Bayesian data analysis for animal scientists

Agustín Blasco Bayesian Data Analysis for Animal Scientists The Basics Bayesian Data Analysis for Animal Scientists Agustı´n Blasco Bayesian Data Analysis for Animal Scientists The Basics Agustı´n Blasco Institute of Animal Science and Technology Universitat Polite`cnica de Vale`ncia Vale`ncia Spain ISBN 978-3-319-54273-7 ISBN 978-3-319-54274-4 DOI 10.1007/978-3-319-54274-4 (eBook) Library of Congress Control Number: 2017945825 # Springer International Publishing AG 2017 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Daniel and Daniel, from whom I have learnt so much in these matters Preface What we now call ‘Bayesian analysis’ was the standard approximation to statistics in the nineteenth century and the first quarter of the twentieth century For example, Gauss’s first deduction of the least squares method was made by using Bayesian procedures A strong reaction against Bayesian methods, led by Ronald Fisher, Jerzy Neyman and Egon Pearson, took place in the 1920s and 1930s with the result that Bayesian statistics was replaced by the current ‘frequentist’ methods of Pvalues, significance tests, confidence intervals and likelihood estimation The reasons were in part philosophical and in part practical; often Bayesian procedures needed to solve complex multidimensional integrals The philosophical issues were mainly related to the use of prior information inherent to Bayesian analysis; it seemed that for most cases, informative priors should be subjective, and moreover, it was not possible to represent ignorance When the computers’ era arrived, the practical problems had a solution using techniques known as ‘Markov Chain Monte Carlo’ (MCMC), and a new interest in Bayesian procedures took place Subjective informative priors were almost abandoned, and ‘objective’ priors with little information were used instead, expecting that having enough data, the priors will not affect the results in practice It was soon discovered that some difficult problems in biological sciences, particularly in genetics, could be approached much easily using Bayesian techniques, and some problems that had no solution using frequentist statistics could be solved with this new tool The geneticists Daniel Gianola and Daniel Sorensen brought in the 1990s the MCMC procedures to solve animal breeding problems using Bayesian statistics, and its use started being more and more common Nowadays, Bayesian techniques are common in animal breeding, and there is no reason why they should not be applied in other fields of animal production The objective of this book is to offer the animal production scientist an introduction to this methodology, offering examples of its advantages for analysing common problems like comparison between treatments, regression or linear mixed models analyses Classical statistical tools are often not well understood by practical scientists Significance tests and P-values are misused so often that in some fields of research it has been suggested to eliminate them I think that Bayesian procedures give much more intuitive results and provide easy tools, helping practical scientists to be more accurate when explaining the results of their research Substituting stars and ‘n.s.’ by probabilities of obtaining a relevant difference vii viii Preface between treatments helps not only in understanding results but in making further decisions Moreover, some difficult problems have a straightforward Bayesian procedure to be solved that does not need new conceptual tools but the use of the principles learnt for simpler problems This book is based on the lecture notes of an introductory Bayesian course I gave in Scotland (Edinburgh, Scottish Agricultural College), France (Toulouse, Institut National de la Recherche Agronomique), Spain (Valencia, Universitat Polite`cnica de Vale`ncia), the USA (Madison, University of Wisconsin), Uruguay (Montevideo, Universidad Nacional), Brazil (Botucatu, Universidade Estadual Paulista; Lavras, Universidade Federal de Lavras) and Italy (Padova, Universita` di Padova) The book analyses common cases that can be found in animal production using practical examples and deriving demonstrations to appendices to facilitate reading the chapters I have preferred to be more intuitive than rigorous, to help the reader in his first contact with Bayesian inference The data are almost always considered to be Normally distributed conditioned to the unknowns, because Normal distribution is the most common in animal production and all derivations are similar using other distributions The book is structured into 10 chapters The first one reviews the classical statistics concepts stressing the common misinterpretations; quite a few practical scientists will discover that the techniques they were applying not mean what they thought Chapter offers the Bayesian possibilities for common analyses like comparisons between treatments; here, the animal production scientists will see new ways of presenting the results: probability of a difference between treatments being relevant, minimum guaranteed value with a chosen probability, when can it be said ‘there is no difference between treatments’ and when ‘we not know whether there is a difference or not’, etc Chapter gives elementary notions about distribution functions Chapter introduces MCMC procedures intuitively Chapters 5, and analyse the linear model from its simplest form (only the mean and the error) to the complex multitrait mixed models Chapter gives some examples of complex problems that have no straightforward solution under classical statistics Chapter deals with all the problems related to prior information, the main object of criticism of Bayesian statistics in the past Chapter 10 deals with the complex problem of model selection from both frequentist and Bayesian points of view Although this is a book for practical scientists, not necessarily interested in the philosophy of Bayesian inference, I thought it would be possible to communicate the philosophical problems that made frequentism and Bayesianism two irreconcilable options in the view of classical statisticians like Fisher, Jeffreys, Pearson (father and son), Neyman, Lindley and many others Following a classical tradition in Philosophy, I wrote three dialogues between a frequentist statistician and a Bayesian one, in which they discuss about the problem of using probability as a degree of belief, the limitations of the classical probability theory, the Bayesian solution and the difficult problem of induction, i.e the reasons why we think our inferences are correct I took Hylas and Philonous, the characters invented by Bishop Berkeley in the eighteenth century for his dialogues, assigning the frequentist role to Hylas and the Bayesian to Philonous, and I also placed the Preface ix dialogues in the eighteenth century, when Bayes published his famous theorem, to make them entertaining and give some sense of humour to a book dedicated to such a rather arid matter as statistics I have to acknowledge many people who have contributed to this book First of all, the colleagues who invited me to give the introductory Bayesian course on which this book is based; the experience teaching this matter in several countries and in several languages has been invaluable for trying to be clear and concise in the explanations of all the statistical concepts and procedures contained in the book I am grateful to Jose´ Miguel Bernardo for his suggestions about how a practical scientist can use the Bayesian credibility intervals for most common problems, developed in Chap 2, and to Luis Varona for the intuitive interpretation of how Gibbs sampling works I should also be grateful to all the colleagues who have read the manuscript, corrected my faults and given advice; I am especially grateful to Manolo Baselga for his detailed reading line by line and also to the colleagues who reviewed all or part of the book, Miguel Toro, Juan Manuel Serradilla, Quim Casellas, Luis Varona, Noelia Iba´~nez and Andre´s Legarra, although all the errors that could be found in the book are my entire responsibility Finally, I am grateful to my son Alejandro, educated in British universities, for his thorough English revision of this book and of course to Concha for her patience and support when I was engaged dedicating so many weekends to writing the book Vale`ncia, Spain Agustı´n Blasco Notation Colour To help in identifying the functions used in this book, when necessary, we have used red colour for the variables and black for the constants or given parameters Thus, À f yjμ; " # y ị2 ẳ pffiffiffiffiffiffiffiffiffiffi exp À 2σ 2πσ is the probability density function of a Normal distribution, because the variable is y, but the conditional À f yjμ; σ Á " # y ị2 ẳ p exp 2σ 2πσ is not a Normal but an Inverted Gamma distribution because the variable is σ This also helps in distinguishing a likelihood f(y| θ) from a probability density function of the data f(y| θ) Probabilities and Probability Distributions The letter ‘P’ is used for probabilities; for example, P(a x b) means the probability of x being between a and b The letter ‘f ’ is used to represent probability density functions; e.g f(x) can be the probability density function of a Normal distribution or a binomial distribution It is equivalent to the common use of the letter ‘p’ for probability density functions, but using ‘f ’ stresses that it is a function We will use the word Normal (with first capital letter) for the Gaussian distribution, to avoid the confusion with the common word ‘normal’ xi xii Notation Scalars, Vectors and Matrixes Bold small letters are column vectors, e.g y ¼ 5 0 y is a transposed (row) vector, e.g y ¼ [1 5] ! À1 Bold capital letters are matrixes, e.g A ¼ Symmetric matrixes are usually written only placing the upper triangle, e.g G¼4 σ 2a σ ac σ 2c σ ak σ ck σ 2k Lower case letters are scalars, e.g y1 ¼ Greek letters are parameters, e.g σ is a variance Proportionality The sign / means ‘proportional to’ For example, if c and k are constants, the distribution N(0,1) is  2  2  2   y y y y2 / exp À / k Á expðcÞ Á exp À / k Á exp c f yị ẳ p exp 2 2 2π Notice that we cannot add constants or multiply the exponent by a constant if f ðyÞ / expðy2 Þ, then f ðyÞ not proportional to k þ expðy2 Þ f ðyÞ not proportional to expðc Á y2 Þ Appendix: The Bayesian Perspective—Three New Dialogues Between Hylas 261 HYLAS—You are right, but for me to fully enjoy the pleasures that you are suggesting, the premise that my understanding is pacified must be an accurate one PHILONOUS—And is it not so? Or you feel new doubts with respect to all the things that we have so much discussed? Do you for any reason think you can be an impartial judge of what you experience? HYLAS—Have no fear, Philonous, for all of that is clear to me What now worries me is the way in which we have resolved the problem of induction We sustain that we can know the probability of an event, or that a variable takes on a value, given on one hand the facts obtained in the experiment and on the other the probability derived from our previous experience PHILONOUS—That is right, and even though this probability is subjective, when the number of facts is large enough, your previous opinion must be irrelevant HYLAS—But there is still something that doesn’t quite fit in the process What would happen if I had obtained the sample incorrectly? What if the facts did not reflect the state of the population correctly? What would happen if they were not distributed in the way I presumed? What would happen if my previous opinion, shared by many other scientists, were to differ so greatly from the truth because of my prejudgements, so common in men, that not even with a reasonably high number of facts could I offset the mistaken opinion given by the experts? PHILONOUS—Look, Hylas, you are letting yourself slide down the hill that leads to scepticism, and from there to solipsism and later to melancholy there is a short stretch HYLAS—Should I, then fool myself and say that I understand what I not, and admit that I find reasonable what seems to me has faulty foundations? PHILONOUS—No, no you should not, but neither should you ignore the limits of human understanding Probabilities a priori not exist alone, floating on the waves created by seas of opinions All probability is based on certain hypotheses, and so is this previous probability In reality we should call it ‘probability a priori given a series of hypotheses’, for only with regard to them is this probability feasible HYLAS—But then the result of our inference is not a probability ‘given experimental evidence’, but a probability ‘given experimental evidence and a group of attached hypotheses’ PHILONOUS—That is the way I understand it HYLAS—And how you then say you have resolved the problem of induction, if you have left it as intact as it was before starting your lucubration? It has maintained its virginity if I am correct! How can you assert that Science already has a firm base if it all depends on the truth of previous hypotheses, and this is a truth that you are unaware of? How can you justify that your knowledge progresses if its foundation is as fragile as the clay of the idol’s feet? Or is it that you have some means of estimating the certainty of the hypotheses that accompany all your reasoning? PHILONOUS—Calm down, Hylas, for your own health comes before all else in the world I not know how certain those hypotheses may be However, if I suspected that the sample was not taken properly or that the worker who weighed 262 Appendix: The Bayesian Perspective—Three New Dialogues Between Hylas the pigs mixed up the facts, or did it badly out of spite because he had an unsettled debt with me, then in such cases, I wouldn’t give credit to my results If I did, it would be because I was convinced that everything was done in an honest and reasonable way And science progresses that way, because I can choose between rival theories even though they are conditioned by the veracity of those hypotheses, for in the end the experimental evidence will make me lean towards one theory or the other HYLAS—They seem to me like poor means for reaching such high objectives If science should progress on the basis of good faith, of the absence of unavoidable error, of the correct interpretation of Nature, and of all that your hypotheses contain, I wouldn’t tell you that I expect quick and constant progress, but rather ups and downs, false starts and errors that should remain unsolved because we trust in the fact that we did things correctly PHILONOUS—Having a way of discerning which theory we prefer is not a poor result, although it is true that science is always conditioned by the veracity of those hypotheses In reality the sword of Damocles is always present, providing us with mistaken hypotheses, but once we admit that they are the most reasonable we can find, we can activate our decision mechanism to make our estimations about all we find in Nature more precise, say how probable it is for one character to have this or that value, and prefer this theory to that one based on how much more probable this one is to its alternatives HYLAS—In how much more probable you think it is, you mean PHILONOUS—Of course Hylas, we have already agreed that probability is nothing other than my state of beliefs HYLAS—That’s it, but I have the impression that it is so much our custom to believe that probability is something objective and that it is outside us that upon talking about how probable this or that hypothesis is, we will end up believing that we are truly talking about objective probability and not about the state of our opinion PHILONOUS—And what you suggest? That scientists send their declarations to the Royal Society using the phrase ‘my probability’? It would be annoying and it would emphasise the author’s ego, something that a man of science should always avoid HYLAS—Yes, but it wouldn’t give the impression of a detached objectivity PHILONOUS—Nor does it intend to Insisting upon subjectivity would also give the impression of something arbitrary being done, and we have already agreed that subjectivity does not in any way imply arbitrariness if the scientist isn’t mad And this without taking into account that most of the time the facts dominate that subjective previous opinion that so disgusts you HYLAS—I admit that science can progress, although I doubt that progress can be so rapid and constant as the application of your principle seems to suggest, but we have left the induction principle untouched In reality no experimental evidence can guarantee us proximity to the truth, for we depend on other hypotheses which we not know to be true Appendix: The Bayesian Perspective—Three New Dialogues Between Hylas 263 PHILONOUS—But you won’t deny that we have established a reasonable behaviour that can be followed by all those men that are interested in learning about the laws of Nature HYLAS—A behaviour that will produce good results because God won’t allow it to be any other way, but not because it intrinsically guarantees an approximation to the truth PHILONOUS—Hylas, you are adducing a comparison that could be quite fortunate Ethics without God are based on the behaviour that is most beneficial for the individual and the community In the same way, scientific progress without God would be based on the premise that a scientist’s behaviour will give results which we consider, under certain hypotheses, to be more probable and better adjusted to what we already know In both cases the individual’s behaviour is determined on uncertain, but experimental, bases HYLAS—I not understand the comparison, Philonous, and I think it is completely forced With regard to ethics, I don’t know how you fuse the individual’s and society’s benefit, when they so often come into conflict; and with regard to science, I can’t see what relation there is between the behaviour of the scientist and what it is I am asking you: how can you guarantee that experimental evidence will lead you nearer to the truth? PHILONOUS—Hylas, If you are so extreme then I shall say no, no I cannot assure that experimental evidence will lead me nearer to the truth; nor can I assure that you are here now and that I am not speaking to a ghost or I am immersed in a dream without knowing it I not even know if I can talk appropriately like myself, since I only have experimental evidence about myself! HYLAS—I beg you not to feel offended, Philonous, my questions arise from my perplexity and not from my desire to ridicule your answers I wanted certainty and I exchanged it with you for probability, and now I don’t even have the consolation of reaching probable knowledge How happy the monks must be, not having to doubt anything and trusting in God to govern and resolve the course of their lives! Is it that blind faith that definitively guides our belief that we know something? PHILONOUS—Hylas, to begin to discuss something you must at least first assume that your interlocutor is truly present To add hypotheses to our experiments, taking for granted they hadn’t been sabotaged, the samples were taken at random and no neglected risk was introduced, does not seem to me to be paying a high price for the conclusions at which we will arrive And if the result permits you to act upon Nature and benefit from it, if you can build bridges that don’t collapse, if your crops yield more, if you can cure the ill, all of this indicates that you cannot be too mistaken HYLAS—Yes, but ordinary practice does not require speculative knowledge You can believe that you are ill because bad humours take over your blood, and that you shall recover by taking certain herbs; and if you take the right ones, health shall return to your body, even though the causes of your illness might not have been the ones you suspected Using theories with scarce foundation because they work well is like making children believe that the bogeyman exists just so they will eat their soup An engineer might not need anything more than rules he can keep in his 264 Appendix: The Bayesian Perspective—Three New Dialogues Between Hylas memory, but a scientist is obliged to think in a different way and ask himself the reasons for things he infers and for the decisions he takes PHILONOUS—I not know if this world is made for you, Hylas, but I think we have good reasons for behaving the way we and believing in all that we believe HYLAS—And what is the nature of those good reasons? 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(2011) Analysis of a genetically structured variance heterogeneity model using the Box-Cox transformation Genet Res 93:33–46 Yates F (1990) Preface of statistical methods, experimental design, and scientific inference by R.A Fisher Oxford University Press, Oxford Zome~no C, Hernandez P, Blasco A (2013) Divergent selection for intramuscular fat content in rabbits I Direct response to selection J Anim Sci 91:4526–4531 Index A Acceptance-rejection method, 98–100 Akaike Information Criterion (AIC), 233–237 Animal breeding programs, 155 B Baby model credibility intervals, 108 inverted gamma distribution, 105 joint posterior distributions (see Joint posterior distributions) mean, 107–108 median, 108 mode, 106, 117–118 student t-distribution, 104 vague informative priors, 112–114 Bayes A, 185–187, 189 Bayes B, 187–189 Bayes C, 188, 189 Bayes Cπ, 188, 189 Bayes factors (BF), 239 model selection, 224–226 test of hypothesis, 52–53, 56–57 Bayes theorem Bayesian inference, 34–36 conditional distributions, 72–73 posterior distribution, 195–196 prior probability, 223–224 Bayesian inference, 1, advantages, 36, 54, 61 Bayes theorem, 34–36 credibility intervals guaranteed value, 46, 60 highest posterior density interval, 44–45, 60 interval estimation, 55 marginalisation, 49–51 positive/negative probability, 45, 46 for ratio of levels, treatment, 48–49 relevance probability, 47 significant differences, 55 similitude probability, 47–48, 60 frequentist properties, estimator, 55 Hylas and Philonous beliefs, 247–264 marginal posterior distributions, 57–61 mean, 42, 43, 62 median, 42, 44, 62–63 mode, 42–44, 63 prior information, 209–210 assumption, 203 Bernardo reference prior, 206–207 definition, 36 exact information (see Exact prior information) flat priors, 40, 204–205 improper priors, 207 Jeffreys prior, 205–206 Mendel’s law of inheritance, 37 objective/noninformative priors, 40 principle of indifference, 40 standard errors, 37 state of beliefs, 39 statistics, 208 vague prior information (see Vague prior information) probability density, 40–42 random variable, 54 test of hypothesis Bayes factors, 52–53, 56–57 model averaging, 53 model choice, 51–52 nested model, 52 with/without sex effect, 56 Bayesian Information Criterion (BIC), 239–241 # Springer International Publishing AG 2017 A Blasco, Bayesian Data Analysis for Animal Scientists, DOI 10.1007/978-3-319-54274-4 271 272 Bayesian least absolute shrinkage and selection operator (Bayesian LASSO), 188–190 Bayesian statistics, 2, Belgian Landrace cross, 58–59 Bernardo reference prior, 206–207 Best linear unbiased predictor (BLUP) animal breeding programs, 155 animal breeding values, 21 in Bayesian context, 156–158 for genetic mixed model, 154 mixed model equations, 154–156 RR-BLUP, 183–185, 190–191 Binomial distribution, 232–233 BLUP See Best linear unbiased predictor (BLUP) C Canalising selection, residuals modelling fixed effects, 175 genetic effects, 176 heteroscedastic model, 175 marginal posterior distributions, 176–177 variance, 177 Central limit theorem, 28, 36, 171, 214–215 Conditional distributions Bayes theorem, 72–73 fixed effects model, 128, 131–135 genetic animal model, 152–154, 164 Gibbs sampling process, 91–92 of mean, 75–76 mixed model, repeated records, 143–144 of sample, 73 of variance, 73–75 Confidence intervals (CI), 13–15 Conjugate density distributions, 113 Credibility intervals Bayesian inference guaranteed value, 46 highest posterior density interval, 44–45, 60 marginalisation, 49–51 positive/negative probability, 45, 46 for ratio of levels, treatment, 48–49 relevance probability, 47 similitude probability, 47–48 posterior distributions, 72 Crossed entropy, 233 Cumulative distribution function, 80–81 D Data augmentation, 160–163, 174 Density of heritability, 70 Index Deviance Information Criterion (DIC), 237–239 Duroc cross, 58–59 E Error of estimation, 16 Exact prior information anti-Bayesian, 193 black and brown mouse, 194 posterior probability flat priors, 197–198 maximum likelihood method, 197 sample of, 195–196 uncertainty, 196 F Fixed effects model bias, 20 conditional distributions, 128, 131–135 confidence interval, 126 conjugated prior, 123 continuous effect, 126 covariate, 125–126 distribution of effects, 18, 19 effect of parity, 120 improper priors, 121 intramuscular fat measurement, 124 inverted gamma distribution, 123 least square estimator, 132 marginal posterior distributions flat priors, 121, 122, 127–130 inferences, 123 vague informative priors, 122, 130–132 multivariate beliefs, 122 probability of relevance, 124 probability of similitude, 124 regression coefficient, 120 residual variance, 126 sampling effect, 126 seasonal effect, 125 treatment effect, 119 true value, 20 unbiased estimators, 20 variance, 20 G Genetic animal model conjugated/flat prior, 148 covariance, 146 genetic effect, 145–149 marginal posterior distributions, 149–154 permanent effect, 145 Index prior information, 148 shrinkage effect, 145 variance, 148 Genetic value, 28 Genome wide association studies (GWAS), 182–183 Genomic selection Bayes A, 185–187, 189 Bayes B, 187–189 Bayes C, 188, 189 Bayes Cπ, 188 Bayes L, 188–190 flat prior, 181 genetic value, 180 GWAS, 182–183 linkage disequilibrium, 180 prior information, 181, 182 RR-BLUP, 183–185, 190–191 SNP, 178–180, 190–191 variance–covariance matrix, 181 Gibbs sampling process, Baby model, 110–112, 114–115 conditional distributions, 91–92 convergence, 95 equal probability density, 93–94 features, 95–98 fixed effects model, 129–130 mixed model, repeated records, 144 sampling error, 94 univariate distributions, 91 Gompertz curve, 168, 170 Growth curves, nested models data augmentation, 174 fitting residuals, 169, 170, 173 Gompertz curve, 168, 170 marginal posterior distributions, 171–173 relationship matrix, 174 weekly weights data, 169 GWAS See Genome wide association studies (GWAS) H Highest posterior density interval, 44–45, 60 I Inverse probability, 2, 208 Inverted gamma distribution, 73–75, 83–84, 105, 115 J Jeffreys prior, 205–206 Joint posterior distributions 273 Baby model flat priors, 109–112 of mean and variance, 105 mode, 105–106, 115–116 conditional distributions, 173 fixed effects model, 127 genetic animal model, 150–152 mixed model, repeated records, 142–143 nested model, growth curves, 171–172 K Kronecker product, 159 Kullback–Leibler distance, 232–233 Kullback–Leibler divergence, 232–233 L Likelihood, definition, 22–24 Linkage disequilibrium phenomenon, 180 M Marginal posterior distributions Baby model credibility intervals, 108 inverted gamma distribution, 105 joint posterior distributions (see Joint posterior distributions) mean, 107–108 median, 108 mode, 106, 117–118 student t-distribution, 104 vague informative priors, 112–114 Bayesian inference, 57–61 definition, 76–77 fixed effects model flat priors, 121, 122, 127–130 inferences, 123 vague informative priors, 122, 130–132 genetic animal model, 149–154 histogram, 86, 87 inferences guaranteed value, 89 probability counting, 88 relevant difference, 88–89 shortest interval, 89 MCMC methods acceptance-rejection method, 98–100 Gibbs sampling, 91–98 Metropolis–Hastings method, 100–101 software, 101–102 mixed model, repeated records conditional distributions, 143–144 Gibbs sampling, 144 274 Marginal posterior distributions (cont.) joint posterior distribution, 142–143 nested models, growth curves, 171–173 Normal distribution of mean, 78–80 of median, 90 random samples, 86 sampling error, 87 of variance, 77–78 Marginalisation, 49–51 Markov chain Monte Carlo (MCMC) methods acceptance-rejection method, 98–100 Gibbs sampling (see Gibbs sampling process) marginal posterior distributions using flat priors, 121, 122, 127–130 vague informative priors, 122, 130–132 Metropolis–Hastings method, 100–101 mixed model, 139 prior information, 207, 208 software, 101–102 Maximum likelihood (ML) estimator, 24–26, 29–30 MCMC methods See Markov chain Monte Carlo (MCMC) methods MCSE See Monte Carlo standard error (MCSE) Mendel’s law of inheritance, 37 Metropolis–Hastings method, 100–101 Mixed model, repeated records fixed effect, 141 flat priors, 138, 140 marginal posterior distributions conditional distributions, 143–144 Gibbs sampling, 144 joint posterior distribution, 142–143 MCMC methods, 139 parity effect, 138 prior information, 139, 140 random effect, 138, 141 season effect, 138, 141 variance, 140 ML estimator See Maximum likelihood (ML) estimator Model selection AIC, 233–237 BIC, 239–241 DIC, 237–239 differentiation, 242–243 fitting data vs new records prediction, 217– 218 hypothesis tests Bayes factors, 224–226 Index frequentist tests, 221 likelihood ratio test, 221–223 prior probability, 223–224 likelihood curves entropy, 231–232 Fisher’s information, 227–230 Kullback–Leibler discrimination information, 232–233 maximum estimators, 226 Shannon information, 231 mean deviance, 244–245 model comparison, 241–242 Normal distribution animal production, 214–215 coefficient of determination, 220 cross-validation, 220–221 dead piglets, 218–219 food conversion, 214 growth rate, 214–215 noise effects, 215–216 non-Normal, 218–219 non-significant effects, 219–220 parameters, 213–214 parsimony, 218, 220 probability density function, 214, 245–246 year-season effects, 216–217 Taylor’s expansion, 243 Monte Carlo standard error (MCSE), 109 Multitrait model data augmentation, 160–163 flat priors, 159 permanent effect, 160 vague priors, 159 N Nested models, growth curves data augmentation, 174 fitting residuals, 169, 170, 173 Gompertz curve, 168, 170 marginal posterior distributions, 171–173 relationship matrix, 174 weekly weights data, 169 Normal distribution, 103 animal production, 214–215 coefficient of determination, 220 cross-validation, 220–221 dead piglets, 218–219 food conversion, 214 growth rate, 214–215 noise effects, 215–216 non-Normal, 218–219 Index non-significant effects, 219–220 parameters, 213–214 parsimony, 218, 220 probability density function, 214, 245–246 vague prior information, 200 year-season effects, 216–217 Null hypothesis, 3–5 P Popper theory of refutation, Posterior distribution Bayes theorem, 195–196 conditional distributions, 72–76 credibility intervals, 72 cumulative distribution function, 80–81 expectation/mean, 71 flat priors, 197–198 marginal distributions, 76–80 maximum likelihood method, 197 median, 71 mode, 72 notation, 67–68 probability density function, 68–71 uncertainty, 196 Probability density function auxiliary function, 40 bivariate distribution, 70 definition, 68 inference, 41–42 for liver flavour intensity of meat, 69 mode selection, 214, 245–246 multivariate case, 70 R Random effects model, 18–21 Relevance probability, 47 Relevant difference, 26–27 Repeated records, mixed model fixed effect, 141 flat priors, 138, 140 marginal posterior distributions conditional distributions, 143–144 Gibbs sampling, 144 joint posterior distribution, 142–143 MCMC methods, 139 parity effect, 138 prior information, 139, 140 random effect, 138, 141 season effect, 138, 141 variance, 140 275 Residual/restricted maximum likelihood estimator (REML), 17–18, 158 Ridge regression/random regression BLUP (RR-BLUP), 183–185, 190–191 S Similitude probability, 47–48 Single Nucleotide Polymorphism (SNP), 178, 179, 190–191 Standard error, 13 Student t-distribution, 104 T Test of hypothesis error level, 7, in experimental research, 12 model selection Bayes factors, 224–226 frequentist tests, 221 likelihood ratio test, 221–223 prior probability, 223–224 non-significant difference, 9, 11–12 null hypothesis, 4–5 P-value, 6–9 relevant value, 10, 12 scientific behaviour, significant difference, 9, 10, 12 Type I error, 6, Type II error, U Unbiased estimators, 16–18 Univariate distributions, 91 V Vague prior information Baby model, 112–114 Bayesian paradigm, 203 conjugate distribution, 200 definition, 198–199 fixed effects model, 129–130 inverted Wishart distributions, 202–203 MCMC methods, 122, 130–132 Normal distribution, 200 ovulation rate, pigs, 200–201 uterine capacity, rabbits, 201–202 Variance of error, 20–21 .. .Bayesian Data Analysis for Animal Scientists Agustı´n Blasco Bayesian Data Analysis for Animal Scientists The Basics Agustı´n Blasco Institute of Animal Science and Technology... 2017 A Blasco, Bayesian Data Analysis for Animal Scientists, DOI 10.1007/978-3-319-54274-4_2 33 34 The Bayesian Choice founders of what we now call ‘classical statistics’ (see, for example, K... now Bayesian techniques were commonly used along the nineteenth and the first few decades of the twentieth # Springer International Publishing AG 2017 A Blasco, Bayesian Data Analysis for Animal