The paper introduces a Bayesian measure of model complexity and fit, called $ p_D $, which quantifies the effective number of parameters in a model. This measure is derived from the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. $ p_D $ is shown to approximate the trace of the product of Fisher's information and the posterior covariance matrix, and in normal models, it corresponds to the trace of the 'hat' matrix projecting observations onto fitted values. The posterior mean deviance is proposed as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can be used to create diagnostic plots of deviance residuals against leverages. Adding $ p_D $ to the posterior mean deviance gives the Deviance Information Criterion (DIC), which is related to other information criteria and has a decision-theoretic justification. The procedure is illustrated with examples and compared with alternative Bayesian and classical proposals. The paper emphasizes that the quantities required for $ p_D $ are trivial to compute in a Markov chain Monte Carlo analysis. The paper also explores the properties of $ p_D $ in exponential families and discusses its application in hierarchical models. The paper concludes that $ p_D $ provides a useful measure of model complexity and fit in Bayesian analysis.The paper introduces a Bayesian measure of model complexity and fit, called $ p_D $, which quantifies the effective number of parameters in a model. This measure is derived from the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. $ p_D $ is shown to approximate the trace of the product of Fisher's information and the posterior covariance matrix, and in normal models, it corresponds to the trace of the 'hat' matrix projecting observations onto fitted values. The posterior mean deviance is proposed as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can be used to create diagnostic plots of deviance residuals against leverages. Adding $ p_D $ to the posterior mean deviance gives the Deviance Information Criterion (DIC), which is related to other information criteria and has a decision-theoretic justification. The procedure is illustrated with examples and compared with alternative Bayesian and classical proposals. The paper emphasizes that the quantities required for $ p_D $ are trivial to compute in a Markov chain Monte Carlo analysis. The paper also explores the properties of $ p_D $ in exponential families and discusses its application in hierarchical models. The paper concludes that $ p_D $ provides a useful measure of model complexity and fit in Bayesian analysis.