In clinical trials, a surrogate outcome variable (S) can be measured before the outcome of interest (T) and may provide early information regarding the treatment (Z) effect on T. Using the principal surrogacy framework introduced by Frangakis and Rubin (2002. Principal stratification in causal inference. Biometrics 58, 21-29), we consider an approach that has a causal interpretation and develop a Bayesian estimation strategy for surrogate validation when the joint distribution of potential surrogate and outcome measures is multivariate normal. From the joint conditional distribution of the potential outcomes of T, given the potential outcomes of S, we propose surrogacy validation measures from this model. As the model is not fully identifiable from the data, we propose some reasonable prior distributions and assumptions that can be placed on weakly identified parameters to aid in estimation. We explore the relationship between our surrogacy measures and the surrogacy measures proposed by Prentice (1989. Surrogate endpoints in clinical trials: definition and operational criteria. Statistics in Medicine 8, 431-440). The method is applied to data from a macular degeneration study and an ovarian cancer study.