0.394 0.01 3.843852e-04 3.494411e-03 0.404 0.01 4.326537e-04 3.933216e-03 0.414 0.01 4.848815e-04 4.408014e-03 0.424 0.01 5.411553e-04 4.919594e-03 0.434 0.01 6.015383e-04 5.468530e-03 0.444 0.01 6.660670e-04 6.055155e-03 0.455 0.01 7.347491e-04 6.679538e-03 0.465 0.01 8.075608e-04 7.341462e-03 0.475 0.01 8.844439e-04 8.040400e-03 0.485 0.01 9.653043e-04 8.775495e-03 0.495 0.01 1.050009e-03 9.545536e-03 0.505 0.01 1.138384e-03 1.034894e-02 0.515 0.01 1.230212e-03 1.118375e-02 0.525 0.01 1.325233e-03 1.204757e-02 0.535 0.01 1.423139e-03 1.293763e-02 0.545 0.01 1.523576e-03 1.385069e-02 0.556 0.01 1.626140e-03 1.478309e-02 0.566 0.01 1.730380e-03 1.573073e-02 0.576 0.01 1.835795e-03 1.668905e-02 0.586 0.01 1.941834e-03 1.765304e-02 0.596 0.01 2.047899e-03 1.861727e-02 0.606 0.01 2.153343e-03 1.957585e-02 0.616 0.01 2.257474e-03 2.052249e-02 0.626 0.01 2.359555e-03 2.145050e-02 0.636 0.01 2.458809e-03 2.235281e-02 0.646 0.01 2.554421e-03 2.322201e-02 0.657 0.01 2.645541e-03 2.405037e-02 0.667 0.01 2.731291e-03 2.482992e-02 0.677 0.01 2.810770e-03 2.555245e-02 0.687 0.01 2.883058e-03 2.620962e-02 0.697 0.01 2.947227e-03 2.679297e-02 0.707 0.01 3.002346e-03 2.729406e-02 0.717 0.01 3.047492e-03 2.770448e-02 0.727 0.01 3.081761e-03 2.801601e-02 0.737 0.01 3.104276e-03 2.822069e-02 0.747 0.01 3.114202e-03 2.831093e-02 0.758 0.01 3.110760e-03 2.827964e-02 0.768 0.01 3.093241e-03 2.812037e-02 0.778 0.01 3.061020e-03 2.782746e-02 0.788 0.01 3.013580e-03 2.739618e-02 0.798 0.01 2.950523e-03 2.682294e-02 0.808 0.01 2.871598e-03 2.610544e-02 0.818 0.01 2.776717e-03 2.524288e-02 0.828 0.01 2.665983e-03 2.423622e-02 0.838 0.01 2.539715e-03 2.308832e-02 0.848 0.01 2.398472e-03 2.180430e-02 0.859 0.01 2.243087e-03 2.039170e-02 0.869 0.01 2.074693e-03 1.886085e-02 0.879 0.01 1.894761e-03 1.722510e-02 0.889 0.01 1.705132e-03 1.550120e-02 0.899 0.01 1.508053e-03 1.370957e-02 0.909 0.01 1.306221e-03 1.187473e-02 0.919 0.01 1.102820e-03 1.002564e-02 0.929 0.01 9.015701e-04 8.196093e-03 0.939 0.01 7.067677e-04 6.425161e-03 0.949 0.01 5.233396e-04 4.757633e-03 0.96 0.01 3.568930e-04 3.244483e-03 0.97 0.01 2.137698e-04 1.943362e-03 0.98 0.01 1.011037e-04 9.191243e-04 0.99 0.01 2.688023e-05 2.443657e-04 1 0.01 0.000000e+00 0.000000e+00 > > ## 6 successes, 8 trials and a non-uniform discrete prior > theta<-seq(0,1,by=0.01) > theta.prior<-runif(101) > theta.prior<-sort(theta.prior/sum(theta.prior)) > binodp(6,8,uniform=FALSE,theta=theta,theta.prior=theta.prior) Conditional distribution of x given theta and n: 0 1 2 3 4 5 6 7 8 0 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.01 0.9227 0.0746 0.0026 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.02 0.8508 0.1389 0.0099 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.03 0.7837 0.1939 0.0210 0.0013 0.0001 0.0000 0.0000 0.0000 0.0000 0.04 0.7214 0.2405 0.0351 0.0029 0.0002 0.0000 0.0000 0.0000 0.0000 0.05 0.6634 0.2793 0.0515 0.0054 0.0004 0.0000 0.0000 0.0000 0.0000 0.06 0.6096 0.3113 0.0695 0.0089 0.0007 0.0000 0.0000 0.0000 0.0000 0.07 0.5596 0.3370 0.0888 0.0134 0.0013 0.0001 0.0000 0.0000 0.0000 0.08 0.5132 0.3570 0.1087 0.0189 0.0021 0.0001 0.0000 0.0000 0.0000 0.09 0.4703 0.3721 0.1288 0.0255 0.0031 0.0002 0.0000 0.0000 0.0000 0.1 0.4305 0.3826 0.1488 0.0331 0.0046 0.0004 0.0000 0.0000 0.0000 0.11 0.3937 0.3892 0.1684 0.0416 0.0064 0.0006 0.0000 0.0000 0.0000 0.12 0.3596 0.3923 0.1872 0.0511 0.0087 0.0009 0.0001 0.0000 0.0000 0.13 0.3282 0.3923 0.2052 0.0613 0.0115 0.0014 0.0001 0.0000 0.0000 0.14 0.2992 0.3897 0.2220 0.0723 0.0147 0.0019 0.0002 0.0000 0.0000 0.15 0.2725 0.3847 0.2376 0.0839 0.0185 0.0026 0.0002 0.0000 0.0000 0.16 0.2479 0.3777 0.2518 0.0959 0.0228 0.0035 0.0003 0.0000 0.0000 0.17 0.2252 0.3691 0.2646 0.1084 0.0277 0.0045 0.0005 0.0000 0.0000 0.18 0.2044 0.3590 0.2758 0.1211 0.0332 0.0058 0.0006 0.0000 0.0000 0.19 0.1853 0.3477 0.2855 0.1339 0.0393 0.0074 0.0009 0.0001 0.0000 0.2 0.1678 0.3355 0.2936 0.1468 0.0459 0.0092 0.0011 0.0001 0.0000 0.21 0.1517 0.3226 0.3002 0.1596 0.0530 0.0113 0.0015 0.0001 0.0000 0.22 0.1370 0.3092 0.3052 0.1722 0.0607 0.0137 0.0019 0.0002 0.0000 0.23 0.1236 0.2953 0.3087 0.1844 0.0689 0.0165 0.0025 0.0002 0.0000 0.24 0.1113 0.2812 0.3108 0.1963 0.0775 0.0196 0.0031 0.0003 0.0000 0.25 0.1001 0.2670 0.3115 0.2076 0.0865 0.0231 0.0038 0.0004 0.0000 0.26 0.0899 0.2527 0.3108 0.2184 0.0959 0.0270 0.0047 0.0005 0.0000 0.27 0.0806 0.2386 0.3089 0.2285 0.1056 0.0313 0.0058 0.0006 0.0000 0.28 0.0722 0.2247 0.3058 0.2379 0.1156 0.0360 0.0070 0.0008 0.0000 0.29 0.0646 0.2110 0.3017 0.2464 0.1258 0.0411 0.0084 0.0010 0.0001 0.3 0.0576 0.1977 0.2965 0.2541 0.1361 0.0467 0.0100 0.0012 0.0001 0.31 0.0514 0.1847 0.2904 0.2609 0.1465 0.0527 0.0118 0.0015 0.0001 0.32 0.0457 0.1721 0.2835 0.2668 0.1569 0.0591 0.0139 0.0019 0.0001 0.33 0.0406 0.1600 0.2758 0.2717 0.1673 0.0659 0.0162 0.0023 0.0001 0.34 0.0360 0.1484 0.2675 0.2756 0.1775 0.0732 0.0188 0.0028 0.0002 0.35 0.0319 0.1373 0.2587 0.2786 0.1875 0.0808 0.0217 0.0033 0.0002 0.36 0.0281 0.1267 0.2494 0.2805 0.1973 0.0888 0.0250 0.0040 0.0003 0.37 0.0248 0.1166 0.2397 0.2815 0.2067 0.0971 0.0285 0.0048 0.0004 0.38 0.0218 0.1071 0.2297 0.2815 0.2157 0.1058 0.0324 0.0057 0.0004 0.39 0.0192 0.0981 0.2194 0.2806 0.2242 0.1147 0.0367 0.0067 0.0005 0.4 0.0168 0.0896 0.2090 0.2787 0.2322 0.1239 0.0413 0.0079 0.0007 0.41 0.0147 0.0816 0.1985 0.2759 0.2397 0.1332 0.0463 0.0092 0.0008 0.42 0.0128 0.0742 0.1880 0.2723 0.2465 0.1428 0.0517 0.0107 0.0010 0.43 0.0111 0.0672 0.1776 0.2679 0.2526 0.1525 0.0575 0.0124 0.0012 0.44 0.0097 0.0608 0.1672 0.2627 0.2580 0.1622 0.0637 0.0143 0.0014 0.45 0.0084 0.0548 0.1569 0.2568 0.2627 0.1719 0.0703 0.0164 0.0017 0.46 0.0072 0.0493 0.1469 0.2503 0.2665 0.1816 0.0774 0.0188 0.0020 0.47 0.0062 0.0442 0.1371 0.2431 0.2695 0.1912 0.0848 0.0215 0.0024 0.48 0.0053 0.0395 0.1275 0.2355 0.2717 0.2006 0.0926 0.0244 0.0028 0.49 0.0046 0.0352 0.1183 0.2273 0.2730 0.2098 0.1008 0.0277 0.0033 0.5 0.0039 0.0313 0.1094 0.2188 0.2734 0.2188 0.1094 0.0313 0.0039 0.51 0.0033 0.0277 0.1008 0.2098 0.2730 0.2273 0.1183 0.0352 0.0046 0.52 0.0028 0.0244 0.0926 0.2006 0.2717 0.2355 0.1275 0.0395 0.0053 0.53 0.0024 0.0215 0.0848 0.1912 0.2695 0.2431 0.1371 0.0442 0.0062 0.54 0.0020 0.0188 0.0774 0.1816 0.2665 0.2503 0.1469 0.0493 0.0072 0.55 0.0017 0.0164 0.0703 0.1719 0.2627 0.2568 0.1569 0.0548 0.0084 0.56 0.0014 0.0143 0.0637 0.1622 0.2580 0.2627 0.1672 0.0608 0.0097 0.57 0.0012 0.0124 0.0575 0.1525 0.2526 0.2679 0.1776 0.0672 0.0111 0.58 0.0010 0.0107 0.0517 0.1428 0.2465 0.2723 0.1880 0.0742 0.0128 0.59 0.0008 0.0092 0.0463 0.1332 0.2397 0.2759 0.1985 0.0816 0.0147 0.6 0.0007 0.0079 0.0413 0.1239 0.2322 0.2787 0.2090 0.0896 0.0168 0.61 0.0005 0.0067 0.0367 0.1147 0.2242 0.2806 0.2194 0.0981 0.0192 0.62 0.0004 0.0057 0.0324 0.1058 0.2157 0.2815 0.2297 0.1071 0.0218 0.63 0.0004 0.0048 0.0285 0.0971 0.2067 0.2815 0.2397 0.1166 0.0248 0.64 0.0003 0.0040 0.0250 0.0888 0.1973 0.2805 0.2494 0.1267 0.0281 0.65 0.0002 0.0033 0.0217 0.0808 0.1875 0.2786 0.2587 0.1373 0.0319 0.66 0.0002 0.0028 0.0188 0.0732 0.1775 0.2756 0.2675 0.1484 0.0360 0.67 0.0001 0.0023 0.0162 0.0659 0.1673 0.2717 0.2758 0.1600 0.0406 0.68 0.0001 0.0019 0.0139 0.0591 0.1569 0.2668 0.2835 0.1721 0.0457 0.69 0.0001 0.0015 0.0118 0.0527 0.1465 0.2609 0.2904 0.1847 0.0514 0.7 0.0001 0.0012 0.0100 0.0467 0.1361 0.2541 0.2965 0.1977 0.0576 0.71 0.0001 0.0010 0.0084 0.0411 0.1258 0.2464 0.3017 0.2110 0.0646 0.72 0.0000 0.0008 0.0070 0.0360 0.1156 0.2379 0.3058 0.2247 0.0722 0.73 0.0000 0.0006 0.0058 0.0313 0.1056 0.2285 0.3089 0.2386 0.0806 0.74 0.0000 0.0005 0.0047 0.0270 0.0959 0.2184 0.3108 0.2527 0.0899 0.75 0.0000 0.0004 0.0038 0.0231 0.0865 0.2076 0.3115 0.2670 0.1001 0.76 0.0000 0.0003 0.0031 0.0196 0.0775 0.1963 0.3108 0.2812 0.1113 0.77 0.0000 0.0002 0.0025 0.0165 0.0689 0.1844 0.3087 0.2953 0.1236 0.78 0.0000 0.0002 0.0019 0.0137 0.0607 0.1722 0.3052 0.3092 0.1370 0.79 0.0000 0.0001 0.0015 0.0113 0.0530 0.1596 0.3002 0.3226 0.1517 0.8 0.0000 0.0001 0.0011 0.0092 0.0459 0.1468 0.2936 0.3355 0.1678 0.81 0.0000 0.0001 0.0009 0.0074 0.0393 0.1339 0.2855 0.3477 0.1853 0.82 0.0000 0.0000 0.0006 0.0058 0.0332 0.1211 0.2758 0.3590 0.2044 0.83 0.0000 0.0000 0.0005 0.0045 0.0277 0.1084 0.2646 0.3691 0.2252 0.84 0.0000 0.0000 0.0003 0.0035 0.0228 0.0959 0.2518 0.3777 0.2479 0.85 0.0000 0.0000 0.0002 0.0026 0.0185 0.0839 0.2376 0.3847 0.2725 0.86 0.0000 0.0000 0.0002 0.0019 0.0147 0.0723 0.2220 0.3897 0.2992 0.87 0.0000 0.0000 0.0001 0.0014 0.0115 0.0613 0.2052 0.3923 0.3282 0.88 0.0000 0.0000 0.0001 0.0009 0.0087 0.0511 0.1872 0.3923 0.3596 0.89 0.0000 0.0000 0.0000 0.0006 0.0064 0.0416 0.1684 0.3892 0.3937 0.9 0.0000 0.0000 0.0000 0.0004 0.0046 0.0331 0.1488 0.3826 0.4305 0.91 0.0000 0.0000 0.0000 0.0002 0.0031 0.0255 0.1288 0.3721 0.4703 0.92 0.0000 0.0000 0.0000 0.0001 0.0021 0.0189 0.1087 0.3570 0.5132 0.93 0.0000 0.0000 0.0000 0.0001 0.0013 0.0134 0.0888 0.3370 0.5596 0.94 0.0000 0.0000 0.0000 0.0000 0.0007 0.0089 0.0695 0.3113 0.6096 0.95 0.0000 0.0000 0.0000 0.0000 0.0004 0.0054 0.0515 0.2793 0.6634 0.96 0.0000 0.0000 0.0000 0.0000 0.0002 0.0029 0.0351 0.2405 0.7214 0.97 0.0000 0.0000 0.0000 0.0000 0.0001 0.0013 0.0210 0.1939 0.7837 0.98 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0099 0.1389 0.8508 0.99 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0026 0.0746 0.9227 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 Joint distribution: 0 1 2 3 4 5 6 7 8 [1,] 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [2,] 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [3,] 0.0010 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [4,] 0.0009 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [5,] 0.0010 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [6,] 0.0011 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [7,] 0.0012 0.0006 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [8,] 0.0012 0.0007 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [9,] 0.0012 0.0008 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [10,] 0.0011 0.0009 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 [11,] 0.0012 0.0010 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 [12,] 0.0013 0.0013 0.0006 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 [13,] 0.0013 0.0014 0.0007 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 [14,] 0.0013 0.0015 0.0008 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 [15,] 0.0012 0.0015 0.0009 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 [16,] 0.0011 0.0015 0.0009 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 [17,] 0.0010 0.0015 0.0010 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 [18,] 0.0010 0.0017 0.0012 0.0005 0.0001 0.0000 0.0000 0.0000 0.0000 [19,] 0.0010 0.0017 0.0013 0.0006 0.0002 0.0000 0.0000 0.0000 0.0000 [20,] 0.0009 0.0016 0.0013 0.0006 0.0002 0.0000 0.0000 0.0000 0.0000 [21,] 0.0008 0.0017 0.0014 0.0007 0.0002 0.0000 0.0000 0.0000 0.0000 [22,] 0.0008 0.0016 0.0015 0.0008 0.0003 0.0001 0.0000 0.0000 0.0000 [23,] 0.0007 0.0016 0.0016 0.0009 0.0003 0.0001 0.0000 0.0000 0.0000 [24,] 0.0007 0.0017 0.0017 0.0010 0.0004 0.0001 0.0000 0.0000 0.0000 [25,] 0.0007 0.0017 0.0019 0.0012 0.0005 0.0001 0.0000 0.0000 0.0000 [26,] 0.0006 0.0017 0.0019 0.0013 0.0005 0.0001 0.0000 0.0000 0.0000 [27,] 0.0006 0.0016 0.0020 0.0014 0.0006 0.0002 0.0000 0.0000 0.0000 [28,] 0.0005 0.0015 0.0020 0.0015 0.0007 0.0002 0.0000 0.0000 0.0000 [29,] 0.0005 0.0015 0.0020 0.0015 0.0007 0.0002 0.0000 0.0000 0.0000 [30,] 0.0004 0.0014 0.0020 0.0016 0.0008 0.0003 0.0001 0.0000 0.0000 [31,] 0.0004 0.0013 0.0020 0.0017 0.0009 0.0003 0.0001 0.0000 0.0000 [32,] 0.0004 0.0013 0.0021 0.0019 0.0010 0.0004 0.0001 0.0000 0.0000 [33,] 0.0003 0.0012 0.0021 0.0019 0.0011 0.0004 0.0001 0.0000 0.0000 [34,] 0.0003 0.0012 0.0020 0.0020 0.0012 0.0005 0.0001 0.0000 0.0000 [35,] 0.0003 0.0011 0.0020 0.0020 0.0013 0.0005 0.0001 0.0000 0.0000 [36,] 0.0002 0.0010 0.0019 0.0021 0.0014 0.0006 0.0002 0.0000 0.0000 [37,] 0.0002 0.0009 0.0019 0.0021 0.0015 0.0007 0.0002 0.0000 0.0000 [38,] 0.0002 0.0009 0.0018 0.0021 0.0016 0.0007 0.0002 0.0000 0.0000 [39,] 0.0002 0.0008 0.0018 0.0022 0.0017 0.0008 0.0003 0.0000 0.0000 [40,] 0.0001 0.0008 0.0017 0.0022 0.0018 0.0009 0.0003 0.0001 0.0000 [41,] 0.0001 0.0007 0.0016 0.0022 0.0018 0.0010 0.0003 0.0001 0.0000 [42,] 0.0001 0.0007 0.0016 0.0023 0.0020 0.0011 0.0004 0.0001 0.0000 [43,] 0.0001 0.0006 0.0016 0.0023 0.0021 0.0012 0.0004 0.0001 0.0000 [44,] 0.0001 0.0006 0.0015 0.0023 0.0022 0.0013 0.0005 0.0001 0.0000 [45,] 0.0001 0.0005 0.0015 0.0023 0.0023 0.0014 0.0006 0.0001 0.0000 [46,] 0.0001 0.0005 0.0014 0.0023 0.0024 0.0016 0.0006 0.0001 0.0000 [47,] 0.0001 0.0004 0.0013 0.0023 0.0024 0.0017 0.0007 0.0002 0.0000 [48,] 0.0001 0.0004 0.0012 0.0022 0.0025 0.0017 0.0008 0.0002 0.0000 [49,] 0.0000 0.0004 0.0012 0.0021 0.0025 0.0018 0.0008 0.0002 0.0000 [50,] 0.0000 0.0003 0.0011 0.0021 0.0025 0.0019 0.0009 0.0003 0.0000 [51,] 0.0000 0.0003 0.0010 0.0021 0.0026 0.0021 0.0010 0.0003 0.0000 [52,] 0.0000 0.0003 0.0010 0.0020 0.0026 0.0022 0.0011 0.0003 0.0000 [53,] 0.0000 0.0002 0.0009 0.0020 0.0027 0.0023 0.0013 0.0004 0.0001 [54,] 0.0000 0.0002 0.0009 0.0019 0.0027 0.0025 0.0014 0.0004 0.0001 [55,] 0.0000 0.0002 0.0008 0.0019 0.0028 0.0026 0.0015 0.0005 0.0001 [56,] 0.0000 0.0002 0.0008 0.0019 0.0029 0.0028 0.0017 0.0006 0.0001 [57,] 0.0000 0.0002 0.0007 0.0019 0.0030 0.0030 0.0019 0.0007 0.0001 [58,] 0.0000 0.0001 0.0007 0.0018 0.0029 0.0031 0.0020 0.0008 0.0001 [59,] 0.0000 0.0001 0.0006 0.0017 0.0030 0.0033 0.0023 0.0009 0.0002 [60,] 0.0000 0.0001 0.0006 0.0016 0.0029 0.0034 0.0024 0.0010 0.0002 [61,] 0.0000 0.0001 0.0005 0.0015 0.0029 0.0034 0.0026 0.0011 0.0002 [62,] 0.0000 0.0001 0.0005 0.0014 0.0028 0.0035 0.0027 0.0012 0.0002 [63,] 0.0000 0.0001 0.0004 0.0013 0.0027 0.0035 0.0029 0.0013 0.0003 [64,] 0.0000 0.0001 0.0004 0.0012 0.0026 0.0035 0.0030 0.0015 0.0003 [65,] 0.0000 0.0001 0.0003 0.0011 0.0025 0.0035 0.0031 0.0016 0.0004 [66,] 0.0000 0.0000 0.0003 0.0010 0.0024 0.0035 0.0033 0.0017 0.0004 [67,] 0.0000 0.0000 0.0002 0.0009 0.0023 0.0035 0.0034 0.0019 0.0005 [68,] 0.0000 0.0000 0.0002 0.0009 0.0022 0.0036 0.0036 0.0021 0.0005 [69,] 0.0000 0.0000 0.0002 0.0008 0.0021 0.0035 0.0037 0.0023 0.0006 [70,] 0.0000 0.0000 0.0002 0.0007 0.0020 0.0035 0.0039 0.0025 0.0007 [71,] 0.0000 0.0000 0.0001 0.0006 0.0018 0.0035 0.0040 0.0027 0.0008 [72,] 0.0000 0.0000 0.0001 0.0006 0.0017 0.0034 0.0041 0.0029 0.0009 [73,] 0.0000 0.0000 0.0001 0.0005 0.0016 0.0033 0.0042 0.0031 0.0010 [74,] 0.0000 0.0000 0.0001 0.0004 0.0015 0.0032 0.0043 0.0033 0.0011 [75,] 0.0000 0.0000 0.0001 0.0004 0.0014 0.0032 0.0045 0.0036 0.0013 [76,] 0.0000 0.0000 0.0001 0.0003 0.0013 0.0030 0.0046 0.0039 0.0015 [77,] 0.0000 0.0000 0.0000 0.0003 0.0011 0.0029 0.0046 0.0041 0.0016 [78,] 0.0000 0.0000 0.0000 0.0002 0.0010 0.0027 0.0046 0.0044 0.0018 [79,] 0.0000 0.0000 0.0000 0.0002 0.0009 0.0026 0.0045 0.0046 0.0020 [80,] 0.0000 0.0000 0.0000 0.0002 0.0008 0.0024 0.0045 0.0048 0.0023 [81,] 0.0000 0.0000 0.0000 0.0001 0.0007 0.0022 0.0044 0.0050 0.0025 [82,] 0.0000 0.0000 0.0000 0.0001 0.0006 0.0020 0.0043 0.0052 0.0028 [83,] 0.0000 0.0000 0.0000 0.0001 0.0005 0.0018 0.0042 0.0054 0.0031 [84,] 0.0000 0.0000 0.0000 0.0001 0.0004 0.0016 0.0040 0.0056 0.0034 [85,] 0.0000 0.0000 0.0000 0.0001 0.0004 0.0015 0.0039 0.0058 0.0038 [86,] 0.0000 0.0000 0.0000 0.0000 0.0003 0.0013 0.0037 0.0060 0.0043 [87,] 0.0000 0.0000 0.0000 0.0000 0.0002 0.0011 0.0035 0.0061 0.0047 [88,] 0.0000 0.0000 0.0000 0.0000 0.0002 0.0010 0.0033 0.0063 0.0053 [89,] 0.0000 0.0000 0.0000 0.0000 0.0001 0.0008 0.0031 0.0064 0.0059 [90,] 0.0000 0.0000 0.0000 0.0000 0.0001 0.0007 0.0028 0.0064 0.0065 [91,] 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0025 0.0063 0.0071 [92,] 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0021 0.0062 0.0078 [93,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0018 0.0060 0.0086 [94,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0015 0.0057 0.0095 [95,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0012 0.0053 0.0104 [96,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0009 0.0048 0.0115 [97,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0042 0.0126 [98,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0035 0.0140 [99,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0025 0.0153 [100,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0014 0.0169 [101,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0189 Marginal distribution of x: 0 1 2 3 4 5 6 7 8 [1,] 0.0302 0.053 0.0731 0.0919 0.111 0.1304 0.1491 0.1667 0.1945 Prior Likelihood Posterior 0 0.0002553486 0.000000e+00 0.000000e+00 0.01 0.0004449172 1.220977e-14 8.186522e-14 0.02 0.0011238562 1.934198e-12 1.296859e-11 0.03 0.0011782407 2.262888e-11 1.517242e-10 0.04 0.0013478226 1.424601e-10 9.551796e-10 0.05 0.0016065566 6.343388e-10 4.253174e-09 0.06 0.0018967819 2.189468e-09 1.468015e-08 0.07 0.0020584439 5.864773e-09 3.932268e-08 0.08 0.0023206187 1.441708e-08 9.666500e-08 0.09 0.0023942879 2.950341e-08 1.978172e-07 0.1 0.0027327600 6.197900e-08 4.155626e-07 0.11 0.0033668701 1.322880e-07 8.869771e-07 0.12 0.0035510989 2.299183e-07 1.541578e-06 0.13 0.0038459977 3.934284e-07 2.637895e-06 0.14 0.0038652642 6.027016e-07 4.041050e-06 0.15 0.0039278568 9.051053e-07 6.068634e-06 0.16 0.0040454772 1.340930e-06 8.990796e-06 0.17 0.0045696418 2.127600e-06 1.426533e-05 0.18 0.0046681910 2.989299e-06 2.004293e-05 0.19 0.0046813726 4.045972e-06 2.712781e-05 0.2 0.0049202819 5.642973e-06 3.783553e-05 0.21 0.0050631493 7.588374e-06 5.087924e-05 0.22 0.0050957966 9.842273e-06 6.599139e-05 0.23 0.0055989047 1.375972e-05 9.225745e-05 0.24 0.0060311812 1.864039e-05 1.249818e-04 0.25 0.0062043418 2.385703e-05 1.599588e-04 0.26 0.0063386401 3.002325e-05 2.013027e-04 0.27 0.0063649610 3.679443e-05 2.467027e-04 0.28 0.0064659920 4.522790e-05 3.032482e-04 0.29 0.0064903260 5.449156e-05 3.653601e-04 0.3 0.0066111223 6.612365e-05 4.433520e-04 0.31 0.0070962613 8.395682e-05 5.629215e-04 0.32 0.0072471264 1.007492e-04 6.755126e-04 0.33 0.0072919930 1.183689e-04 7.936507e-04 0.34 0.0073247119 1.380098e-04 9.253410e-04 0.35 0.0073630489 1.601219e-04 1.073600e-03 0.36 0.0074369523 1.856641e-04 1.244858e-03 0.37 0.0076277405 2.174931e-04 1.458268e-03 0.38 0.0077580970 2.514191e-04 1.685738e-03 0.39 0.0078201475 2.866951e-04 1.922260e-03 0.4 0.0078428470 3.238130e-04 2.171131e-03 0.41 0.0082887911 3.837571e-04 2.573050e-03 0.42 0.0083543453 4.319388e-04 2.896103e-03 0.43 0.0086819107 4.992681e-04 3.347539e-03 0.44 0.0087542089 5.577853e-04 3.739891e-03 0.45 0.0090838371 6.388926e-04 4.283706e-03 0.46 0.0091005959 7.039833e-04 4.720133e-03 0.47 0.0091080246 7.721846e-04 5.177415e-03 0.48 0.0091256759 8.450398e-04 5.665902e-03 0.49 0.0091930845 9.266935e-04 6.213381e-03 0.5 0.0094116451 1.029399e-03 6.902008e-03 0.51 0.0094909353 1.122743e-03 7.527870e-03 0.52 0.0098901582 1.261432e-03 8.457766e-03 0.53 0.0101015510 1.384832e-03 9.285149e-03 0.54 0.0105461922 1.549287e-03 1.038781e-02 0.55 0.0109240958 1.714528e-03 1.149573e-02 0.56 0.0114334923 1.911483e-03 1.281629e-02 0.57 0.0115358478 2.048300e-03 1.373363e-02 0.58 0.0119969656 2.255776e-03 1.512474e-02 0.59 0.0122481929 2.431699e-03 1.630428e-02 0.6 0.0123391919 2.579124e-03 1.729275e-02 0.61 0.0124118523 2.723346e-03 1.825975e-02 0.62 0.0124271709 2.853958e-03 1.913549e-02 0.63 0.0124853364 2.992299e-03 2.006305e-02 0.64 0.0126011627 3.142342e-03 2.106907e-02 0.65 0.0126241851 3.265710e-03 2.189624e-02 0.66 0.0127474065 3.410369e-03 2.286616e-02 0.67 0.0131012645 3.613665e-03 2.422924e-02 0.68 0.0132101274 3.744712e-03 2.510790e-02 0.69 0.0135608113 3.937872e-03 2.640301e-02 0.7 0.0135873840 4.028326e-03 2.700950e-02 0.71 0.0136847121 4.128001e-03 2.767781e-02 0.72 0.0138008926 4.220620e-03 2.829881e-02 0.73 0.0139649446 4.313819e-03 2.892370e-02 0.74 0.0144373641 4.487278e-03 3.008672e-02 0.75 0.0146132531 4.551479e-03 3.051718e-02 0.76 0.0146805832 4.562526e-03 3.059125e-02 0.77 0.0148232102 4.576151e-03 3.068260e-02 0.78 0.0148255848 4.524628e-03 3.033715e-02 0.79 0.0148536068 4.458524e-03 2.989393e-02 0.8 0.0149302301 4.383535e-03 2.939113e-02 0.81 0.0150527232 4.297248e-03 2.881259e-02 0.82 0.0151458522 4.177148e-03 2.800733e-02 0.83 0.0152043762 4.022472e-03 2.697025e-02 0.84 0.0154629872 3.893741e-03 2.610712e-02 0.85 0.0156551337 3.719725e-03 2.494036e-02 0.86 0.0157776946 3.503066e-03 2.348769e-02 0.87 0.0160078083 3.284673e-03 2.202338e-02 0.88 0.0164229373 3.075158e-03 2.061860e-02 0.89 0.0164826251 2.775299e-03 1.860809e-02 0.9 0.0165846738 2.467857e-03 1.654672e-02 0.91 0.0166920450 2.149814e-03 1.441428e-02 0.92 0.0167101207 1.815702e-03 1.217409e-02 0.93 0.0170138830 1.510272e-03 1.012621e-02 0.94 0.0171319497 1.191336e-03 7.987784e-03 0.95 0.0173191773 8.911831e-04 5.975287e-03 0.96 0.0174081968 6.104628e-04 4.093088e-03 0.97 0.0178244734 3.741516e-04 2.508647e-03 0.98 0.0180145982 1.787307e-04 1.198370e-03 0.99 0.0183186200 4.829053e-05 3.237828e-04 1 0.0189152721 0.000000e+00 0.000000e+00 > > ## 5 successes, 6 trials, non-uniform prior > theta<-c(0.3,0.4,0.5) > theta.prior<-c(0.2,0.3,0.5) > results<-binodp(5,6,uniform=FALSE,theta=theta,theta.prior=theta.prior,ret=TRUE) Conditional distribution of x given theta and n: 0 1 2 3 4 5 6 0.3 0.1176 0.3025 0.3241 0.1852 0.0595 0.0102 0.0007 0.4 0.0467 0.1866 0.3110 0.2765 0.1382 0.0369 0.0041 0.5 0.0156 0.0938 0.2344 0.3125 0.2344 0.0938 0.0156 Joint distribution: 0 1 2 3 4 5 6 [1,] 0.0235 0.0605 0.0648 0.0370 0.0119 0.0020 0.0001 [2,] 0.0140 0.0560 0.0933 0.0829 0.0415 0.0111 0.0012 [3,] 0.0078 0.0469 0.1172 0.1562 0.1172 0.0469 0.0078 Marginal distribution of x: 0 1 2 3 4 5 6 [1,] 0.0453 0.1634 0.2753 0.2762 0.1706 0.06 0.0092 Prior Likelihood Posterior 0.3 0.2 0.0020412 0.03403395 0.4 0.3 0.0110592 0.18439560 0.5 0.5 0.0468750 0.78157044 > > ## plot the results from the previous example using a side-by-side barplot > results.matrix<-rbind(results$theta.prior,results$posterior) > colnames(results.matrix)<-theta > barplot(results.matrix,col=c("red","blue"),beside=TRUE + ,xlab=expression(theta),ylab=expression(Probability(theta))) > box() > legend(1,0.65,legend=c("Prior","Posterior"),fill=c("red","blue")) > > > > cleanEx(); ..nameEx <- "binogcp" > > ### * binogcp > > flush(stderr()); flush(stdout()) > > ### Name: binogcp > ### Title: Binomial sampling with a general continuous prior > ### Aliases: binogcp > ### Keywords: misc > > ### ** Examples > > ## simplest call with 6 successes observed in 8 trials and a continuous > ## uniform prior > binogcp(6,8) > > ## 6 successes, 8 trials and a Beta(2,2) prior > binogcp(6,8,density="beta",params=c(2,2)) > > ## 5 successes, 10 trials and a N(0.5,0.25) prior > binogcp(5,10,density="normal",params=c(0.5,0.25)) > > ## 4 successes, 12 trials with a user specified triangular continuous prior > theta<-seq(0,1,by=0.001) > theta.prior<-rep(0,length(theta)) > theta.prior[theta<=0.5]<-4*theta[theta<=0.5] > theta.prior[theta>0.5]<-4-4*theta[theta>0.5] > results<-binogcp(4,12,"user",theta=theta,theta.prior=theta.prior,ret=TRUE) > > ## find the posterior CDF using the previous example and Simpson's rule > cdf<-sintegral(theta,results$posterior,n.pts=length(theta),ret=TRUE) > plot(cdf,type="l",xlab=expression(theta[0]) + ,ylab=expression(Pr(theta<=theta[0]))) > > ## use the cdf to find the 95% credible region. Thanks to John Wilkinson for this simplified code. > lcb<-cdf$x[with(cdf,which.max(x[y<=0.025]))] > ucb<-cdf$x[with(cdf,which.max(x[y<=0.975]))] > cat(paste("Approximate 95% credible interval : [" + ,round(lcb,4)," ",round(ucb,4),"]\n",sep="")) Approximate 95% credible interval : [0.1733 0.6129] > > ## find the posterior mean, variance and std. deviation > ## using Simpson's rule and the output from the previous example > dens<-theta*results$posterior # calculate theta*f(theta | x, n) > post.mean<-sintegral(theta,dens) > > dens<-(theta-post.mean)^2*results$posterior > post.var<-sintegral(theta,dens) > post.sd<-sqrt(post.var) > > # calculate an approximate 95% credible region using the posterior mean and > # std. deviation > lb<-post.mean-qnorm(0.975)*post.sd > ub<-post.mean+qnorm(0.975)*post.sd > > cat(paste("Approximate 95% credible interval : [" + ,round(lb,4)," ",round(ub,4),"]\n",sep="")) Approximate 95% credible interval : [0.165 0.6123] > > > > cleanEx(); ..nameEx <- "normdp" > > ### * normdp > > flush(stderr()); flush(stdout()) > > ### Name: normdp > ### Title: Bayesian inference on a normal mean with a discrete prior > ### Aliases: normdp > ### Keywords: misc > > ### ** Examples > > ## generate a sample of 20 observations from a N(-0.5,1) population > x<-rnorm(20,-0.5,1) > > ## find the posterior density with a uniform prior on mu > normdp(x,1) > > ## find the posterior density with a non-uniform prior on mu > mu<-seq(-3,3,by=0.1) > mu.prior<-runif(length(mu)) > mu.prior<-sort(mu.prior/sum(mu.prior)) > normdp(x,1,uniform=FALSE,mu=mu,mu.prior=mu.prior) > > ## Let mu have the discrete distribution with 5 possible > ## values, 2, 2.5, 3, 3.5 and 4, and associated prior probability of > ## 0.1, 0.2, 0.4, 0.2, 0.1 respectively. Find the posterior > ## distribution after a drawing random sample of n = 5 observations > ## from a N(mu,1) distribution y = [1.52, 0.02, 3.35, 3.49, 1.82] > mu<-seq(2,4,by=0.5) > mu.prior<-c(0.1,0.2,0.4,0.2,0.1) > y<-c(1.52,0.02,3.35,3.49,1.82) > normdp(y,1,uniform=FALSE,n.mu=5,mu,mu.prior) > > > > cleanEx(); ..nameEx <- "normgcp" > > ### * normgcp > > flush(stderr()); flush(stdout()) > > ### Name: normgcp > ### Title: Bayesian inference on a normal mean with a general continuous > ### prior > ### Aliases: normgcp > ### Keywords: misc > > ### ** Examples > > ## generate a sample of 20 observations from a N(-0.5,1) population > x<-rnorm(20,-0.5,1) > > ## find the posterior density with a uniform U[-3,3] prior on mu > normgcp(x,1,params=c(-3,3)) > > ## find the posterior density with a non-uniform prior on mu > mu<-seq(-3,3,by=0.1) > mu.prior<-rep(0,length(mu)) > mu.prior[mu<=0]<-1/3+mu[mu<=0]/9 > mu.prior[mu>0]<-1/3-mu[mu>0]/9 > normgcp(x,1,density="user",mu=mu,mu.prior=mu.prior) > > ## find the CDF for the previous example and plot it > results<-normgcp(x,1,density="user",mu=mu,mu.prior=mu.prior,ret=TRUE) > cdf<-sintegral(mu,results$posterior,n.pts=length(mu),ret=TRUE) > plot(cdf,type="l",xlab=expression(mu[0]) + ,ylab=expression(Pr(mu<=mu[0]))) > > ## use the CDF for the previous example to find a 95% > ## credible interval for mu. Thanks to John Wilkinson for this simplified code > > lcb<-cdf$x[with(cdf,which.max(x[y<=0.025]))] > ucb<-cdf$x[with(cdf,which.max(x[y<=0.975]))] > cat(paste("Approximate 95% credible interval : [" + ,round(lcb,4)," ",round(ucb,4),"]\n",sep="")) Approximate 95% credible interval : [-0.7969 0.0469] > > ## use the CDF from the previous example to find the posterior mean > ## and std. deviation > dens<-mu*results$posterior > post.mean<-sintegral(mu,dens) > > dens<-(mu-post.mean)^2*results$posterior > post.var<-sintegral(mu,dens) > post.sd<-sqrt(post.var) > > ## use the mean and std. deviation from the previous example to find > ## an approximate 95% credible interval > lb<-post.mean-qnorm(0.975)*post.sd > ub<-post.mean+qnorm(0.975)*post.sd > > cat(paste("Approximate 95% credible interval : [" + ,round(lb,4)," ",round(ub,4),"]\n",sep="")) Approximate 95% credible interval : [-0.7254 0.1379] > > > > > cleanEx(); ..nameEx <- "normnp" > > ### * normnp > > flush(stderr()); flush(stdout()) > > ### Name: normnp > ### Title: Bayesian inference on a normal mean with a normal prior > ### Aliases: normnp > ### Keywords: misc > > ### ** Examples > > ## generate a sample of 20 observations from a N(-0.5,1) population > x<-rnorm(20,-0.5,1) > > ## find the posterior density with a N(0,1) prior on mu > normnp(x,1) Posterior mean : -0.2947392 Posterior std. deviation : 0.2182179 Prob. Quantile ------ --------- 0.005 -0.8568312 0.01 -0.8023899 0.025 -0.7224384 0.05 -0.6536757 0.5 -0.2947392 0.95 0.0641973 0.975 0.13296 0.99 0.2129116 0.995 0.2673529 > > ## find the posterior density with N(0.5,3) prior on mu > normnp(x,1,0.5,3) Posterior mean : -0.3050039 Posterior std. deviation : 0.2229882 Prob. Quantile ------ --------- 0.005 -0.8793835 0.01 -0.8237521 0.025 -0.7420528 0.05 -0.6717869 0.5 -0.3050039 0.95 0.0617791 0.975 0.132045 0.99 0.2137443 0.995 0.2693758 > > ## Find the posterior density for mu, given a random sample of 4 > ## observations from N(mu,sigma^2=1), y = [2.99, 5.56, 2.83, 3.47], > ## and a N(3,sd=2)$ prior for mu > y<-c(2.99,5.56,2.83,3.47) > normnp(y,1,3,2) Posterior mean : 3.6705882 Posterior std. deviation : 0.4850713 Prob. Quantile ------ --------- 0.005 2.4211275 0.01 2.5421438 0.025 2.7198661 0.05 2.872717 0.5 3.6705882 0.95 4.4684594 0.975 4.6213104 0.99 4.7990327 0.995 4.920049 > > > > cleanEx(); ..nameEx <- "sintegral" > > ### * sintegral > > flush(stderr()); flush(stdout()) > > ### Name: sintegral > ### Title: Numerical integration using Simpson's Rule > ### Aliases: sintegral > ### Keywords: misc > > ### ** Examples > > ## integrate the normal density from -3 to 3 > x<-seq(-3,3,length=100) > fx<-dnorm(x) > estimate<-sintegral(x,fx) > true.val<-diff(pnorm(c(-3,3))) > cat(paste("Absolute error :",round(abs(estimate-true.val),7),"\n")) Absolute error : 8.1e-06 > cat(paste("Relative percentage error :", 100*round((abs(estimate-true.val)/true.val),6),"%\n")) Relative percentage error : 8e-04 % > > > > cleanEx(); ..nameEx <- "sscsample" > > ### * sscsample > > flush(stderr()); flush(stdout()) > > ### Name: sscsample > ### Title: Simple, Stratified and Cluster Sampling > ### Aliases: sscsample > ### Keywords: misc > > ### ** Examples > > ## Draw 200 samples of size 20 using simple random sampling > sscsample(20,200) Sample Mean Stratum 1 Stratum 2 Stratum 3 ------ ------- --------- --------- --------- 1 8.6622 8 7 5 2 9.4229 10 7 3 3 9.1091 7 11 2 4 8.9554 5 10 5 5 9.6778 10 8 2 6 9.1381 9 5 6 7 8.6038 5 11 4 8 9.0082 10 10 0 9 8.5104 7 10 3 10 9.4496 8 8 4 11 9.489 7 9 4 12 9.0726 11 7 2 13 8.1903 6 7 7 14 9.4222 12 6 2 15 9.0209 12 4 4 16 8.7743 8 9 3 17 8.9269 8 7 5 18 9.575 10 8 2 19 9.4204 10 7 3 20 8.51 8 6 6 21 8.0331 5 8 7 22 9.1312 8 8 4 23 8.6813 10 5 5 24 9.2441 8 9 3 25 8.8631 11 5 4 26 9.1506 4 14 2 27 8.4566 11 4 5 28 8.9483 9 6 5 29 8.7779 7 8 5 30 8.8151 7 10 3 31 9.0852 9 4 7 32 9.7672 10 7 3 33 8.9782 8 7 5 34 9.9522 10 9 1 35 8.8938 7 10 3 36 8.7591 8 7 5 37 7.969 8 7 5 38 9.2396 8 7 5 39 8.8153 6 8 6 40 8.5527 7 9 4 41 8.6683 8 6 6 42 7.7887 5 10 5 43 8.5634 9 6 5 44 8.4282 8 7 5 45 8.7474 10 6 4 46 8.9232 9 7 4 47 8.1349 7 6 7 48 9.708 7 8 5 49 9.002 8 7 5 50 9.5617 11 6 3 51 9.1597 8 8 4 52 8.8433 8 7 5 53 8.3989 9 9 2 54 9.3509 9 5 6 55 8.729 8 11 1 56 8.8508 9 8 3 57 9.2596 10 7 3 58 8.9809 12 5 3 59 8.5617 11 7 2 60 8.6512 10 6 4 61 8.2904 7 8 5 62 9.6128 10 5 5 63 8.1623 9 7 4 64 9.2797 9 8 3 65 9.4126 10 8 2 66 8.6353 9 9 2 67 9.2193 9 9 2 68 8.9068 10 5 5 69 9.22 9 8 3 70 8.7532 7 7 6 71 9.5983 11 8 1 72 9.212 8 7 5 73 8.7754 8 8 4 74 8.7958 10 8 2 75 8.6738 8 9 3 76 9.2328 9 8 3 77 8.7389 5 9 6 78 9.1247 8 9 3 79 9.1996 8 8 4 80 9.1067 7 10 3 81 8.4344 10 5 5 82 8.4276 8 7 5 83 8.7144 8 7 5 84 9.1489 7 8 5 85 8.6467 7 6 7 86 9.1523 13 6 1 87 8.7855 7 8 5 88 8.1847 9 7 4 89 9.3723 8 8 4 90 9.6553 10 7 3 91 8.7851 8 8 4 92 9.4223 6 10 4 93 9.0978 7 7 6 94 8.4406 9 7 4 95 10.1049 10 6 4 96 9.2016 9 8 3 97 9.1549 9 5 6 98 9.59 11 7 2 99 9.256 6 9 5 100 8.9676 9 8 3 101 8.2574 8 6 6 102 8.9522 11 4 5 103 8.5347 9 8 3 104 9.6387 8 11 1 105 7.8958 5 10 5 106 9.1675 8 9 3 107 8.8193 11 4 5 108 9.1815 11 5 4 109 9.5379 7 9 4 110 9.3397 7 8 5 111 9.4981 8 8 4 112 9.1161 10 9 1 113 9.1644 13 4 3 114 9.1795 7 9 4 115 8.2678 8 8 4 116 9.6186 11 7 2 117 8.583 8 9 3 118 8.4399 7 9 4 119 9.0378 7 8 5 120 7.9806 11 5 4 121 9.4438 7 11 2 122 8.774 5 12 3 123 8.9517 7 10 3 124 9.1335 8 8 4 125 9.2435 11 5 4 126 8.9456 6 8 6 127 9.297 8 8 4 128 8.6661 6 8 6 129 9.0915 7 8 5 130 8.4431 4 7 9 131 10.1527 9 9 2 132 8.0642 9 8 3 133 9.1083 12 6 2 134 9.0962 9 7 4 135 8.558 9 4 7 136 9.5 8 6 6 137 8.3753 11 6 3 138 9.4504 7 10 3 139 9.6651 12 6 2 140 8.8378 8 9 3 141 8.7776 9 2 9 142 8.5071 5 13 2 143 8.9758 7 7 6 144 9.4339 7 7 6 145 9.0367 5 10 5 146 8.9752 5 11 4 147 8.8449 9 6 5 148 8.7691 10 7 3 149 9.4006 8 9 3 150 9.0513 8 7 5 151 8.7615 6 7 7 152 9.3279 12 5 3 153 9.6178 14 3 3 154 8.2611 5 9 6 155 9.9021 6 13 1 156 9.415 9 7 4 157 9.0172 7 6 7 158 8.6915 6 8 6 159 8.7625 8 9 3 160 8.9248 9 9 2 161 8.1824 5 12 3 162 8.8059 7 8 5 163 8.6766 7 7 6 164 8.8441 10 6 4 165 9.4665 4 11 5 166 9.0637 7 9 4 167 8.796 9 6 5 168 8.2304 9 8 3 169 8.2867 6 13 1 170 9.2026 9 7 4 171 9.6162 10 5 5 172 8.9342 10 4 6 173 8.7011 5 9 6 174 9.2244 8 7 5 175 8.8161 6 9 5 176 9.6898 9 9 2 177 7.6997 5 10 5 178 9.9048 10 8 2 179 9.8132 12 7 1 180 8.6626 11 5 4 181 9.4864 9 9 2 182 9.3652 6 7 7 183 8.2622 6 10 4 184 8.0684 11 4 5 185 8.7937 5 9 6 186 9.0172 9 7 4 187 8.7494 8 7 5 188 9.0436 12 5 3 189 8.5754 9 8 3 190 9.4385 8 11 1 191 8.2175 7 6 7 192 8.4886 9 6 5 193 8.5794 9 9 2 194 9.2497 8 8 4 195 9.3179 9 8 3 196 8.8312 8 8 4 197 8.2743 9 7 4 198 8.6393 10 8 2 199 8.7134 9 5 6 200 8.5928 7 10 3 > > ## Draw 200 samples of size 20 using simple random sampling and store the > ## results. Extract the means of all 200 samples, and the 50th sample > res<-sscsample(20,200,ret=TRUE) Sample Mean Stratum 1 Stratum 2 Stratum 3 ------ ------- --------- --------- --------- 1 9.4863 9 8 3 2 8.6533 8 9 3 3 8.7836 9 7 4 4 9.6997 10 4 6 5 8.9729 7 7 6 6 9.1877 10 5 5 7 9.4656 9 9 2 8 8.3907 5 10 5 9 8.4478 7 11 2 10 10.3547 11 5 4 11 9.0014 9 6 5 12 8.8108 8 6 6 13 8.6727 7 7 6 14 8.8854 11 7 2 15 8.4782 10 7 3 16 8.8776 8 7 5 17 9.476 8 7 5 18 8.6036 9 7 4 19 9.5443 10 7 3 20 9.2135 6 9 5 21 9.8451 9 5 6 22 9.6243 8 7 5 23 8.7605 7 6 7 24 8.7291 9 9 2 25 9.0578 9 8 3 26 9.7057 12 4 4 27 9.4043 6 11 3 28 8.5616 11 5 4 29 8.9044 5 12 3 30 8.9648 8 9 3 31 8.9774 8 9 3 32 8.6963 5 10 5 33 9.0671 7 9 4 34 8.955 10 9 1 35 8.9291 7 8 5 36 9.9137 8 8 4 37 8.5105 6 8 6 38 8.7797 6 12 2 39 8.9932 6 10 4 40 9.2841 7 10 3 41 8.701 6 11 3 42 9.4415 8 10 2 43 8.9116 12 4 4 44 8.9389 6 10 4 45 9.0241 7 9 4 46 9.063 9 7 4 47 8.4735 6 8 6 48 9.3749 7 9 4 49 8.2386 4 8 8 50 8.8762 8 7 5 51 8.4559 11 5 4 52 9.4079 8 10 2 53 9.3928 9 8 3 54 9.2893 9 8 3 55 8.6606 7 10 3 56 8.698 9 8 3 57 8.8879 10 7 3 58 8.6991 8 7 5 59 9.048 6 11 3 60 9.3507 7 9 4 61 9.7158 9 6 5 62 8.2017 5 9 6 63 9.7734 9 7 4 64 9.3463 10 5 5 65 8.9293 5 12 3 66 8.0254 3 12 5 67 8.9181 9 9 2 68 9.1991 8 5 7 69 9.0505 7 10 3 70 8.6754 7 9 4 71 8.468 8 9 3 72 8.6153 8 7 5 73 9.3528 7 10 3 74 9.0786 6 8 6 75 9.1784 6 9 5 76 9.304 7 10 3 77 8.184 7 8 5 78 8.4211 7 11 2 79 8.1604 2 11 7 80 9.3803 7 9 4 81 9.059 9 8 3 82 9.0318 10 6 4 83 8.3925 5 12 3 84 8.9867 9 7 4 85 8.3381 5 10 5 86 9.101 9 8 3 87 8.9063 5 11 4 88 9.1489 6 9 5 89 9.0295 8 8 4 90 8.3839 4 14 2 91 8.8296 6 10 4 92 9.8552 8 9 3 93 9.4238 11 8 1 94 8.7712 6 8 6 95 10.1326 10 9 1 96 9.6485 7 9 4 97 9.1037 9 9 2 98 8.8005 8 8 4 99 8.8423 6 9 5 100 9.7982 9 6 5 101 8.7422 7 8 5 102 8.4462 8 8 4 103 9.1879 7 11 2 104 8.8439 4 10 6 105 8.4098 7 8 5 106 9.4888 10 7 3 107 8.5291 4 12 4 108 8.3135 6 7 7 109 9.0385 9 8 3 110 9.1492 8 6 6 111 8.8593 8 11 1 112 8.0788 6 7 7 113 9.929 7 10 3 114 8.9839 8 5 7 115 8.2334 4 10 6 116 9.4493 10 5 5 117 8.8599 8 9 3 118 8.3468 6 10 4 119 8.9048 8 9 3 120 9.4872 7 9 4 121 9.0251 7 9 4 122 8.6434 8 7 5 123 10.2725 11 5 4 124 9.1342 7 7 6 125 8.6342 7 9 4 126 9.0658 8 9 3 127 8.9256 11 5 4 128 9.3343 9 8 3 129 9.2614 8 7 5 130 8.5318 5 10 5 131 9.3652 10 8 2 132 8.2462 6 6 8 133 9.1295 8 9 3 134 8.5595 5 10 5 135 8.8283 8 7 5 136 8.6845 8 9 3 137 8.5538 8 7 5 138 8.9558 9 6 5 139 8.6588 5 11 4 140 9.2658 11 8 1 141 9.4837 9 10 1 142 8.7831 5 9 6 143 8.3021 8 8 4 144 9.0955 5 12 3 145 8.6961 4 12 4 146 9.1454 6 7 7 147 9.8267 9 6 5 148 9.4111 8 8 4 149 9.4847 5 9 6 150 8.3499 9 3 8 151 8.5449 5 11 4 152 9.8904 9 6 5 153 9.2638 10 5 5 154 8.7941 8 5 7 155 8.9406 8 7 5 156 8.5486 8 7 5 157 8.4053 7 10 3 158 9.1784 10 4 6 159 9.4133 9 6 5 160 8.982 9 4 7 161 8.9184 9 4 7 162 8.7582 7 8 5 163 8.5003 12 5 3 164 8.412 7 7 6 165 8.4121 7 8 5 166 8.4454 4 9 7 167 8.863 8 7 5 168 10.0257 7 8 5 169 8.5717 7 11 2 170 9.4035 7 8 5 171 9.1742 9 7 4 172 8.8994 8 7 5 173 9.6792 9 8 3 174 8.5737 12 6 2 175 9.3472 11 5 4 176 10.088 13 4 3 177 8.3262 8 6 6 178 9.153 6 11 3 179 9.1781 8 9 3 180 9.952 11 6 3 181 9.6041 9 9 2 182 9.6503 8 8 4 183 8.442 7 6 7 184 8.5559 8 7 5 185 8.4689 4 12 4 186 8.8371 8 8 4 187 9.1285 7 9 4 188 9.7369 10 6 4 189 9.2645 9 6 5 190 9.3722 10 6 4 191 9.2217 6 12 2 192 10.0535 12 5 3 193 9.3676 9 9 2 194 8.9871 11 5 4 195 8.6927 4 11 5 196 8.6958 8 6 6 197 9.3348 7 11 2 198 8.7216 10 8 2 199 9.4217 8 8 4 200 9.1843 9 10 1 > res$means [1] 9.486300 8.653255 8.783595 9.699665 8.972930 9.187670 9.465560 [8] 8.390655 8.447820 10.354720 9.001370 8.810780 8.672670 8.885385 [15] 8.478230 8.877605 9.476040 8.603620 9.544335 9.213545 9.845140 [22] 9.624335 8.760540 8.729095 9.057845 9.705700 9.404275 8.561595 [29] 8.904440 8.964790 8.977430 8.696285 9.067085 8.955035 8.929140 [36] 9.913710 8.510450 8.779745 8.993245 9.284100 8.700995 9.441470 [43] 8.911550 8.938945 9.024100 9.063030 8.473455 9.374880 8.238650 [50] 8.876220 8.455940 9.407945 9.392800 9.289325 8.660585 8.697960 [57] 8.887895 8.699110 9.048050 9.350700 9.715780 8.201680 9.773435 [64] 9.346295 8.929300 8.025405 8.918130 9.199050 9.050455 8.675395 [71] 8.467960 8.615320 9.352845 9.078590 9.178430 9.303980 8.184040 [78] 8.421075 8.160405 9.380270 9.058970 9.031780 8.392490 8.986735 [85] 8.338130 9.101020 8.906345 9.148900 9.029495 8.383900 8.829605 [92] 9.855180 9.423765 8.771185 10.132550 9.648495 9.103715 8.800480 [99] 8.842310 9.798160 8.742175 8.446230 9.187935 8.843910 8.409770 [106] 9.488830 8.529115 8.313480 9.038545 9.149190 8.859340 8.078810 [113] 9.928990 8.983860 8.233420 9.449255 8.859855 8.346835 8.904825 [120] 9.487245 9.025070 8.643435 10.272515 9.134215 8.634190 9.065805 [127] 8.925615 9.334295 9.261445 8.531750 9.365200 8.246215 9.129465 [134] 8.559530 8.828325 8.684535 8.553840 8.955850 8.658840 9.265810 [141] 9.483745 8.783130 8.302105 9.095505 8.696080 9.145425 9.826680 [148] 9.411090 9.484665 8.349950 8.544915 9.890415 9.263820 8.794070 [155] 8.940595 8.548600 8.405290 9.178360 9.413295 8.981960 8.918440 [162] 8.758230 8.500285 8.412005 8.412065 8.445425 8.863020 10.025725 [169] 8.571695 9.403475 9.174185 8.899400 9.679250 8.573735 9.347225 [176] 10.087980 8.326150 9.153040 9.178120 9.952045 9.604095 9.650275 [183] 8.441970 8.555890 8.468940 8.837080 9.128455 9.736945 9.264535 [190] 9.372215 9.221725 10.053480 9.367560 8.987090 8.692700 8.695845 [197] 9.334775 8.721555 9.421675 9.184330 > res$samples[,50] [1] 54 91 20 25 43 71 15 68 82 66 39 70 9 2 48 90 88 35 8 97 > > > > cleanEx(); ..nameEx <- "xdesign" > > ### * xdesign > > flush(stderr()); flush(stdout()) > > ### Name: xdesign > ### Title: Monte Carlo study of randomized and blocked designs > ### Aliases: xdesign > ### Keywords: misc > > ### ** Examples > > # Carry out simulations using the default parameters > xdesign() Variable N Mean Median TrMean StDev SE Mean X 80 0.106 0.074 0.107 0.901 0.101 Y 80 0.027 0.084 0.055 0.77 0.086 Variable Minimum Maximum Q1 Q3 X -2.215 2.402 -0.452 0.637 Y -1.895 1.852 -0.477 0.607 The Pearson correlation between Y and Y is: 0.725 Variable N Mean Median TrMean StDev SE Mean Randomized 2000 0.027 0.025 0.027 0.145 0.003 Blocked 2000 0.027 0.029 0.027 0.085 0.002 Variable Minimum Maximum Q1 Q3 Randomized -0.442 0.558 -0.075 0.126 Blocked -0.253 0.314 -0.032 0.085 > # Carry out simulations using a simulated response with 5 treaments, groups of size 25, and a correlation of -0.6 between the response and lurking variable > xdesign(corr=-0.6,size=25,n.treatments=5) Variable N Mean Median TrMean StDev SE Mean X 125 -0.041 -0.026 -0.047 0.912 0.082 Y 125 -0.025 0.012 -0.03 0.978 0.087 Variable Minimum Maximum Q1 Q3 X -2.564 2.342 -0.677 0.5 Y -2.08 2.73 -0.711 0.611 The Pearson correlation between Y and Y is: -0.591 Variable N Mean Median TrMean StDev SE Mean Randomized 2500 -0.025 -0.029 -0.026 0.177 0.004 Blocked 2500 -0.025 -0.028 -0.026 0.143 0.003 Variable Minimum Maximum Q1 Q3 Randomized -0.729 0.603 -0.145 0.096 Blocked -0.522 0.464 -0.12 0.073 > > > > ### *