The 2nd edition is completely rewritten from cover to cover, with all new programs too!
Here is a partial list of what's new:

There are all new programs in JAGS and Stan. The new programs are designed to be much easier to use than the scripts in the first edition. In particular, there are now compact highlevel scripts that make it easy to run the programs on your own data sets. This new programming was a major undertaking by itself.

The introductory Chapter 2, regarding the basic ideas of how Bayesian inference reallocates credibility across possibilities, is completely rewritten and greatly expanded.

There are completely new chapters on the programming languages R (Ch. 3), JAGS (Ch. 8), and Stan (Ch. 14). The lengthy new chapter on R includes explanations of data files and structures such as lists and data frames, along with several utility functions. (It also has a new poem that I am particularly pleased with.) The new chapter on JAGS includes explanation of the RunJAGS package which executes JAGS on parallel computer cores. The new chapter on Stan provides a novel explanation of the concepts of Hamiltonian Monte Carlo. The chapter on Stan also explains conceptual differences in program flow between it and JAGS.

Chapter 5 on Bayes’ rule is greatly revised, with a new emphasis on how Bayes’ rule reallocates credibility across parameter values from prior to posterior. The material on model comparison has been removed from all the early chapters and integrated into a compact presentation in Chapter 10.

What were two separate chapters on the Metropolis algorithm and Gibbs sampling have been consolidated into a single chapter on MCMC methods (as Chapter 7).

There is extensive new material on MCMC convergence diagnostics in Chapters 7 and 8. There are explanations of autocorrelation and effective sample size. There is also exploration of the stability of the estimates of the HDI limits. New computer programs display the diagnostics, as well.

Chapter 9 on hierarchical models includes extensive new and unique material on the crucial concept of shrinkage, along with new examples.

All the material on model comparison, which was spread across various chapters in the first edition, in now consolidated into a single focused chapter (Ch. 10) that emphasizes its conceptualization as a case of hierarchical modeling.

Chapter 11 on null hypothesis significance testing is extensively revised. It has new material for introducing the concept of sampling distribution. It has new illustrations of sampling distributions for various stopping rules, and for multiple tests.

Chapter 12, regarding Bayesian approaches to null value assessment, has new material about the region of practical equivalence (ROPE), new examples of accepting the null value by Bayes factors, and new explanation of the Bayes factor in terms of the SavageDickey method.

Chapter 13, regarding statistical power and sample size, has an extensive new section on sequential testing, and making the research goal be precision of estimation instead of rejecting or accepting a particular value.

Chapter 15, which introduces the generalized linear model, is fully revised, with more complete tables showing combinations of predicted and predictor variable types.

Chapter 16, regarding estimation of means, now includes extensive discussion of comparing two groups, along with explicit estimates of effect size.

Chapter 17, regarding regression on a single metric predictor, now includes extensive examples of robust regression in JAGS and Stan. New examples of hierarchical regression, including quadratic trend, graphically illustrate shrinkage in estimates of individual slopes and curvatures. The use of weighted data is also illustrated.

Chapter 18, on multiple linear regression, includes a new section on Bayesian variable selection, in which various candidate predictors are probabilistically included in the regression model.

Chapter 19, on onefactor ANOVAlike analysis, has all new examples, including a completely worked out example analogous to analysis of covariance (ANCOVA), and a new example involving heterogeneous variances.

Chapter 20, on multifactor ANOVAlike analysis, has all new examples, including a completely worked out example of a splitplot design that involves a combination of a withinsubjects factor and a betweensubjects factor.

Chapter 21, on logistic regression, is expanded to include examples of robust logistic regression, and examples with nominal predictors.

There is a completely new chapter (Ch. 22) on multinomial logistic regression. This chapter fills in a case of the generalized linear model (namely, a nominal predicted variable) that was missing from the first edition.

Chapter 23, regarding ordinal data, is greatly expanded. New examples illustrate singlegroup and twogroup analyses, and demonstrate how interpretations differ from treating ordinal data as if they were metric.

There is a new section (25.4) that explains how to model censored data in JAGS.

Many exercises are new or revised.
And the cover is different! The correspondence of the doggies to Bayes’ rule is now made explicit: The halfup ears of the posterior doggie are a compromise between the perky ears and floppy ears of the likelihood and prior doggies. The marginal likelihood is not usually computed in MCMC methods, so the doggie in the denominator gets sleepy with nothing much to do. Let's hope that what’s between the covers of the book is as friendly and engaging as the doggies on the cover.
The first review on Amazon.com:
The author does not know these reviewers, but extends his thanks to them for taking the time to post reviews. And, of course, he's pleased that the reviews are so positive!

