Sure. I think a better thing to do (which I think what Carl is suggesting) is to have a prior distribution over x (the effectiveness of a randomly chosen intervention), and interventionDistribution (a categorical distribution over different shapes you think the space of interventions might have). So P(x, ‘Pareto’) = P(‘Pareto’) P(x | ‘Pareto’) = w_{Pareto} P_{Pareto}(x) and P(x, ‘logNormal’) = P(‘logNormal’) P(x | ‘logNormal’) = w_{logNormal} P_{logNormal}(x). Then, for the first intervention you see, your prior density over effectiveness is indeed P(x) = w_{Pareto} P_{Pareto}(x) + w_{logNormal} P_{logNormal}(x), but after measuring a bunch of interventions, you can update your beliefs about the empirical distribution of effectivenesses.
Sure. I think a better thing to do (which I think what Carl is suggesting) is to have a prior distribution over x (the effectiveness of a randomly chosen intervention), and interventionDistribution (a categorical distribution over different shapes you think the space of interventions might have). So P(x, ‘Pareto’) = P(‘Pareto’) P(x | ‘Pareto’) = w_{Pareto} P_{Pareto}(x) and P(x, ‘logNormal’) = P(‘logNormal’) P(x | ‘logNormal’) = w_{logNormal} P_{logNormal}(x). Then, for the first intervention you see, your prior density over effectiveness is indeed P(x) = w_{Pareto} P_{Pareto}(x) + w_{logNormal} P_{logNormal}(x), but after measuring a bunch of interventions, you can update your beliefs about the empirical distribution of effectivenesses.