Subscription services typically offer duration discounts, rewarding longer commitments with lower per-period costs. The “menu” of contract plan prices must be balanced to encourage potential customers to select into subscription overall and to nudge those that do to more profitable contracts. Because subscription menu prices change infrequently, they are difficult to optimize using historical pricing data alone.

We propose that firms solve this problem via an experiment and a model that jointly accounts for whether to opt-in and, conditionally, which plan to choose. To that end, we conduct a randomized online pricing experiment that orthogonalizes the “elevation” and “steepness” of price menus for a major dating pay site. Users’ opt-in and plan choice decisions are analyzed using a novel model for correlated binary selection and multinomial conditional choice, estimated via Hamiltonian Monte Carlo. Benchmark comparisons suggest that common models of consumer choice may systematically misestimate price substitution patterns, and that a key consideration is the distinctiveness of the opt-out (e.g., non-subscription) option relative to others available.

Our model confirms several anticipated pricing effects (e.g., on the margin, raising prices discourages both opt-in overall and choice of any higher-priced plans), but also some that alternative models fail to capture, most notably that across-the-board pricing increases have a far lower negative impact than standard random-utility models would imply. Joint optimization of the price menu’s component prices suggests the firm has set them too low overall, particularly for its longest-duration plan, and adjusting the price menu accordingly should lead to roughly 10% greater revenue.

Making sense of massive, individual-level data is challenging: marketing researchers and analysts need flexible models that can accommodate rich patterns of heterogeneity and dynamics, work with and link diverse data types, and scale to modern data sizes. Practitioners also need tools that can quantify uncertainty in models and predictions of consumer behavior to inform optimal decision-making. In this paper, we demonstrate the promise of probabilistic machine learning (PML), which refers to the pairing of probabilistic modeling and machine learning methods, in pushing the frontier of combining flexibility, scalability, interpretability, and uncertainty quantification for building better models of consumers and their choices. Specifically, we overview both PML models and inference methods, and highlight their utility for addressing four common classes of marketing problems: (1) uncovering heterogeneity, (2) flexibly modeling nonlinearities and dynamics, (3) handling high-dimensional and unstructured data, and (4) addressing missingness, often via data fusion. We also discuss promising directions in enriching marketing models, reflecting recent developments in representation learning, causal inference, experimentation and decision-making, and theory-based behavioral modeling.

Data fusion combines multiple datasets to make inferences that are more accurate, generalizable, and useful than those made with any single dataset alone. However, data fusion poses a privacy hazard due to the risk of revealing user identities. We propose a privacy preserving data fusion (PPDF) methodology intended to preserve user-level anonymity while allowing for a robust and expressive data fusion process. PPDF is based on variational autoencoders and normalizing flows, together enabling a highly expressive, nonparametric, Bayesian, generative modeling framework, estimated in adherence to differential privacy – the state-of-the-art theory for privacy preservation. PPDF does not require the same users to appear across datasets when learning the joint data generating process and explicitly accounts for missingness in each dataset to correct for sample selection. Moreover, PPDF is model-agnostic: it allows for downstream inferences to be made on the fused data without the analyst needing to specify a discriminative model or likelihood a priori.

We undertake a series of simulations to showcase the quality of our proposed methodology. Then, we fuse a large-scale customer satisfaction survey to the customer relationship management (CRM) database from a leading U.S. telecom carrier. The resulting fusion yields the joint distribution between survey satisfaction outcomes and CRM engagement metrics at the customer level, including the likelihood of leaving the company’s services. Highlighting the importance of correcting selection bias, we illustrate the divergence between the observed survey responses vs. the imputed distribution on the customer base. Managerially, we find a negative, nonlinear relationship between satisfaction and future account termination across the telecom carrier’s customers, which can aid in segmentation, targeting, and proactive churn management.

When is the most effective time to post updates during a crowdfunding campaign? Answering this question requires a framework that accounts for the dynamic effects of updates, as well as a provision point mechanism. Crowdfunding platforms like Kickstarter involve an all-or-nothing structure such that no money exchanges hands if the target goal is not met.  Addressing this question raises three interrelated challenges. First, entrepreneurs post updates strategically in response to evolving campaign conditions. Second, the provision point itself is an endogenous decision. Third, the effects of updates are not static but evolve nonlinearly across time. We tackle these challenges by developing a Bayesian provision point model that builds on the insight that entrepreneurs benchmark their decisions against similar past campaigns. This benchmarking provides exogenous variation that allows us to instrument for the non-random timing of updates and correct for endogenous goal-setting. We incorporate Gaussian Process priors to capture smooth, stage-specific dynamics in update effects. We find that updates have a modest and positive effects before the goal is reached, but their effectiveness becomes time-dependent once the goal is surpassed. Our framework also reveals that campaigns setting higher-than-benchmarked goals face weaker pre-goal and stronger post-goal support.  Our research emphasizes the need for entrepreneurs to consider the sequence of updates as these have consequential effects on funding outcomes, and the need to consider the provision point when studying dynamics in crowdfunding.