I consider the problem of estimating and projecting the effect of policy interventions on demographic outcomes and develop a conditional Bayesian hierarchical model for probabilistic projections of the outcome of interest given a set of interventions. Under specified assumptions, I show that the estimated effect is causal. The motivating question is that of identifying policy interventions to accelerate fertility decline in high-fertility countries. Education and family planning are two factors that can be influenced by policy and are thought to accelerate fertility decline. Quantifying their effects on fertility decline in a probabilistic way is thus of interest to policymakers. I first identify the mechanisms by which education and family planning can accelerate fertility decline using a generalized least squares framework inspired by Granger causality. I build on these findings to apply my conditional Bayesian hierarchical model for projections of the Total Fertility Rate (TFR) given education and family planning policy interventions. Unlike existing methods for intervention-based projections of fertility, my model simultaneously accounts for uncertainty in covariate projections, produces probabilistic projections of TFR, and uses demographic background knowledge to inform causal assumptions and ensure demographic plausibility of projections. I assess the model using out-of-sample predictive validation. As an example, I create probabilistic projections of TFR conditional on achieving the Sustainable Development Goals for universal secondary education and universal access to family planning.