512018-08-20 09:42:10PeterWatson502018-08-20 09:40:44PeterWatson492018-08-20 09:37:47PeterWatson482015-03-11 11:56:16PeterWatson472014-08-11 15:01:35PeterWatson462014-08-11 15:01:23PeterWatson452014-08-11 15:00:57PeterWatson442014-08-11 14:56:09PeterWatson432013-09-30 14:34:09PeterWatson422013-05-15 11:37:23PeterWatson412013-05-15 11:37:05PeterWatson402013-05-15 11:36:48PeterWatson392013-05-15 11:36:23PeterWatson382013-05-15 11:33:15PeterWatson372013-05-14 15:22:23PeterWatson362013-05-13 15:05:51PeterWatson352013-05-13 15:02:25PeterWatson342013-05-13 14:58:41PeterWatson332013-03-08 10:17:23localhostconverted to 1.6 markup322012-08-10 10:34:45PeterWatson312012-08-10 10:33:54PeterWatson302012-08-10 10:33:34PeterWatson292012-08-10 10:32:06PeterWatson282012-08-10 10:31:47PeterWatson272012-08-10 10:30:53PeterWatson262012-08-10 10:30:31PeterWatson252012-08-10 10:23:55PeterWatson242012-08-10 10:18:34PeterWatson232012-08-10 09:20:55PeterWatson222012-08-10 09:19:54PeterWatson212012-08-10 09:19:27PeterWatson202012-08-10 09:18:35PeterWatson192012-08-10 09:15:33PeterWatson182012-08-10 09:14:29PeterWatson172012-04-26 14:04:35PeterWatson162012-04-26 14:04:17PeterWatson152011-04-19 12:47:03PeterWatson142011-04-19 12:46:15PeterWatson132011-04-19 12:08:07PeterWatson122009-11-23 12:12:31PeterWatson112009-11-23 12:11:49PeterWatson102007-08-13 16:16:29PeterWatson92007-08-13 16:15:33PeterWatson82007-08-13 16:15:15PeterWatson72007-08-13 16:14:22PeterWatson62007-08-13 16:14:06PeterWatson52007-08-13 16:10:14PeterWatson42007-08-13 15:54:11PeterWatson32007-08-13 15:50:17PeterWatson22007-08-13 15:50:04PeterWatson12007-08-13 15:41:10PeterWatson

Gigerenzer (2002) suggests a way to obtain conditional probabilities using frequencies in a decision tree. An illustrated example (Wininger and Johnson, 2018) using this method in prosthetics is is here. Cortina and Dunlap (1997) give an example evaluating the detection rate of a test (positive/negative result) to detect schizophrenia (disorder). To do this one fixes the following: The base rate of schizophrenia in adults (2%) The test will correctly identify schizophrenia (give a positive result) on 95% of people with schizophrenia The test will correctly identify normal individuals (give a negative result) on 97% of normal people. Despite this we can show the test is unreliable. This is a more intuitive way of illustrating the equivalent Bayesian equation: $$\mbox{P(No disorder|+ result) = }\frac{\mbox{P(No disorder) * P(+ result | No disorder)}}{\mbox{P(No disorder) * P(+ result | No disorder) + P(Disorder) * P(- result | Disorder)}}$$ A talk with subtitles further illustrating aspects of conditional probabilities given by Ted Donnelly (Oxford), a geneticist, is available for viewing here. More on Bayes theorem:Illustration of priors and likelihoods Using statistical distributions of likelihoods and priors to obtain posterior distributions Baguley (2012, p.393-395) gives formulae for the posterior mean ($$u_text{post}$$) and variance ($$\sigma_text{post}^text{2}$$) for a normal distribution, of form N(u, sigma2 ), with an assumed prior distribution of form N(u_p, sigma_p2 ) and an obtained likelihood distribution (obtained using sample data) equal to a N(u_lik, sigma_lik2 ). In particular sigma_post2 = [ 1 /sigma_lik2 + 1 /sigma_p2 ] -1 u_post = (sigma_post2 / sigma_lik2 ) u_lik + (sigma_post2 / sigma_p2 ) u_p Zoltan Dienes also has a comprehensive website featuring a range of on-line Bayesian calculators including one that will evaluate posterior means and sds for Normal distributions here. Baguley also gives references for obtaining posterior distributions for data having a binomial distribution which assumes a beta distribution as its prior distribution. For this reason the posterior distribution, in this case, is called a beta-binomial distribution. WINBUGS is freeware for fitting a range of models using simulation (via the Gibbs sampler) and is available from here. Using conditional probabilities to compute False Discovery Rates (article) References Andrews M and Baguley T (2013) Prior approval: The growth of Bayesian methods in psychology British Journal of Mathematical and Statistical Psychology 66(1) 1–7. Primer article free on-line to CBSU users. Baguley T (2012) Serious Stats. A guide to advanced statistics for the behavioral sciences. Palgrave Macmillan:New York. Cortina JM, Dunlap WP (1997) On the logic and purpose of significance testing. Psychological Methods 2(2) 161-172. Gelman A and Shalizi CR (2013) Philosophy and the practice of Bayesian statistics British Journal of Mathematical and Statistical Psychology 66(1) 8–38. Primer article free to access on-line to CBSU users. Gigerenzer G (2002) Reckoning with risk: learning to live with uncertainty. London: Penguin. Krushchk JK (2011) Doing bayesian data analysis: a tutorial using R and BUGS. Academic Press:Elsevier. For further reading: genuinely accessible to beginners illustrating using prior and posterior probabilities in inference for ANOVAs and other regression models. Wininger M and Johnson R (2018) Prosthetic hand signals:how Bayesian inference can decode movement intentions and control the next generation of powered prostheses. Significance 15(4) 30-35.