Silverman 2020 - Bayesian-Threshold Method

Summary

Put flat priors on the intercept and main effects and $N(0,1/\lambda )$ priors on pairwise interactions, run MCMC (MCMCpoisson, MCMCpack), drop interactions whose |posterior mean| / posterior SD falls below a threshold $\tau$, and refit. Recommended defaults: variance 1 ($\lambda =1$) and $\tau =2$.

Overview

Idea: allow the prior to shrink interactions toward zero without constraining the intercept or main effects (which determine list sizes and the dark figure). A true spike-and-slab prior (atom at zero plus a continuous part) is the ideal; thresholding approximates it (Johnstone and Silverman 2004) and standard MCMC software cannot do the mixed prior, so exact implementation is “a topic for future research”.

Main Content

Bayesian/threshold algorithm

Model: Poisson log-linear (Eq. 1) of Silverman 2020 - Poisson Log-Linear Model for Multiple Lists. Step 1 (prior). $\mu ,{\alpha }_i,{\beta }_{ij}$ independent; $\mu$ and ${\alpha }_i$ have improper uniform priors on $(-\infty ,\infty )$; ${\beta }_{ij}\sim N(0,1/\lambda )$, $\lambda \ge 0$ ($\lambda =0$ means improper uniform). Fit by MCMC. Step 2 (threshold). Constrain to zero every ${\beta }_{ij}$ for which |posterior mean| / posterior SD does not exceed $\tau$, then rerun the MCMC with those interactions omitted. $\tau =0$ means no thresholding. Tuning: $\exp {\beta }_{ij}$ is the factor by which a case is more or less likely to be on both $i$ and $j$ than under independence, so ${\beta }_{ij}$ far outside $\pm 1$ is implausible; prior variance $1/\lambda$ with $\lambda$ in 1 to 10 suggested. Liberal threshold $\tau =2$ (include interactions not clearly spurious); parsimonious $\tau =5$ (admittedly arbitrary).

Implementation (Sec. 6.2).

• MCMCpoisson does not handle a mixture of improper-uniform and finite-variance priors; if $\lambda >0$, MCMCfit in modslavmse gives the intercept and main effects a large finite variance $10^4$.
• With $\lambda =0$ (improper) and empty overlaps, the interaction posterior is an atom at $-\infty$ and those cells must be removed (Silverman 2020 - Existence, Identifiability and Empty Overlaps; routine removeemptyoverlaps).
• With a proper prior the empty-overlap problem disappears because the posterior is proper; improper uniform is computationally fastest.
• The total population estimate is posterior draws of $e^{\mu }$ plus observed cases (quantiles reported include observed data).

Examples

Sketch in PyMC (untested)

A Bayesian log-linear MSE with Gaussian interaction prior; the thresholding step is applied post hoc from the first fit.

import numpy as np, pymc as pm, itertools
K = 4
cells = [h for h in itertools.product([0,1], repeat=K) if any(h)]
H = np.array(cells)                       # (2^K-1, K)
pairs = list(itertools.combinations(range(K), 2))
P = np.stack([H[:, i]*H[:, j] for i, j in pairs], 1)   # pair indicators
y = np.array([...])                       # observed counts, same order as cells
with pm.Model() as m:
    mu = pm.Flat("mu"); alpha = pm.Flat("alpha", shape=K)
    beta = pm.Normal("beta", 0, 1.0, shape=len(pairs))  # variance 1, Silverman default
    log_lam = mu + H @ alpha + P @ beta
    pm.Poisson("y", pm.math.exp(log_lam), observed=y)
    pm.Deterministic("dark", pm.math.exp(mu))
    idata = pm.sample()
# threshold: keep pairs with |mean|/sd > 2, refit with only those columns

The same model with a regularised horseshoe on beta replaces hard thresholding with continuous shrinkage; see Horseshoe and Regularized Horseshoe Priors.

Relevance to marketing measurement

Pairwise “interaction” terms among overlapping data sources (panels, CRM lists, ad-server logs) are the analogue of list dependence. Shrinking weak interactions while leaving main effects unpenalised mirrors a standard MMM practice: do not shrink the quantities the decision depends on (baseline and main effects), do regularise the long tail of possible interactions.

Connections

• Spike-and-slab and horseshoe alternatives: Global-Local Shrinkage Priors, Horseshoe and Regularized Horseshoe Priors.
• Prior predictive checks of ${\beta }_{ij}$ scale: Prior Predictive Checking.

See Also

• Silverman 2020 - Bayesian-Threshold Results and Conclusions
• Silverman 2020 - Multiple Systems Analysis for Modern Slavery - Overview