In this brief note, we introduce a Bayesian approach to estimating parameters for IRT using a freeware called WinBUGS. We use simple Rasch model below to illustrate such an approach and summarize its benefits at the end, as compared with the use of proprietary software (e.g. WINSTEPS and BILOG).
Simple Dichotomous Rasch Model
A student i will score 1 from answering an item k correctly; 0 otherwise. Let y_{ik} be the score. Using Simple Rasch Model, we have
y_{ik} ~ Bernoulli(p_{ik})
logit(p_{ik}) = θ_{i}  d_{k}
where θ_{i} is the ability of student i
d_{k} is the difficulty of item k.
Formulation of the Rasch Model in WinBUGS
The BUGS (Bayesian inference Using Gibbs Sampling) project is concerned with flexible software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods. WinBUGS is a freeware, which provides graphical interface to access all these modeling utilities.
The first step using WinBUGS is to specify the model concerned and the prior distributions for the unknown parameters. For the simple Rasch model, this is shown in the box below.
The posterior distribution of the unknown parameters can then be obtained by running the model in WinBUGS with the response data.
Bayesian Graphical Modeling of the Rasch Model
In Bayesian graphical modeling, the simple Rasch model is represented in Figure 1.
Figure 1. Bayesian Graph of the Rasch Model 
The known data response[i,j] is represented in rectangular form. The unknown parameters (θ[i], d[i,j], τ) are represented in circular form. The dependency amongst the data and parameters are shown using directed arrows.
Such a graphical illustration can enhance understanding of the model by others; especially for a more complex model.

Empirical Results and Model Checking
We illustrate our approach using the classical example in educational testing  the Law School Admission Test (LSAT) data, which is available in the R package called ltm (Latent Trait Model). The data contain responses of 1000 individuals to five items which were designed to measure a single latent ability. Here are the results obtained using WinBUGS. "ltm" are the R statistics as estimates for reference.
Estimates of Item difficulty  

Item  mean  sd  2.5%  median  97.5%  ltm 
1  2.74  0.13  3.00  2.74  2.49  2.87 
2  1  0.08  1.15  1  0.84  1.06 
3  0.24  0.07  0.38  0.24  0.1  0.26 
4  1.31  0.08  1.47  1.31  1.14  1.39 
5  2.1  0.11  2.31  2.1  1.9  2.22 
We can see that the estimated values from WinBUGS are close to the ones from ltm which uses a Marginal Maximum Likelihood (MMLE) approach. As the observed data are discrete, one common method of model checking in Bayesian approach is to draw samples from posterior predictive distribution and compare the simulated frequencies of different possible outcomes with the observed ones. Here are the results of model checking.
The model checking statistics are displayed in the graph below. The observed frequencies are shown by a dashed line. The expected frequencies are shown by vertical bars. We can conclude that the observed outcomes are very close to the predicted ones.
Obs Freq  Expected Frequency  
Score  mean  sd  2.5%  median  97.5%  
0  3  2.4  1.6  0  2  6 
1  21  20.6  5.1  11  20  31 
2  87  88.2  9.7  70  88  107 
3  240  228.1  14.5  200  228  256 
4  361  366.0  17.1  333  366  399 
5  303  294.8  17.6  261  295  330 
Figure 2. Observed and Expected Frequencies 
Flexibility in Enhancing the Model
WinBUGS allows a great flexibility in modeling. For example, we could easily enhance the modeling of student abilities θ_{i} with other covariates X_{ti}, if such information is available. One of the possible formulations could be:
θ_{i} ~ N(μ_{i}, σ_{θ}²)
where μ_{i} = β_{0} + Σ_{t}β_{t}X_{ti} and σ_{θ}²~IG(0.001,0.001).
The WinBUGS code above could be modified easily to incorporate such an enhancement. Parameter estimation in the enhanced model could be automatically taken care by WinBUGS.
Summary
As compared with the proprietary software, the advantages of using the WinBUGS include the following:
(1) the Rasch model can be displayed in a graphical display to facilitate communication and understanding;
(2) testing statistics for model checking could be tailored for the problem at hand; and
(3) a great flexibility in modeling is provided.
Dr. Fung Tzeho
ManagerAssessment Technology & Research,
Hong Kong Examinations and Assessment Authority
www.hkeaa.edu.hk/en
Tzeho F. (2009) Bayesian Estimation for the Rasch Model using WinBUGS, Rasch Measurement Transactions, 2009, 23:1, 11901
Rasch Publications  

Rasch Measurement Transactions (free, online)  Rasch Measurement research papers (free, online)  Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch  Applying the Rasch Model 3rd. Ed., Bond & Fox  Best Test Design, Wright & Stone 
Rating Scale Analysis, Wright & Masters  Introduction to Rasch Measurement, E. Smith & R. Smith  Introduction to ManyFacet Rasch Measurement, Thomas Eckes  Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr.  Statistical Analyses for Language Testers, Rita Green 
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar  Journal of Applied Measurement  Rasch models for measurement, David Andrich  Constructing Measures, Mark Wilson  Rasch Analysis in the Human Sciences, Boone, Stave, Yale 
in Spanish:  Análisis de Rasch para todos, Agustín Tristán  Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez 
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Coming Raschrelated Events  

Oct. 11  Nov. 8, 2019, Fri.Fri.  Online workshop: Practical Rasch Measurement  Core Topics (E. Smith, Winsteps), www.statistics.com 
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Jan. 24  Feb. 21, 2020, Fri.Fri.  Online workshop: Practical Rasch Measurement  Core Topics (E. Smith, Winsteps), www.statistics.com 
May 22  June 19, 2020, Fri.Fri.  Online workshop: Practical Rasch Measurement  Core Topics (E. Smith, Winsteps), www.statistics.com 
June 26  July 24, 2020, Fri.Fri.  Online workshop: Practical Rasch Measurement  Further Topics (E. Smith, Winsteps), www.statistics.com 
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Oct. 9  Nov. 6, 2020, Fri.Fri.  Online workshop: Practical Rasch Measurement  Core Topics (E. Smith, Winsteps), www.statistics.com 
June 25  July 23, 2021, Fri.Fri.  Online workshop: Practical Rasch Measurement  Further Topics (E. Smith, Winsteps), www.statistics.com 
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