Correlated Voting Models

rstan code for fitting the models from Valence in a statistical mechanics of voting.

Setup

To setup the code, first install rstan

install.packages("rstan", repos = "https://cloud.r-project.org/", dependencies = TRUE)

Then import the necessary functions

library(rstan)
source("fitting_fns.r")

Running

To reproduce model fits from the paper, run in bash

Rscript run_all_fits.r

or from within the R-terminal

source('run_all_fits.r')

For any specific data set, one can run

S = read.csv('myData.csv', header=FALSE)
fit_all_models(S, "myData")

where myData.csv is a csv of +/-1's and each row corresponds to a different case and "myData" is the name of the models. The model parameters are saved into CSV files into the out directory.

The data synthetic.csv is generated from the model without interactions, and so the model evidence (correctly) picks model 1. We also provide the voting data for the Second Rehnquist Court and the Roberts Court as rehnquist1607.csv and roberts1709.csv for reproduction. The corresponding justice initials are in rehnquist1607_initials.txt and roberts1709_initials.txt. These have been taken from the SCDB Version 01 from 2022 and the four digit suffix is the natural court identifier.

Compiled output

Individual fields from model 3 are in the CSV file analysis/h_i.csv. This can be read in R using

df_read <- read.csv("analysis/h_i.csv", row.names = 1)

and in Python using pandas

df = pd.read_csv('analysis/h_i.csv', index_col=0)

The column named H is the unanimity field.

Note that there is ambiguity with respect to the sign of the individual field, so we have oriented unanimity field H to be positive.

Equations

Models are defined as below.

Model 1

$$ P( \boldsymbol{s} \vert H, \boldsymbol{h} ) = \frac{1}{2} \left( \frac{\cosh(\sum_i (H+h_i)s_i )}{ \prod_{i} 2 \cosh(H+h_i) } + \frac{\cosh(\sum_i (H - h_i)s_i )}{ \prod_{i} 2 \cosh(H - h_i) } \right) $$

Model 2

$$ P( \boldsymbol{s} \vert J ) = \frac{1}{Z} \exp \Big( {\textstyle \sum_{i < j}} J_{ij} s_i s_j / \sqrt{n} \Big) $$

Model 3

$$ P( \boldsymbol{s} \vert H, \boldsymbol{h}, J ) = \exp \big( {\textstyle \sum_{i < j}} J_{ij} s_i s_j / \sqrt{n} \big) \left( \frac{\cosh(\sum_i (H+h_i)s_i )}{ Z_1 } + \frac{\cosh(\sum_i (H - h_i)s_i )}{ Z_2 } \right) $$

Model 4

$$ P( \boldsymbol{s} \vert H, \boldsymbol{h}, J, f ) = \exp \big( {\textstyle \sum_{i < j}} J_{ij} s_i s_j / \sqrt{n} \big) \left( f\frac{\cosh(\sum_i (H+h_i)s_i )}{ Z_1 } + (1-f)\frac{\cosh(\sum_i (H - h_i)s_i )}{ Z_2 } \right) $$