A method is proposed for combining information from several emergent constraints into a probabilistic estimate for a climate sensitivity proxy $Y$ such as equilibrium climate sensitivity (ECS) or the climate feedback parameter $\lambda$. The method is based on fitting a multivariate Gaussian PDF for $Y$ and the emergent constraints using an ensemble of global climate models (GCMs). For a single perfectly-observed constraint $X$, it reduces to a linear regression-based estimate of $Y$. The method accounts for uncertainties in sampling this multidimensional PDF with a small number of models, for observational uncertainties in the constraints, and for overconfidence about the correlation of the constraints with the climate sensitivity. Two methods are presented. Method C accounts for correlations between emergent constraints but can fail if some constraints are too strongly related. Method U assumes constraints are uncorrelated except through their mutual relationship to the climate proxy; it is robust to small GCM sample size and is appealingly interpretable. These methods are applied to ECS and $\lambda$ using a previously-published set of 11 possible emergent constraints derived from climate models in the Coupled Model Intercomparison Project (CMIP). This study corroborates and quantifies past findings that most constraints predict higher climate sensitivity than the CMIP mean. The $\pm2\sigma$ posterior range of ECS for Method C with no overconfidence adjustment is $4.1 \pm 0.8$ K. For Method U with a large overconfidence adjustment, it is $4.0 \pm 1.3$ K.