Hahn-Carvalho-Scott (2012) A Sparse Factor Analytic Probit Model for Congressional Voting Patterns

Summary

Hahn, Carvalho, and Scott propose a sparse factor analytic probit model for high-dimensional binary outcomes, extending the Chib-Greenberg (1998) multivariate probit by placing a spike-and-slab prior on the factor loading matrix $B$ . The model decomposes the $p\times p$ covariance matrix as $\Sigma = BVB^T + I$ , where $B$ is lower triangular (Geweke-Zhou 1996 identification) and $V$ is free to be non-diagonal, allowing correlated latent factors that foster a sparser loading structure. Applied to US Senate roll-call votes 1949–2009, the model recovers a dominant partisanship factor and documents the rise of polarisation after 1979. In simulation at $p=100$ , the sparse factor model achieves Stein loss of 43.4 versus 503.1 for the Wishart benchmark.

Key Claims

Latent variables: $z_i \sim N(\alpha + \lambda\gamma_i + Bf_i, I_p)$ ; $\Sigma = BVB^T + I$ with $B$ lower triangular, $f_i \sim N(0, V)$
$\gamma_i$ is a scalar partisanship factor: truncated to $\mathbb{R}^+$ for Republicans, $\mathbb{R}^-$ for Democrats; $\lambda$ is the $p$ -vector of partisanship loadings
Spike-and-slab prior on each loading $b_{js}$ : $(b_{js} \mid \nu_s, q_s) \sim q_s\,N(0,\nu_s) + (1-q_s)\,\delta_0(b_{js})$ ; $\nu_s \sim$ IG (inverse-gamma); $q_s \sim \text{Beta}(1,1)$
$\text{Beta}(1,1)$ on $q_s$ induces automatic multiplicity correction (Scott-Berger 2006, 2010): posterior inclusion odds shrink when many candidates are tested
$B$ lower triangular with $b_{jj} > 0$ identifies the model (Geweke-Zhou 1996); $b_{js} = 0$ for $s > j$
Free factor covariance $V \neq I$ fosters a sparser $B$ because correlation across factors can be absorbed into $V$ rather than off-diagonal $B$ entries; $\tilde B = BL$ (orthogonal rotation) would generally be denser
Posterior inclusion probability in Step 7: $\hat q_{js} = \hat r_{js} / [(1-q_s)/q_s + \hat r_{js}]$ where $\hat r_{js}$ is the Bayes factor ratio
Factor scores Step 5: $f_i \sim N(m, M)$ with $M = (V^{-1} + B^T B)^{-1}$ , $m = M\,B^T(z_i - \alpha - \lambda\gamma_i)$
Factor covariance Step 6: $V \sim$ IW $(V_0 + FF^T, n + h)$ (inverse-Wishart)
12-step Gibbs sampler cycling over $\alpha$ , $\lambda$ , $\gamma$ , partisanship hyperparameters, $f$ , $V$ , $B$ , $\nu$ , $q$ , $\sigma^2$ , $\beta$ (regression coefficients for covariates)
Simulation (Table 1): Stein loss at ( $n=50$ , $p=100$ , $k=3$ true factors) — sparse-6-factor: 43.4; factor (no sparsity): 136.7; Wishart: 503.1
Senate application: three-factor model on 30 closest votes per term; dimensionality of bills stable over time; partisanship proportion lowest ~1970, increased sharply from 1979 onward
Three partisanship metrics: (19) proportion of variance explained by $\gamma$ ; $q_\lambda$ = mean posterior inclusion prob. of $\lambda_j$ ; (20) mean bill dimensionality

Concepts Introduced or Extended

Entities Mentioned

Quotes

"Allowing the factors to be correlated appears to foster a sparser loading matrix."

"The Beta(1,1) prior on the inclusion probabilities provides automatic multiplicity correction in the sense of Scott and Berger (2006, 2010)."

"The pattern of partisan voting did not decrease monotonically: it was relatively low in the late 1960s and early 1970s, but increased sharply starting around 1979."

My Take

The key insight is that freeing $V$ (factor covariance) trades off density in $V$ against sparsity in $B$ , yielding a more parsimonious loading structure than orthogonal-factor models. The multiplicity correction via $\text{Beta}(1,1)$ on $q_s$ is elegant and practically important when $p$ is large. The Senate application is cleanly motivated but relies on a deliberate data selection (30 closest votes per term), which conditions on high-information votes and may exaggerate polarisation trends. The 12-step Gibbs is conjugate throughout and straightforward to implement.