Consideration Set Mixed Logit in PyMC Marketing

Abstract

Bayesian Estimation of a Two-Stage Consideration-then-Choice Mixed Logit Model

Consideration Set Mixed Logit

This contribution extends the existing MixedLogit class with a two-stage consideration-then-choice structure. The core insight is that consumers do not evaluate every available alternative before deciding — they first form a consideration set (which products are even on the table) and only then choose among the products they actually considered. The implementation can be found in pymc-marketing here.

The Model

Stage 1 — Consideration:

\[ \pi_{nj} = \sigma\!\left(\gamma_{0j} + \sum_k \gamma_{zjk} \cdot \tilde{z}_{njk} + \eta_n\right) \]

where \(\tilde{z}_{njk}\) are mean-centred consideration instruments satisfying an exclusion restriction: they must be structurally separate from the utility covariates. At \(\tilde{z} = 0\), \(\pi = 0.5\) — only deviations from population-average screening drive consideration.

Stage 2 — Choice:

\[ P(j \mid n) = \operatorname{softmax}\!\left(\log \pi_{nj} + V_{nj}\right) \]

The bridge formula \(U^{\text{avail}}_{nj} = V_{nj} + \log \pi_{nj}\) integrates consideration directly into the utility index, so the mixed logit likelihood structure from the parent class is preserved without modification.

This is the discrete choice analogue of the key/query separation in transformer attention: the consideration instruments \(Z\) play the role of the query while the utility covariates \(X\) play the role of the value.

Identification

• Exclusion restriction: \(Z\) instruments must not appear in \(V_{nj}\). This identifies the consideration stage separately from the preference stage.
• Mean-centring: At average screening behaviour (\(\tilde{z} = 0\)) the consideration probability is exactly \(0.5\), so the utility intercept \(\alpha_j\) absorbs baseline alternative-specific effects. When consideration_intercept=True, \(\gamma_{0j}\) and \(\alpha_j\) compete to explain baseline effects — use informative priors or constrain one set.
• Random consideration (random_consideration=True) adds a per-individual intercept \(\eta_n \sim \mathcal{N}(0, \sigma_{\text{consider}})\), capturing unobserved heterogeneity in “visibility” across all alternatives.

Implementation

from pymc_marketing.customer_choice.consideration_set_logit import (
    ConsiderationSetMixedLogit,
    ConsiderationInstruments,
)

instruments: ConsiderationInstruments = {
    "Z_tilde": Z_tilde,             # (N, J) mean-centred, or (N, J, K_z)
    "z_instrument_names": ["adspend", "shelf_position"],
}

model = ConsiderationSetMixedLogit(
    choice_df=df,
    utility_equations=[
        "brand_a ~ price_a + quality_a | income",
        "brand_b ~ price_b + quality_b | income",
        "brand_c ~ price_c + quality_c | income",
    ],
    depvar="choice",
    covariates=["price", "quality"],
    consideration_instruments=instruments,
    consideration_intercept=False,   # alpha_j absorbs baseline consideration
    random_consideration=True,       # per-individual consideration heterogeneity
)

idata = model.fit(target_accept=0.97, tune=2000)

The class inherits the full MixedLogit interface: Wilkinson-style formula specification, non-centered parameterisation, panel data support, control-function endogeneity correction, apply_intervention, and sample_posterior_predictive.

Key Design Choices

• Numerically stable log-sigmoid: \(\log(\sigma(x)) = x - \operatorname{softplus}(x)\), avoiding the catastrophic cancellation in \(\log(\sigma(x) + \varepsilon)\) for large negative \(x\).
• Multi-instrument support: \(Z\) can be 2-D \((N, J)\) for a single instrument per alternative or 3-D \((N, J, K_z)\) for multiple instruments, with named coordinates surfaced in the posterior.
• Dimension-switch guard: switching \(Z\) from 2-D to 3-D after the model is built raises a clear ValueError, preventing silent model mismatches on apply_intervention.