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Appropriate and defensible statistical frameworks are required in order to make credible inferences about future climate based on projections derived from multiple climate models. It is shown that a two-way analysis of variance framework can be used to estimate the response of the actual climate, if all the climate models in an ensemble simulate the same response. The maximum likelihood estimate of the expected response provides a set of weights for combining projections from multiple climate models. Statistical F tests are used to show that the differences between the climate response of the North Atlantic storm track simulated by a large ensemble of climate models cannot be distinguished from internal variability. When climate models simulate different responses, the differences between the re- sponses represent an additional source of uncertainty. Projections simulated by climate models that share common components cannot be considered independent. Ensemble thinning is advocated in order to obtain a subset of climate models whose outputs are judged to be exchangeable and can be modelled as a random sample. It is shown that the agreement between models on the climate response in the North Atlantic storm track is overestimated due to model dependence. Correlations between the climate responses and historical climates simulated by cli- mate models can be used to constrain projections of future climate. It is shown that the estimate of any such emergent relationship will be biased, if internal variability is large compared to the model uncertainty about the historical climate. A Bayesian hierarchical framework is proposed that is able to separate model uncertainty from internal variability, and to estimate emergent constraints without bias. Conditional cross-validation is used to show that an apparent emergent relationship in the North Atlantic storm track is not robust. The uncertain relationship between an ensemble of climate models and the actual climate can be represented by a random discrepancy. It is shown that identical inferences are obtained whether the climate models are treated as predictors for the actual climate or vice versa, provided that the discrepancy is assumed to be sym- metric. Emergent relationships are reinterpreted as constraints on the discrepancy between the expected response of the ensemble and the actual climate response, onditional on observations of the recent climate. A simple method is proposed for estimating observation uncertainty from reanalysis data. It is estimated that natural variability accounts for 30-45% of the spread in projections of the climate response in the North Atlantic storm track.