2 Method details 2.1 Theoretical basis The goal is to sample a parameter vector q from a posterior density $_{p}$($_{q}$ | y ) , where $_{p}$($_{q}$ ) is the prior on q , f ( y |$_{q}$ ) is the data likelihood conditional on q , and L ( y ) is the marginal likelihood of the data. Therefore, where D ($_{q}$ , y ) is the joint density of the data and the parameters (of the unnormalized posterior density). In a marketing research context, under the conditional independence assumption, the likelihood can be factored as where i indexes households. 1 Each y$_{i}$ is a vector of observed data, each b i is a vector of hetero- geneous parameters, and a is a vector of homogeneous population-level parameters. The b i are distributed across the population of households according to a mixing distribution $_{p}$($_{b}$$_{i}$ |$_{a}$ ) , which also serves as the prior on each b $_{i}$. The elements of a may influence either the household-level data likelihoods, or the mixing distribution (or both). In this example, q includes all b $_{1}$. . . b N and all elements of a . The prior itself can be factored as Let q ∗ be the mode of D ($_{q}$ , y ) , which is also the mode of $_{p}$($_{q}$ | y ) , since L ( y ) is a constant that does not depend on q . One will probably use some kind of iterative numerical optimizer to find q $^{∗}$, such as a quasi-Newton line search or trust region algorithm. Define c$_{1}$ = D ($_{q}$ $^{∗}$, y ) , and choose a proposal distribution g ($_{q}$) that also has its mode at q $^{∗}$. Define c$_{2}$ = g ($_{q}$ $^{∗}$) , and define the function $^{1}$For brevity, we use the term “household” to describe any heterogeneous unit. F ($_{q}$ | y ) = f ( y |$_{q}$ )$_{p}$($_{q}$) · c$_{2}$ g ($_{q}$) · c$_{1}$ (4) $_{p}$($_{q}$ ) = N GLYPH<213> i = 1 p $_{i}$($_{b}$$_{i}$ |$_{a}$ ) ×$_{p}$ ($_{a}$) . (3) $_{p}$($_{q}$ | y ) = f ( y |$_{q}$ )$_{p}$($_{q}$) L ( y ) = D ($_{q}$ , y ) L ( y ) (1) f ( y |$_{q}$ ) = N GLYPH<213> i = 1 f$_{i}$ ( y$_{i}$ |$_{b}$$_{i}$ , $_{a}$) (2) 4