The collected data are y1 , The data are shown in Table 5. Numbers of subjects recalling one stressful event. Approximate these moments by a normal approximation about the posterior mode function laplace. Compare your estimates with the estimates using the normal approximation.
Compare the estimates with the normal approximation estimates found in part a. Mixture of exponential data Suppose a company obtains boxes of electronic parts from a particular supplier. Suppose y1 , The following function log. Batch Batch Yield in grams 1 2 3 4 5 6 Let yij denote the jth observation in batch i. To determine the relative importance of between-batch variation versus sampling variation, the fol- lowing multilevel model is applied N denotes the number of batches and n denotes the number of observations per batch.
The input y in the function is a matrix with N rows and n columns, where a row contains the measurements for a particular batch. Markov chains are introduced in the dis- crete state space situation in Section 6. Through a simple random walk example, we illustrate some of the important properties of a special Markov chain, and we use R to simulate from the chain and move toward the sta- tionary distribution.
In Section 6. We describe one strategy for sum- marizing a posterior distribution and illustrate it for three problems. MCMC algorithms are very attractive in that they are easy to set up and program and require relatively little prior input from the user. R is a convenient language for programming these algorithms and is also very suitable for performing out- put analysis, where one does several graphical and numerical computations to check if the algorithm is indeed producing draws from the target posterior distribution.
If the person is currently at an interior value 2, 3, 4, or 5 , in the next second she is equally likely to remain at that number or move to an adjacent J. If she does move, she is equally likely to move left or right. If the person is currently at one of the end values 1 or 6 , in the next second she is equally likely to stay still or move to the adjacent location. This is a simple example of a discrete Markov chain. A Markov chain de- scribes probabilistic movement between a number of states.
Here there are six possible states, 1 through 6, corresponding to the possible locations of the walker. The probability that she moves to another location depends only on her current location and not on previous locations visited. We describe movement between states in terms of transition proba- bilities — they describe the likelihoods of moving between all possible states in a single step in a Markov chain.
There are several important properties of this particular Markov chain. It is possible to go from every state to every state in one or more steps — a Markov chain with this property is said to be irreducible. Given that the person is in a particular state, if the person can only return to this state at regular intervals, then the Markov chain is said to be periodic.
This example is aperiodic since it is not a periodic Markov chain. Then w is said to be the stationary distribution. If a Markov chain is irreducible and aperiodic, then it has a unique stationary distribution. We can empirically demonstrate the existence of the stationary distribu- tion of our Markov chain by running a simulation experiment.
The relative fre- quencies of our traveler in the six locations after many steps will eventually approach the stationary distribution w. We start our simulation in R by reading in the transition matrix P and setting up a storage vector s for the locations of our traveler in the random walk. We convert the counts to relative frequencies by dividing by the number of steps.
This essentially is a continuous- valued generalization of the discrete Markov chain setup described in the pre- vious section. The MCMC sampling strategy sets up an irreducible, aperiodic Markov chain for which the stationary distribution equals the posterior distri- bution of interest.
A general way of constructing a Markov chain is by using a Metropolis-Hastings algorithm. In this section, we focus on two particular variants of Metropolis-Hastings algorithms, the independence chain and the random walk chain, that are applicable to a wide variety of Bayesian inference problems. Desirable features of the proposal density in an algorithm depend on the MCMC algorithm employed. For an independence chain, we desire that the proposal density p approximate the posterior density g, suggesting a high acceptance rate.
This advice also applies when one monitors the Metropolis within Gibbs algorithm described in Section 6. The idea behind Gibbs sampling is that we can set up a Markov chain simulation algorithm from the joint posterior distribution by successfully simulating individual parameters from the set of p conditional distributions.
Simulating one value of each individual parameter from these distributions in turn is called one cycle of Gibbs sampling. In situations where it is not convenient to sample directly from the condi- tional distributions, one can use a Metropolis algorithm such as the random walk type to simulate from each distribution.
The output of gibbs is a list; the component par is a matrix of simulated draws and accept is a vector of acceptance rates for the individual Metropolis steps. In typical practice, one monitors the performance of an MCMC algorithm by inspecting the value of the acceptance rate, constructing graphs, and com- puting diagnostic statistics on the stream of simulated draws. We call this investigation an MCMC output analysis.
One issue in understanding MCMC output is detecting the size of the burn-in period. Trace plots are especially important when MCMC algorithms are initialized with parameter values that are far from the center of the posterior distribution.
A second concern in analyzing output from MCMC algorithms is the de- gree of autocorrelation in the sampled values.
If there is strong correlation between successive values in the chain, then two consecutive val- ues provide only marginally more information about the posterior distribution than a single simulated draw. Also, a strong correlation between successive iterates may prevent the algorithm from exploring the entire region of the parameter space.
A standard graph is to plot the values of the autocorrelation against the lag L. If the chain is mixing ade- quately, the values of the autocorrelation will decrease to zero as the lag value is increased. Another issue that arises in output analysis is the choice of the simulated sample size and the resulting accuracy of calculated posterior summaries.
One simple method of computing standard errors for correlated output is the method of batch means. Following the recommendation of Chapter 11 in Gelman et al. The Metropolis-Hastings random walk and independence chains and the Gibbs sampling algorithm are attractive Markov chains since they are easy to program and require rela- tively little prior input. These initial guesses can be found by non-Bayesian methods such as the method of moments or maximum likelihood.
In our examples, we illustrate the use of the function laplace to locate the posterior density. We can check the accuracy of the normal approxima- tion in the two-parameter case by constructing a contour graph of the joint posterior. These examples show that there can be some errors in the normal approximation. For example, suppose one is interested in learning about the mean and standard deviation of the heights in inches of men from a local college. One is given the summary frequency data shown in Table 6.
One sees that 14 men were shorter than 66 inches, 30 men had heights between 66 and 68 inches, and so on. Table 6. Grouped frequency data for heights of male students at a college. Height Interval in. Frequency less than 66 14 between 66 and 68 30 between 68 and 70 49 between 70 and 72 70 between 72 and 74 33 over 74 15 We are observing multinomial data with unknown bin probabilities p1 , The associated posterior standard deviations of the parameters can be esti- mated by computing the square roots of the diagonal elements of the variance- covariance matrix.
For the proposal density, we use the variance-covariance matrix obtained from laplace and we set the scale parameter equal to 2. We run 10, iterations of the random walk algorithm starting at the value start.
The output fit2 is a list with two components: par is a matrix of simulated values where each row corresponds to a single draw of the parameter vector, and accept gives the acceptance rate of the random walk chain. The last simulated draws from the random walk Metropolis algorithm are drawn on top in Figure 6. A simulated sample of draws of the posterior is also shown. Suppose we rerun the Metropolis ran- dom walk algorithm for the grouped data posterior with poor choices of start- ing value and proposal density.
Figure 6. One can observe the strong correlation structure of the sequences by us- ing autocorrelation plots produced by the autocorr. If we assume naively that this simulated sample represented independent draws, then the standard error of this esti- mate is 0. However, a more accurate estimate of the standard error is the Batch SE given by 0. The lag one autocorrelations are high, but the autocorrelation values dissipate rapidly as a function of the lag.
As before, we can compute summary statistics for this stream of MCMC output. To simplify the code, we use the R function dt, which computes the density of the t random variable.
The Cauchy density is the t density with a single degree of freedom. This dataset can be found in the LearnBayes library with the name darwin. We read in the dataset and attach the data frame so we can access the variable difference. The output also gives the associated variance-covariance matrix and an estimate of the log integral.
We can use these estimates of center and spread to construct a rectangle that covers essentially all of the posterior probability of the parameters. As an initial guess at this rectangle, we take for each parameter the posterior mode plus and minus four standard deviations, where the standard deviations are obtainable from the diagonal elements of the variance-covariance matrix.
The contours of the exact posterior distribution have an interesting shape and one may wonder how these contours compare with those for a bivari- ate normal approximation. Using these values as inputs, we draw contours of a bivariate normal density in Figure 6. Although the normal approximation may not be the best summary of the posterior distribution, the estimated variance-covariance matrix is helpful in setting up a Metropolis random walk chain.
The simulation algorithm is run using the function rwmetrop. One can use a Metropolis-Hastings independence chain, where the proposal density is mul- tivariate normal with mean and variance given by the normal approximation. Alternatively, one can apply a Gibbs sampling algorithm with a vector of scale parameters equal to 12,.
All the simulation algorithms were run with a simulation sample size of 50, The R code for the implementation of the four simulation algorithms follows. In addition, the acceptance rates for the Metropolis- Hastings random walk and independence chains and the Gibbs sampler are shown. For the M patients that have a heart transplant, let yj and zj denote the time to transplant and survival time; m of these patients died during the study.
In the following code, we generally follow the earlier notation. The numbers of nontransplant and transplant patients are denoted by N and M. We divide the data into two groups using the transplant indicator variable t. For the nontransplant patients, the survival times and censoring indicators are denoted by xnt and dnt, and for the transplant patients, the waiting times, survival times, and censoring indicators are denoted by y, z, and dt.
The algorithm converges and we obtain the posterior mode and an estimate at the variance-covariance matrix. Then we 1 set up a grid of values of t and storage vectors p5, p50, and p95; 2 simulate a sample of values of S t for each value of t on the grid; and 3 summarize the posterior sample by computing the 5th, 50th, and 95th percentiles. These percentiles are stored in the variables p5, p50, and p In Figure 6. Posterior distribution of probability of survival S t for heart transplant patients.
Lines correspond to the 5th, 50th, and 95th percentiles of the posterior of S t for each time t. Chapter 11 of Gelman et al. Robert and Casella and Givens and Hoeting give more detailed descriptions of MCMC algorithms within the context of computational statistical methods. Introductory discussions of Metropolis and Gibbs sampling are provided, respectively, in Chib and Green- berg and Casella and George Using the sample command, simulate steps of the Markov chain using the probabilities given in the transition matrix.
Store the locations of the walk in a vector. Guess at the value of the stationary distribution vector w. Use a Metropolis-Hastings random walk algorithm to simulate from the posterior density. In the algorithm, let s be equal to twice the approximate posterior standard deviation found in the normal approximation. Compare your answers with those obtained using the normal approximation in Exercise 1 of Chapter 5.
Use a normal proposal density. Modeling data with Cauchy errors As in Section 6. Con- sider the following hypothetical test scores from a class that is a mixture of good and poor students. Check that you have indeed found the posterior mode by trying several starting values in the Nelder and Mead algorithm. Poisson regression Exercise 4 of Chapter 5 describes an experiment from Haberman involving subjects reporting one stressful event. Flour beetle mortality data. Dosage Number Killed Number Exposed wi yi ni 1.
Use the function rwmetrop to run this chain for 10, iterations. Compute the acceptance rate and the 5th and 95th percentiles for each parameter. Use this estimate in the proposal den- sity of rwmetrop and run the chain for 10, iterations.
Mixture of exponentials model In Exercise 6 of Chapter 5, a mixture of exponential densities was used to model the lifetimes of electronic parts sampled from a mixture of accept- able and unacceptable. In that exercise, a function log. Inference about the Box-Cox transformation Suppose one observes the positive values y1 , We begin by giving a brief introduction to hierarchical modeling. Then we consider the simultaneous estimation of the true mortal- ity rates from heart transplants for a large number of hospitals.
Some of the individual estimated mortality rates are based on limited data, and it may be desirable to combine the individual rates in some way to obtain more accurate estimates. We describe a two-stage model, a mixture of gamma distributions, to represent prior beliefs that the true mortality rates are exchangeable. We describe the use of R to simulate from the posterior distribution. Once we simulate hyperparameters, we can simulate from the posterior distributions of the true mortality rates from gamma distribu- tions.
To illustrate, consider the following three problems described in this chapter and the chapters to follow. Simultaneous estimation of hospital mortality rates In the main example of this chapter, one is interested in learning about the mortality rates due to heart transplant surgery for 94 hospitals.
It is reasonable to believe a priori that the true rates are similar in size, which implies a dependence structure J. One wishes to learn about the collection of population mean GPAs, with the ultimate goal of making predictions about the success of fu- ture students that attend the university. One believes that the population GPAs can be represented as a simple linear function of the high school rank and standardized test score.
Estimating career trajectories In an example in Chapter 11, one is learning about the pattern of perfor- mance of athletes as they age during their sports careers. In particular, one wishes to estimate the career trajectories of the batting performances of a number of baseball players.
Note that the shapes of these trajectories are similar; a player generally will increase in performance until his late 20s or early 30s and then decline until retirement. The prior belief is that the true trajectories will be similar between players, which again implies a prior distribution with dependence. In many-parameter situations such as the ones described here, it is natural to construct a prior distribution in a hierarchical fashion. In this type of model, the observations are given distributions conditional on parameters, and the parameters in turn have distributions conditional on additional parameters called hyperparameters.
One general way of constructing a hierarchical prior is based on the prior belief of exchangeability. This particular form of hierarchical prior will be used for the mortality rates example of this chapter and the career trajectories example of Chapter The number of deaths within 30 days of heart transplant surgery is recorded for each of 94 hospitals. In addition, we record for each hospital an expected number of deaths called the exposure, denoted by e.
We let yi and ei denote the respective observed number of deaths and exposure for the ith hospital. In R, we read in the relevant dataset hearttransplants in the LearnBayes package. In Figure 7. Plot of death rates against log exposure for all hospitals. Each point is labeled by the number of observed deaths. Note that the estimated rates are highly variable, especially for programs with small exposures. The programs experiencing no deaths a plotting label of 0 also are primarily associated with small exposures.
Since the individual death rates can be poor, it seems desirable to combine the individual estimates in some way to obtain improved estimates.
This is questionable since one would expect some variation in the true rates. We will see that this shrinkage estimate is a natural by-product of the application of an exchangeable prior model to the true mortality rates. Did this hospital have an unusually high number of deaths? The actual number of transplant deaths is shown by a vertical line. We can check the consistency of the observed yi with its posterior pre- dictive distribution for all hospitals. This prior distribution induces positive correlation between the true death rates.
See Figure 7. The output of laplace includes the mode and the corresponding estimate at the variance-covariance matrix. The resulting graph is displayed in Figure 7. Figure 7. In contrast to the wide variation in the observed death rates, note the similarity in the locations of the probability intervals for the true rates. This indicates that these Bayesian estimates are shrinking the individual rates toward the pooled estimate.
Plot of the posterior shrinkages against the log exposures for the heart transplant example. Suppose we wish to compare hospital i with hospital j. Using a simple function compare.
Note that hospital 85 is better than most of these hospitals since most of the posterior probabilities are close to zero. The SIR algorithm, described in Section 5. We write an R function sir. The output is a sample from the new posterior sample. The thick solid and dotted lines represent the corresponding posterior densities. Here we use the same methodology to check the ap- propriateness of the exchangeable model. Again we consider hospital 94, which experienced 17 deaths.
Histogram of the posterior predictive distribution of y94 for hospital 94 from the exchangeable model. The observed value of y94 is indicated by the vertical line. Again this exchangeable model can check the consistency of the observed yi with its posterior predictive distribution for all hospitals. Note that only two of the observations have a probability smaller than 0.
Chapter 5 of Carlin and Louis , introduces hierarchical model- ing from an empirical Bayes perspective. Posterior predictive model checking is described as a general method for model checking in Chapter 6 of Gelman et al. The use of hierarchical modeling to analyze the heart transplant data is described in Christiansen and Morris Is the posterior analysis sensitive with respect to the choice of exchangeable model?
Gelman et al. To illustrate this model, Gelman et al. For all observations, compute the shrinkage size E Bj y from the simulated draws of the hyperparameters.
The dataset is available as cancermortality in the LearnBayes package. To model a prior belief of exchangeability, it is assumed that p1 , We introduce the notion of a Bayes factor in the setting where one is comparing two hypotheses about a parameter.
In the setting where one is testing hypotheses about a population mean, we illustrate the computation of Bayes factors in both the one-sided and two-sided settings. We then generalize to the setting where one is comparing two Bayesian models, each consisting of a choice of prior and sampling density.
In this case, the Bayes factor is the ratio of the marginal densities for the two models. We illustrate Bayes factor computations in two examples. In the second application, we illustrate the computation of Bayes factors against independence in a two-way contingency table. The author weighed himself ten times and obtained the measurements in pounds , , , , , , , , , and Using this normal posterior density, we calculate the odds of the null hypothesis.
There is an interesting connection between this Bayesian measure of evi- dence and the frequentist p-value. In the R output, we compute the p-value using the pnorm function.
The func- tion mnormt. This illustrates the general result that a Bayesian probability of a hypothesis is equal to the p-value for one-sided testing problems when a vague prior distribution is placed on the parameter.
The alternative hypothesis H1 is that his weight is now either larger or smaller than The function mnormt. The R code for the computation in this example is shown here. Note that the values. He should conclude that his current weight is not Then the Bayes factor in support of model M0 is the ratio of the respective marginal densities or prior predictive densities of the data for the two models. By applying this method for several models, one can use the computed values of m y to compute a Bayes factor.
You observe the number of goals scored y1 , Prior 1. Prior 2. Prior 3. This prior says that you believe teams score a lot of goals in Major League Soccer. Prior 4. The number of goals was observed for a particular team in Major League Soccer for the season. The dataset is available as soccergoals in the LearnBayes package.
Figure 8. Priors 1 and 2 seem pretty similar in location and shape. The arguments to the function are theta and datapar, a list that contains the data vector data and the parameters of the gamma prior par. Note that we use the R function dgamma in computing both the likelihood and the prior. This function uses both the R functions dgamma and dnorm.
For each of the four priors, we use the function laplace to summarize the posterior. Does it matter if we use a gamma 4. In sports, we observe much streaky behavior in players and teams. We observe many streaky patterns in the performance of players. The interesting question is what these streaky data say about the ability of a player to be streaky. Suppose we divide all of the at-bats in a particular baseball season into N periods.
On the other hand, if the player is truly streaky, then the probability of a hit pi will change across the season. A convenient way to model this variation in the probabilities is to assume that p1 , We can index this streaky model by the parameter K; we represent the streaky model by MK : p1 , To compare the models M0 and MK , we need to compute the associated marginal densities. Under the model M0 , the numbers of hits y1 , The inputs to this function are theta and a list datapar with components data a matrix with columns y and n and K.
To illustrate the use of this method, we consider the hitting data for the New York Yankee player Derek Jeter for the baseball season.
Hitting data for Jeter were collected for each of the games he played in that particular season. The original game-by-game data are available as jeter in the LearnBayes package. These data are listed in Table 8. Table 8. Hitting data of Derek Jeter for baseball season. In this example, we write a short wrapper function that computes the log Bayes factor for a single value of log K. By using the sapply function, the corresponding values of the log Bayes factor log BK are stored in the variable log.
BF,BF ,2 log. K K log. This particular streaky model is approximately two and a half times as likely as the consistent model. This indicates that Jeter did indeed display some true streakiness in his hitting behavior for this particular baseball season. To illustrate this situation, consider the following example presented in Moore in which North Carolina State University looked at student performance in a course taken by chemical engineering majors.
Researchers wished to learn about the relationship between the time spent in extracurricular activities and the grade in the course.
Data on these two categorical variables were collected from students, and the responses are presented using the contingency table in Table 8. Two-way table relating student performance and time spent in extracur- ricular activities. The usual non-Bayesian approach of testing the independence hypothesis is based on a Pearson chi-squared statistic that contrasts the observed counts with expected counts under an independence model.
In R, we read in the table of counts and use the function chisq. From a Bayesian viewpoint, there are two possible models — the model MI that the two categorical variables are independent and the model MD that the two variables are dependent in some manner. To describe the Bayesian models, assume that these data represent a random sample from the population of interest and the counts of the table have a multinomial distribution with proportion values as shown in Table 8.
Under the dependence model MD , the proportion values p11 ,. Probabilities of the table under the hypothesis of dependence. Probabilities of the table under the hypothesis of independence. The R function ctable will compute this Bayes factor for a two-way con- tingency table. One inputs a matrix a of prior parameters for the matrix of probabilities. The output of this problem is the value of the Bayes factor.
This structure of prior means is illustrated for our example in Table 8. This model will be labeled MK , as it is indexed by the Dirichlet precision parameter K.
One straightforward way of computing the Bayes factor is by importance sampling. This density is a convenient choice for an importance sampler since it is easy to simulate draws from a Dirichlet distribution.
One inputs the data matrix y, the Dirichlet precision parameter K, and the size of the simulated sample m. The output is a list with two components: bf, the value of the Bayes factor, and nse, an estimate of the simulation standard error of the computed value of BF. In the following R input, we compute the Bayes factor for a sequence of values of log K.
We write a short function compute. BF to compute the Bayes factor for a single value of log K. The output gives the value of the log Bayes factor and the Bayes factor for six values of log K.
We used the R function spm in the SemiPar package to smooth out the simulation errors in the computed log Bayes factors before plotting. Note that this maximum value of the Bayes factor is 2.
K log. Berger and Sellke and Casella and Berger describe the relationship between Bayesian and frequentist measures of evidence in the two-sided and one-sided testing situations, respectively. Gunel and Dickey describe the use of Dirichlet distributions in the development of tests for contingency tables, and Albert and Gupta introduce the use of mixtures of Dirichlet distributions for contingency tables.
A one-sided test of a binomial probability In , the St. Louis Post Dispatch was interested in measuring public support for the construction of a new indoor stadium. The newspaper conducted a survey in which they interviewed registered voters. Let p denote the proportion of all registered voters in the St. Louis voting district opposed to the stadium. Compute the prior odds of the hypothesis K. Using the R function pbeta, compute the posterior odds of the hypothesis K.
Use the R function mnormt. The arguments to mnormt. Compare the values of the posterior probability with the value of the p-value computed in part a. Comparing Bayesian models using a Bayes factor Suppose that the number of births to women during a month at a partic- ular hospital has a Poisson distribution with parameter R.
During a given year at a particular hospital, 66 births were recorded in January and 48 births were recorded in April. Is a basketball player streaky? Kobe Bryant is one of the most famous players in professional basketball. Use the function laplace together with the function bfexch to compute the logarithm of the Bayes factor in support of the streaky hypothesis MK. Based on your work, is there much evidence that Bryant displayed true streakiness in his shooting performance in these 15 games?
Happiness and family income from General Social Survey. Based on the p-value, is there evidence to suggest that the level of happiness is dependent on the family income? Using the R function ctable, compute the Bayes factor in support of the dependence hypothesis. Which type of measure, frequentist or Bayesian, indicates more evidence against independence? We then illustrate the R Bayesian computations in an ex- ample where one is interested in explaining the variation of extinction times of birds in terms of their nesting behavior, their size, and their migratory status.
Zellner proposed a simple way of inputting prior information in a regression model. Since the two component distributions inverse gamma and multivariate nor- mal are convenient functional forms, it is relatively easy to construct an al- gorithm in R such as the one programmed in the function blinreg to perform this simulation.
Likewise, the representation of the posterior predictive distribution of fu- ture response values suggests a simple algorithm for simulation. If yi is in the tail of the distribution, that indicates that this observation is a potential outlier. After data y are observed, we can compute the posterior probability that each observation is an outlier. The R function bayesresiduals can be used to compute the posterior outlying probabilities for a linear regression model.
Measurements on breeding pairs of land- bird species were collected from 16 islands around Britain over the course of several decades. This dataset is available as birdextinct in the LearnBayes package. Figures 9. Note that there is a positive relationship between the average number of nesting pairs and time to extinction.
Plot of logarithm of the extinction time against the average number of nesting pairs for the bird study. Plot of the logarithm of the extinction time against the bird size for the bird study. The bird size variable is coded 0 for small and 1 for large. Plot of the logarithm of the extinction time against the bird status for the bird study. The bird status variable is coded 0 for migrant and 1 for resident.
The values of the four sets of covariates are shown in Table 9. Table 9. Four sets of covariates of interest in the bird study. The output of the function is a matrix where a column contains the simulated draws for a given covariate set. We construct histograms of the simulated draws for each of the mean extinction times, and the plots are displayed in Figure 9.
The function blinregpred will produce a simulated sample of future response values for a regression model. Comparing Figure 9. Histograms of simulated draws of the posterior of the mean extinction time for four sets of covariate values. In the R code, we summarize each predictive distribution by the 5th and 95th quantiles and graph these distributions as line plots using the matplot command see Figure 9. We place the actual log extinction times y1 , Histograms of simulated draws of the predictive distribution for a future extinction time for four sets of covariate values.
Following the strategy described in Section 9. These probabilities can be com- puted using the function bayesresiduals. The output is a vector of posterior outlying probabilities. We use the plot command to construct a scat- terplot of the probabilities against the nesting covariate; the resulting display is in Figure 9. Carlin,Hal S. Stern,David B. Dunson,Aki Vehtari,Donald B.
Computational Bayesian Statistics Author : M. Carlin,Thomas A. Provides an accessible approach to Bayesian data analysis, as material is explained clearly with concrete examples. The book begins with the basics, including essential concepts of probability and random sampling, and gradually progresses to advanced hierarchical modeling methods for realistic data.
Winner of the De Groot Prize from the International Society for Bayesian Analysis Now in its third edition, this classic book is widely considered the leading text on Bayesian methods, lauded for its accessible, practical approach to analyzing data and solving research problems. Presenting a comprehensive resource for the mastery of network analysis in R, the goal of Network Analysis with R is to introduce modern network analysis techniques in R to social, physical, and health scientists.
The mathematical foundations of network analysis are emphasized in an accessible way and readers are guided through the basic steps of network studies: network conceptualization, data collection and management, network description, visualization, and building and testing statistical models of networks. As with all of the books in the Use R! Appendices will describe the R network packages and the datasets used in the book. An R package developed specifically for the book, available to readers on GitHub, contains relevant code and real-world network datasets as well.
As the world becomes increasingly complex, so do the statistical models required to analyse the challenging problems ahead. These simple, but powerful statistical techniques, take Bayesian statistics beyond the need to specify overly simplified models, to the setting where the model is defined only as a process that generates data. This process can be arbitrarily complex, to the point where standard Bayesian techniques based on working with tractable likelihood functions would not be viable.
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The Handbook of ABC provides illuminating insight into the world of Bayesian modelling for intractable models for both experts and newcomers alike.
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