The paper introduces a Bayesian measure of model complexity and fit, denoted as \( p_D \), which is derived from an information-theoretic argument. \( p_D \) is defined as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. This measure is generally equivalent to 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, \( \hat{D} \), is suggested as a Bayesian measure of fit or adequacy. The contributions of individual observations to the fit and complexity can be visualized through a diagnostic plot of deviance residuals against leverages. By adding \( p_D \) to \( \hat{D} \), a diagnostic measure of model complexity is obtained, which is independent of the choice of information criteria and has an approximate decision-theoretic justification. The paper illustrates the use of this technique on several examples and draws conclusions about the proposed techniques.The paper introduces a Bayesian measure of model complexity and fit, denoted as \( p_D \), which is derived from an information-theoretic argument. \( p_D \) is defined as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. This measure is generally equivalent to 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, \( \hat{D} \), is suggested as a Bayesian measure of fit or adequacy. The contributions of individual observations to the fit and complexity can be visualized through a diagnostic plot of deviance residuals against leverages. By adding \( p_D \) to \( \hat{D} \), a diagnostic measure of model complexity is obtained, which is independent of the choice of information criteria and has an approximate decision-theoretic justification. The paper illustrates the use of this technique on several examples and draws conclusions about the proposed techniques.