Bayesian State-Space Aggregation of Brazilian Presidential Polls
As presidential elections approach, Brazilian voters are confronted
with a growing volume of conflicting polling data from various
institutes, each employing distinct methodologies and sampling designs.
agregR provides the public with a rigorous framework to
process the surfeit of data and estimate the underlying level of support
for each candidate.
The package implements a set of Bayesian state-space models in Stan to aggregate and normalize polling
data, extracting a stable signal from diverse, noisy, and possibly
biased data sources. agregR is able to automatically
down-weight institutes with historically poor accuracy while maintaining
the flexibility to update their evaluation based on current-cycle
performance. It also features specialized methods to account for:
agregR is built on CmdStan, the
state-of-the-art backend for Stan. Since CmdStan is not available on
CRAN (and
will likely never be), it needs to be installed separately. This
one-time setup yields substantial gains in compilation speed and
sampling performance.
We recommend following these installation steps in order:
Windows users must first install RTools to
enable C++ compilation. MacOS requires Xcode
Command Line Tools, and Linux users should install
the distribution-specific compiler (e.g., Ubuntu:
sudo apt install build-essential).
The most convenient way to install CmdStan is via the
cmdstanr interface.
# Install cmdstanr interface
install.packages("cmdstanr", repos = c("https://mc-stan.org/r-packages/", getOption("repos")))
# Install CmdStan
cmdstanr::install_cmdstan()Optional: make sure everything is in place.
cmdstanr::check_cmdstan_toolchain()You can install the release version of agregR from CRAN
with:
install.packages("agregR", type = "source")Experimental: the development (and possibly
unstable) version of agregR can be installed with:
if (!require(pak)) install.packages("pak")
pak::pak("rnmag/agregR")The main function rodar_agregador() centralizes data
preparation, model compilation, and sampling. It returns the full
CmdStanMCMC objects for diagnostics, along with tidy data
frames for house effects and daily voting estimates.
library(agregR)
# Execute the aggregation pipeline for a 2nd round scenario
result <- rodar_agregador(
data_inicio = "01/01/2025",
turno = 2,
cenario = "Lula vs Tarcísio",
modelo = "Viés Empírico"
)
# Daily voting estimates + poll data in tidy format
result$votos_estimados
# House effects in tidy format
result$vies_institutos
# Raw model object
result$modelo_brutoThe package includes a suite of plots designed for public communication.
Visualizes the estimated voting intention for each candidate overlaying the raw polling data.
grafico_agregador(result)
Visualizes the systematic bias for each institute, identifying outliers and consistent directional skews.
grafico_vies(result, candidaturas = c("Lula", "Tarcísio"))
Visualizes how the data has informed the model by comparing prior vs. posterior distributions for selected parameters.
grafico_priori_posteriori(result, tipo = "Viés", candidaturas = c("Lula", "Tarcísio"))
The package offers configuration functions for fine-grained control
over plots and models. Configuration values can be stored in new objects
using the functions configurar_agregador(),
configurar_prioris() and configurar_grafico().
Alternatively, they can be passed directly as lists to the appropriate
arguments.
# Config passed as list: longer run with tighter priors for non-sampling error
result_custom <- rodar_agregador(
turno = 2,
cenario = "Lula vs Tarcísio",
config_agregador = list(stan_chains = 4,
stan_iter = 2000,
stan_warmup = 2000),
config_prioris = list(sd_tau_priori = 0.01)
)
# Config passed as function: custom color and custom symbols
grafico_agregador(
result,
config_grafico = configurar_grafico(
cores_candidaturas = c("Tarcísio" = "yellow"),
simbolos = c("Presencial" = 19, "Online" = 2, "Telefônica" = 4)
)
)
# Config passed as object: custom color
config_custom <- configurar_grafico(cores_candidaturas = c(Lula = "green"))
grafico_agregador(result, config_grafico = config_custom)We are interested in performing inference on the latent state of public opinion: the dynamic, unobserved level of support for each candidate. Polls are periodic snapshots of this state, but the pictures are distorted and grainy.
An apt analogy is a GPS receiver navigating an area with spotty connectivity. It receives sparse, conflicting pings from different satellites, each with its own uncertainty due to equipment miscalibration or inherent manufacturer bias. The system must achieve three objectives:
Much like satellites, polling institutes are often miscalibrated.
Their readings contain noise introduced by different sampling designs,
weighting protocols, and question wording, among other factors.
agregR shares the same objectives as the GPS receiver:
Data collection is deliberately unselective. Instead of subjectively deciding which institutes produce high quality polls, we trust the models to separate the wheat from the chaff.
Polls enter the model with checks on their sample size in order to avoid undue influence from institutes claiming inflated precision. We calculate an implied \(n\) derived from the published margin of error and compare it to the reported sample size. We use the most conservative figure \(n_{eff}\) to compute specific standard errors for each candidate \(c\) according to their vote share \(v_{i, c}\) in poll \(i\):
\[ \sigma_{i, c} = \sqrt\frac{v_{i, c}(1-v_{i, c})}{n_{eff[i]}} \]
Historical data is sourced from Poder360’s polling database via Base dos Dados.
The methods implemented by agregR build on Jackman
(2009, Chapter 9). They are variously known as state-space models (SSM),
dynamic linear models (DLM) or Kalman filters and consist of two
integrated components:
The latent voting intention for each candidate updates daily according to a local linear trend. The evolution of the latent state through time \(t\) for candidate \(c\) is governed by the level component \(\mu_{t, c}\) and influenced by the trend component \(\nu_{t, c}\).
The level \(\mu_{t, c}\) is defined by the previous state \(\mu_{t - 1, c}\) plus the trend \(\nu_{t - 1, c}\), subject to stochastic level innovations \(\eta_{t, c}\). The trend itself evolves as a random walk, allowing the momentum of the campaign to shift over time, controlled by trend innovations \(\zeta_{t, c}\).
\[ \begin{pmatrix}\mu_{t, c} \\ \nu_{t, c}\end{pmatrix} = \begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix} \begin{pmatrix}\mu_{t - 1, c} \\ \nu_{t - 1, c}\end{pmatrix} + \begin{pmatrix}\eta_{t, c} \\ \zeta_{t, c}\end{pmatrix} \]
The volatility parameters govern the “stiffness” of the aggregator, where daily innovations \(\eta_{t, c}\) are regularized by a candidate-specific scale \(\omega_{\eta, c}\). Pooling accross the time series prevents over-fitting to noise while allowing the model to adapt when consistent evidence of a shift in public opinion emerges.
\[ \begin{align} \eta_{t, c} &\sim N\left(0, \omega^2_{\eta, c}\right) \\ \zeta_{t, c} &\sim N\left(0, \omega^2_{\zeta, c}\right) \end{align} \]
When polling data \(i\) for candidate \(c\) is available, the observed result \(y_{i, c}\) from institute \(j\) at time \(t\) is modeled as a function of the latent state \(\mu_{t(i), c}\) and house effects \(\delta_{j(i), k(i), p(c)}\):
\[ y_{i, c} = \begin{pmatrix}1 & 0\end{pmatrix} \begin{pmatrix}\mu_{t(i), c} \\ \nu_{t(i), c}\end{pmatrix} + \delta_{j(i), k(i), p(c)} + \varepsilon_{i, c} \]
where
\[ \varepsilon_{i, c} \sim N\left(0, \sqrt{\sigma_{i, c}^2 + \tau_{j(i), k(i), p(c)}^2}\right) \]
with subscripts linking poll \(i\) and candidate \(c\) to relevant covariates:
In the error term \(\varepsilon\), \(\sigma\) represents a theoretical lower bound of uncertainty, whereas \(\tau\) captures the excess empirical variance required to account for the data’s observed dispersion.
Computationally, the measurement model is designed to prioritize high sampling efficiency and convergence stability (see Model Validation). The normal likelihood provides a convenient approximation of latent support for competitive candidates whose polling numbers do not approach the 0% boundary. Compared to the full multinomial implementation with Cholesky-factorized covariance proposed by Stoetzer et al. (2019), this normal approximation yields nearly identical inferences for leading candidates, samples significantly faster, and is far less prone to divergent transitions.
In summary, the measurement model identifies three sources of uncertainty for polls:
Based on the methods described above, agregR offers a
set of specialized models that differ in their assumptions regarding
house effects (\(\delta\)) and
non-sampling error (\(\tau\))
estimation:
| Model | House Effects Anchor (\(\delta\)) | Non-Sampling Error (\(\tau\)) |
|---|---|---|
| Viés Relativo com Pesos (Weighted Relative Bias) | Consensus \(\left(\sum_j \delta_{j, k, p} = 0\right)\) | Last election \(\tau_{j,k,p}\) (past RMSE \(\rightarrow \tau\) prior) |
| Viés Relativo sem Pesos (Unweighted Relative Bias) | Consensus \(\left(\sum_j \delta_{j, k, p} = 0\right)\) | Global \(\tau\) shared by all institutes |
| Viés Empírico (Empirical Bias) | Last election \(\delta_{j,k,p}\) (past bias \(\rightarrow \delta\) prior) | Last election \(\tau_{j,k,p}\) (past RMSE \(\rightarrow \tau\) prior) |
| Retrospectivo (Retrospective) | Actual election result \(\left(\mu_T\right)\) | Global \(\tau\) shared by all institutes |
| Naive | None | None |
Early stages of election campaigns are frequently characterized by extreme data sparsity. In such low-information environments, fully hierarchical models struggle to identify group-level variances, often leading to pathological behavior (e.g., complete shrinkage) or convergence failures.
Anchoring the scales for \(\delta\)
and \(\tau\) keeps the models robust
and identifiable throughout the entire cycle, transitioning gracefully
from a prior-dominated regime to a data-dominated one as the volume of
polling increases. Specific values for priors can be accessed (and
modified) by the configurar_prioris() function, and details
are available in the function’s documentation.
Every Stan model in agregR includes a
generated quantities block, enabling Posterior Predictive
Checks (PPC). By simulating \(y_{rep}\)
from the posterior distribution, users can verify the model’s
calibration against real-world data (Gabry et al., 2019). The example
below demonstrates this using the bayesplot package.
library(bayesplot)
# Setup
cand <- "Lula"
modelo_cand <- result$modelo_bruto[[cand]]
color_scheme_set("mix-brightblue-darkgray")
# Observed data
y <- result$votos_estimados |>
filter(!is.na(percentual_pesquisa) & candidatura == cand) |>
pull(percentual_pesquisa)
# Simulated data
y_rep <- modelo_cand$draws("perc_simulado", format = "matrix")
# Prepare plot labels
pesquisa_id <- result$votos_estimados |>
filter(!is.na(percentual_pesquisa) & candidatura == cand) |>
pull(pesquisa_id)
# Plot observed vs simulated data
ppc_intervals(y, y_rep, prob = 0.67, prob_outer = 0.95) +
scale_x_continuous(labels = pesquisa_id,
breaks = seq_along(pesquisa_id)) +
scale_y_continuous(labels = scales::label_percent()) +
labs(title = "Simulated vs Observed Data") +
xaxis_title(FALSE) +
coord_flip() +
theme_minimal() +
theme(plot.title = element_text(face = "bold", size = 18, hjust = .5),
panel.grid = element_blank(),
panel.grid.major.y = element_line(linetype = "dotted", color = "gray80"),
axis.text.y = element_text(size = 8),
legend.position = "top")
Parameter distributions are standardized using Non-Centered Parametrization (NCP). This flattens posterior geometry and addresses the “funnel” problem common in hierarchical models, significantly improving sampling efficiency and virtually eliminating divergent transitions in standard scenarios (Stan Development Team, Efficiency Tuning: Reparametrization).
# Posterior geometry for selected mu and delta parameters
mcmc_scatter(modelo_cand$draws(),
pars = c("mu[1]", "delta[1]"),
np = nuts_params(modelo_cand), # no divergences to display
alpha = 0.1) +
stat_density_2d(color = "black")
The MCMC chains demonstrate robust convergence, with the following plot illustrating typical Effective Sample Size (ESS) and R-hat values. Notably, many parameters exhibit an ESS exceeding the nominal number of post-warmup iterations (blue line), a result of anti-correlated draws that further underscores high sampling efficiency.
# ESS (bulk) vs R-hat
ggplot(modelo_cand$summary(), aes(x = ess_bulk, y = rhat)) +
geom_point(alpha = 0.3) +
geom_hline(yintercept = 1.01, linetype = "dashed", color = "red") +
geom_vline(xintercept = 400, linetype = "dashed", color = "red") +
geom_vline(xintercept = 2000, linetype = "dashed", color = "blue") +
labs(title = "Convergence Diagnostics",
subtitle = "Reference values: R-hat < 1.01 | ESS (bulk) > 4 x 100 | Iterations (post-warmup): 4 x 500)",
x = "Effective Sample Size (bulk)",
y = "R-hat") +
theme_minimal() +
theme(text = element_text(family = "Fira Sans"),
plot.title = element_text(face = "bold", size = 18, hjust = .5),
plot.subtitle = element_text(hjust = .5, color = "#777777"))
Gabry, J., Simpson, D., Vehtari, A., Betancourt, M., & Gelman, A. (2019). Visualization in Bayesian Workflow. Journal of the Royal Statistical Society Series A: Statistics in Society.
Heidemanns, H., Gelman, A., & Morris, G. (2020). An Updated Dynamic Bayesian Forecasting Model for the 2020 Election. Harvard Data Science Review.
Jackman, S. (2009). Bayesian Analysis for the Social Sciences. Wiley.
Stan Development Team. Stan User’s Guide (Efficiency Tuning: Reparametrization). Retrieved from https://mc-stan.org/docs/stan-users-guide/efficiency-tuning.html#reparameterization.section
Stoetzer, L. F., et al. (2019). Forecasting Elections in Multiparty Systems: A Bayesian Approach Combining Polls and Fundamentals. Political Analysis.