3.1 Background on Gaussian Process Regression

Our model for latent public opinion over time is a linear trend with smooth nonlinear deviations. Here, we subsume both components into a single GP model of latent opinion. GPs offer a flexible Bayesian framework for nonlinear regression widely adopted in machine learning (Rasmussen and Williams Reference Rasmussen and Williams2006). GP models have not been used widely in political science, although they have appeared under the label Bayesian kriging (Gill Reference Gill2020; Monogan and Gill Reference Monogan and Gill2016). However, mathematically they can be considered a Bayesian variant of kernal regularized least squares (KRLS) (Hainmueller and Hazlett Reference Hummel and Rothschild2014; Mohanty and Shaffer Reference Mohanty and Shaffer2019).

To define a GP, consider a function $f\colon \mathcal {X} \to \mathbb {R}$ on some arbitrary domain $\mathcal {X}$ ; for our model of latent opinion, $\mathcal {X} = (-\infty , 0]$ is the span of times at which we may wish to predict. The defining property of a Gaussian process is that if $\mathbf {X} \subset \mathcal {X}$ is a finite vector of input locations, then the associated function values $f(\mathbf {X})$ has a multivariate normal distribution. The moments of this distribution are provided by a mean function $\ \mu (x) = \mathbb {E}[f(x) \mid x]$ and covariance function $K(x, x') = \text {cov}[f(x), f(x') \mid x, x']$ ; evaluating these pointwise provides the mean vector and covariance matrix for any desired vector of function values $f(\mathbf {X})$ . Modeling with the GP entails designing the mean and covariance functions to encoding the desired statistical properties of f such as correlations over the domain.

A critical property of GPs is that they enable exact, closed-form inference for regression for observations corrupted by additive Gaussian noise. Let $f \sim \mathcal {GP}(\mu , K)$ have a GP prior and suppose we obtain a vector of observed values $\mathbf {y}$ at locations $\mathbf {X}$ , where $y_i = f(x_i) + \varepsilon $ , $\varepsilon \sim \mathcal {N}(0,\sigma ^2)$ . Then the posterior belief of f given $D = (\mathbf {X}, \mathbf {y})$ is again a GP with updated mean and covariance function:

$$ \begin{align*} \mu_{f\mid D}(x)&=\mu(x)+K(x,\mathbf{X})(K(\mathbf{X},\mathbf{X})+\sigma^2\mathbf{I})^ {-1}(\mathbf{y}-\mu(\mathbf{X}))\\ K_{f\mid D}(x,x')&=K(x,x')-K(x,\mathbf{X})(K(\mathbf{X},\mathbf{X})+\sigma^2\mathbf{I})^ {-1}K(\mathbf{X},x'). \end{align*} $$

Hence, appealing to the definition above, the posterior predictive distribution of any function value $f(x^*)$ is normal:

$$ \begin{align*} f(x^*)\mid D,x^*\sim\mathcal{N}\big(\mu_{f\mid D}(x^*),K_{f\mid D}(x^*,x^*)\big). \end{align*} $$

This final point is important for our application. When modeling the latent opinion with a Gaussian process, our prediction for latent public opinion on Election Day, $f_i(0)$ , is a normal distribution that can be directly derived. This in turn becomes a normal prior for public opinion that is passed directly into the election-level model. This allows us to include our uncertainty about where public opinion will be on election day into the election-level model while at the same time significantly reducing computation time relative to Linzer (Reference Linzer2013).

3.2 Projecting Public Support via GP Regression

We next outline our approach for forecasting voter preferences throughout an election given polling results. Our approach entails building independent GP models for each race conditioned on available polling outcomes.Footnote ⁶ The model includes only polls, hyperparameters, and priors (discussed below).

Denote by $\mathcal {C}$ the set of all candidates in all races we wish to reason about. We will consider the unknown proportion of voters preferring candidate $i \in \mathcal {C}$ in a given race a function of time, writing $f_{i}\colon (-\infty , 0] \to [0,1]$ . Here, the domain of the function is time (measured in days), where the election is defined to occur at time $t = 0\,\text {days}$ . Let $\mathcal {T}_i$ be the set of times when opinion polls for candidate i were conducted.Footnote ⁷

We model the trend of voter preferences $f_{i}$ as a sum of an underlying linear trend $a_i + b_i t$ , with smooth nonlinear deviations from this trend, $\eta _i(t)$ . We place independent Gaussian priors on the intercept $a_i$ (i.e., the prior mean of the voter preferences on Election Day) and slope $b_i$ of the linear trend, and will place an independent, zero-mean GP prior on the nonlinear component $\eta _i$ . The covariance function K determines the correlation of deviations from the linear trend as a function of time and was taken to be identical across all races. Here, we used the Matérn covariance function with $\nu =\tfrac {3}{2}$ , which models isotropic, once-differentiable functions (Rasmussen and Williams Reference Rasmussen and Williams2006). This covariance function has two hyperparameters that we will estimate from training data: a length scale $\rho $ determining the scale of correlations, and an output scale $\lambda $ determining the pointwise variance of the process. Intuitively, we can think of $\rho $ as determining the “window” of days over which nonlinear deviations are estimated and $\lambda $ as controlling the degree of nonlinearity we expect such that higher values lead to more dramatic deviations.

The model can be summarized as:

(1) $$ \begin{align} f_{i}(t)&=a_i + b_i t + \eta_i(t) \end{align} $$

(2) $$ \begin{align} a_i&\sim\mathcal{N}(\bar{a}_i, \sigma_a^2 = 0.1^2) \end{align} $$

(3) $$ \begin{align} b_i&\sim\mathcal{N}(0, \sigma_b^2 = 0.002^2) \end{align} $$

(4) $$ \begin{align} \eta_{i}(t)&\sim \mathcal{GP}(0, K), \end{align} $$

where the covariance function for the nonlinear deviations is

$$ \begin{align*} K(t,t'; \rho, \lambda)=\lambda^2\Biggl(1+\frac{\sqrt{3}d}{\rho}\Biggr)\exp\Biggl(-\frac{\sqrt{3}d}{\rho}\Biggr); \qquad d=|t-t'|. \end{align*} $$

The priors on the linear parameters are constructed to be broad for the slope (so that over a time period of roughly 100 days the linear trend could plausibly assume any possible value) and vaguely informative for the intercept; we will discuss the intercept mean parameter $\bar {a}_i$ shortly.

The above prior choices induce the following joint prior over the voter preference, as shown in (5). Notice that our model provides an automatic marginalization over the linear slope parameters, since the covariance function in our GP model has absorbed the hyperparameters controlling the prior distribution of the linear function parameters.

(5) $$ \begin{align} f_{i}(t)&\sim\mathcal{N}\bigg(\mu_i(t),V(t,t')\bigg), \end{align} $$

(6) $$ \begin{align} \mu_i(t)&=\bar{a}_i, \end{align} $$

(7) $$ \begin{align} V(t,t')&=\sigma_a^2 + tt'\sigma_b^2 +K(t,t'). \end{align} $$

Our goal is to infer the latent voter preference trend from opinion poll outcomes, which are by their nature noisy. Our approach is to model the observed poll outcomes as binomially distributed, then approximate each binomial with a Gaussian for mathematical convenience. This will allow closed-form exact inference, yielding a posterior GP belief about underlying voter trends conditioned on available data. As discussed in Section 3.4, this step is an important innovation, allowing us to exactly solve for the posterior predictive distribution of $f_i(t)$ .

For a candidate $i\in \mathcal {C}$ with $S_i$ conducted opinion polls, let $\mathcal {D}_i=\{t_{is}, n_{is}, x_{is}\},(s=1,\dots ,S_i)$ denote the outcomes of all available polls involving that candidate. Here, $t_{is}$ is the time of the poll, $n_{is}$ is the sample size of the poll, and $x_{is}$ is the number of polled people expressing support for the candidate. Dropping subscripts momentarily, consider one such polling outcome $(t, x, n)\in \mathcal {D}$ . We make the natural assumption that the number of supporters x is binomially distributed given the sample size n and the true (unknown) voter support f at time t:

(8) $$ \begin{align} x\sim \text{Binomial}\big(n,f(t)\big). \end{align} $$

Unfortunately, it is not possible to condition a GP exactly on observations with a binomial likelihood. However, sample sizes for election polls tend to be large enough (often in the hundreds) that we can safely make a Gaussian approximation to the likelihood by moment matching. Here, we also explicitly consider an additional general noise term $\sigma ^2$ , which designates another level of noise stemming from the polling data. Let $p = x / n$ to be the observed proportion of support in a poll, so (8) could be approximated with

(9) $$ \begin{align} p\sim \mathcal{N}\big(f,\hat{p}(1-\hat{p})/n+\sigma^2\big), \end{align} $$

where we have substituted the estimated $\hat {p}$ for the true unknown proportion $f(t)$ in the variance (in our case, ${\hat {p}=p\ }$ after observation). This likelihood is now conjugate to our GP prior on f and allows exact inference.

Let us define the vector $\mathbf {p}$ to entail a set of polling outcomes observed at times $\mathbf {t}$ , $p_{s}=x_{s}/n_{s}$ , and further define $\mathbf {B}$ to be a $S \times S$ diagonal matrix with $B_{ss} = p_{s}(1 - p_{s})/n_{s}+\sigma ^2$ . This is the approximate noise variance for each of these measurements appearing in (9). Using the results in Section 3.1, the posterior predictive distribution of the voter preference at any time t is:

(10) $$ \begin{align} f(t) \mid D \sim\mathcal{N}\big(\mu_{{f}\mid\mathcal{D}}(t), {K}_{{f}\mid\mathcal{D}}(t,t)\big), \end{align} $$

(11) $$ \begin{align} \mu_{{f}\mid\mathcal{D}_i}(t)=\mu(t)+{V}(t,\mathbf{t})\big(\mathbf{V}+\mathbf{B}\big)^{-1}(\mathbf{p}-\boldsymbol\mu), \end{align} $$

(12) $$ \begin{align} {K}_{{f}\mid\mathcal{D}}(t,t')={V}(t,t')-{V}(t,\mathbf{t})\big(\mathbf{V}+\mathbf{B}\big)^{-1}{V}(\mathbf{t},t'), \end{align} $$

where $\boldsymbol \mu $ and $\mathbf {V}$ are the prior mean and covariance of $f(\mathbf {t})$ , respectively. Although we may make forecasts for any time t, we are especially interested in public opinion on Election Day, $f(t=0)$ . This will also be normal following Equation (10). For notational convenience below, we will again use subscripts for candidates and write $f_i(0) \sim \mathcal {N}\big (\mu _{f_i}, K_{f_i} \big )$ .

The candidate-level model is completed by choosing values for the intercepts $\{\bar {a}_i\}$ and the set of shared hyperparameters $\boldsymbol \omega = (\rho , \lambda , \sigma ^2)$ . Here, $\sigma ^2$ represents the level of unmodeled noise remaining in the polling data, $\lambda $ controls the degree to which the time trend deviates from linearity, and $\rho $ represents the “bandwidth” of the smoothing window for these nonlinear deviations.

We chose informative, but wide hyperpriors for $\{{a}_i\}$ so that projections could be made in races with few or zero polls but that polling data would quickly swamp the prior when plentiful. Since the standard deviation for the hyperprior is set at $0.1$ , any vote share within $\pm $ 30 points of the prior should be well supported.Footnote ⁸ To set $\{\bar {a}_i\}$ , we ran a simple regression in the training set with normalized vote share as the dependent variable and party, lagged partisan vote index (PVI), and level or prior experience as covariates.Footnote ⁹ While not an accurate model by itself, it proved to be an adequate prior.Footnote ¹⁰

For $\boldsymbol \omega $ , we adopt a leave-one-year-out (loyo) cross-validation approach using the training period from 1992 to 2016. The motivation is to choose hyperparameters that maximize predictive performance for election results even at the expense of choosing parameters that reduce fit for the polling data.

First, we define the search region of output scale and shared noise both to be $[0,0.05]$ . We search length scale with a minimum of 7 days and a maximum of 56 days.Footnote ¹¹ Empirically, we generate potential $\boldsymbol \omega $ ’s for the validation procedure from a low-discrepancy Sobol sequence (Sobol Reference Sobol1979) in the search region, since it covers the space more efficiently than a grid. We fit the complete model, including the election-level model, for each of 100 values of $\boldsymbol {\omega }$ at each time horizon ( $\tau $ ) leaving out each year in turn.

For example, for choosing the hyperparameters for the model predicting 4 weeks in advance of the election, we used all of the polling data up to day $t=-28$ . We then fit the GP models and trained the election-level model described below leaving out each cycle in turn. We then generate out-of-sample predictions for vote shares and choose the hyperparameter setting that maximized the loyo log-likelihood averaged across all cycles.

The chosen hyperparameters for each time horizon are shown in Table 1, and examples of the resulting candidate-level models for one candidate (John McCain in 2016) are shown at various time horizons in Figure 2. (Supplementary Appendix E shows another example for Democrat Katie McGinty [PA-2016].) This approach yields models that begin as linear far from Election Day but become increasingly nonlinear as $\tau $ approaches zero. Note also that the uncertainty in $f_i(0)$ narrows considerably in run up to Election Day.

Table 1

Learned model hyperparameters from leave-one-year-out cross validation.

3.3 Election-Level Model

The goal of the candidate-level model (Section 3.2) is to project forward at any time horizon a predictive distribution of latent public support on Election Day, $f_i(0)$ . The election-level model in this section takes $f_i(0)$ as an input and combines it with additional contextual factors to generate a predictive distribution. Our method is based on Dirichlet regression, that allows prediction of the election vote shares for multiple candidates.Footnote ¹²

Figure 2

Voter Preference Estimate for John McCain in 2016 Arizona race at various time horizons. Points represent individual polls, the distribution on the right side of each panel is the prior, the dark gray region is the 95% confidence intervals for the estimated latent trend, and the light-gray region is the projected latent trajectory.

In our setting, the vast majority of races involve only two credible candidates.Footnote ¹³ Indeed, in the 1992–2018 period, we included more than 2 candidates in only 11 elections.Footnote ¹⁴ However, we retain the Dirichlet presentation here as being more general and races with third parties can be critical in any given cycle.

Relying on the Dirichlet likelihood contrasts with some work in political science for multi-party elections, which builds on the logistic normal distribution (or t-distribution) applied to log-ratios of the votes (Katz and King Reference Katz and King1999) or seemingly unrelated regression (Tomz, Tucker, and Wittenberg Reference Tomz, Tucker and Wittenberg2002). The primary criticism of Dirichlet regression is that it assumes that ratios of outcomes are independent (Aitchison Reference Aitchison1982; Katz and King Reference Katz and King1999; Philips, Rutherford, and Whitten Reference Philips, Rutherford and Whitten2016), which is unrealistic in more standard settings such as multi-party elections. However, while the outcome in U.S. Senate races is always compositional, the meaning of the categories do not correspond across races as these alternative models require. That is, the “third choices” are typically idiosyncratic to each race. So, for instance, the third-party candidate in the 2018 New Mexico race was Libertarian Gary Johnson while in the 2008 Minnesota race it was Independence Party candidate Dean Barkely. In other cases, even the major party candidate labels can be confused. So, for instance, in the 2012 Maine election Cynthia Dill was the official Democratic nominee while Independent Angus King garnered a significant amount of support from Democrats and caucused with the party once he joined the senate. Indeed, in some instances the category meanings are unstable even when there are only two choices. For example, the 2016 California race featured two Democrats. We therefore retain the Dirichlet regression approach despite the independence assumption since modeling the dependence between the choice categories is impossible when the choice set itself changes from observation to observation.Footnote ¹⁵

We model the parameters in the Dirichlet distribution as a linear function of voter preferences and “fundamentals.” This is similar in nature to other generalized linear models where a linear combination of terms is passed to a link function. In this case, each candidate is represented by a “concentration parameter,” $\alpha _i$ , which we model as a linear combination of both $f_i(0)$ and other covariates. The unique feature of the Dirichlet regression is that each race is characterized by a vector of concentration parameters, $\mathbf {\alpha }$ , where we have one $\alpha _i$ for each candidate. When the concentration parameter for candidate i is relatively large, she is expected to earn a higher proportion of the vote. Furthermore, the predictive density is more concentrated around this expected value when the individual components of $\mathbf {\alpha }_j$ are large.

More formally, consider arbitrary race j with $m_j$ candidates and a specific candidate i. We assume a simple linear model that maps the voter preference $f_{i}(0)$ to the underlying concentration parameters $\alpha _{i}$ . Although there are many possible covariates we could include, we found that very few actually improved out-of-sample predictive performance.Footnote ¹⁶ We therefore include only the lagged PVI generated by Cooks political report (Campbell Reference Campbell2018; MacWilliams Reference MacWilliams2015), and an indicator for the experience of the candidate where a one indicates the candidate has held elected office before and it is coded zero otherwise (Jacobson Reference Jacobson1989; Jacobson and Carson Reference Jacobson and Carson2019). We also include a year-level random effect to accommodate unmodeled electoral “swings” associated with specific election cycles. PVI and the year random effects are reverse coded by party.

More formally, collect $\boldsymbol {\alpha }_j=(\alpha _{1j}+\tilde {\alpha },\dots ,\alpha _{m_j j}+\tilde {\alpha })$ from all candidates in the race ( $\tilde {\alpha } \ge 0$ ). The base parameter $\tilde {\alpha }$ here is introduced for two reasons. First, it can reduce variance of samples and thus stabilize the MCMC sampling. Second, $\tilde {\alpha }$ encodes the prior belief on how equally the vote shares should be distributed without any additional information. We assume that the actual vote share vector $\mathbf {y}_j=(y_{1j},\dots ,y_{m_j j})^\top $ is distributed with a Dirichlet distribution $\mathbf {y}_j\sim \text {Dir}(\mathbf {y}_j; \boldsymbol {\alpha }_j)$ , where $\alpha _{ij}$ is a linear function of $f_i(0)$ and contextual predictors. We also need to integrate over the distribution of $f_i(0)$ ; in our case, the distribution of $f_i(0)$ is a truncated Gaussian. In total, we assume the election outcomes follow the following data generating process:

(13) $$ \begin{align} f_i(0) \sim \mathcal{N}\big(\mu_{f_i}, K_{f_i} \big),\quad 0\le f_{i}\le 1, \end{align} $$

(14) $$ \begin{align} \alpha_{ij}= \theta_1 f_{i}(0) + \theta_2 \text{party}_{ij}\times\text{pvi}_j+\theta_3\text{experience}_{ij}+\text{party}_{ij}\times \gamma_{\text{year}},\quad \alpha_{ij}\ge 0, \end{align} $$

(15) $$ \begin{align} \gamma_{\text{year}} \sim\mathcal{N}(0,\sigma^2_{\text{year}}), \end{align} $$

(16) $$ \begin{align} (y_{j},\dots,y_{m_j})^\top\sim \text{Dirichlet}(\tilde{\alpha}+\alpha_{1},\dots,\tilde{\alpha}+\alpha_{m_j}),\quad \tilde{\alpha}\ge 0. \end{align} $$

Here, we allow party to be equal to 1 for Democratic candidates, $-1$ for Republicans, and 0 for independents.Footnote ¹⁷ This allows for PVI and the year random effects to have opposite effects by party.

The model is completed by placing proper but vague priors across all parameters. The priors for the $\theta $ parameters are set to be wide based on the scale of the relevant variable. Specifically, we set independent truncated Gaussian priors on the $\theta $ coefficients and the year-level random effects.

$$ \begin{gather*} \begin{array}{c c} \theta_1 \sim \mathcal{N}(0, 100^2);\ \theta_1> 0 & \theta_2 \sim \mathcal{N}(0, 10^2);\ \theta_2 > 0, \\ \theta_3 \sim \mathcal{N}(0, 10^2);\ \theta_3 > 0 & \sigma_{\text{year}}^2 \sim \text{Gamma}(1, 0.5). \end{array} \end{gather*} $$

We can combine all of these parameters together in $\boldsymbol \Theta = \big (\{\theta \}, \{\gamma _{\text {year}}\}, \sigma _{\text {year}}^2, \tilde {\alpha }\big )$ and let $\mathbf {z}_j$ be the vector of contextual factors for election j. We obtain the posterior $p(\boldsymbol \Theta \mid \{\mathbf {y}_j \},\{\mathbf z_j\}, \{f_i(0)\})$ using MCMC estimation. Specifically, we use no-U-turn sampling in Stan (Carpenter et al. Reference Carpenter, Gelman, Hoffman, Lee, Goodrich, Betancourt, Brubaker, Guo, Li and Riddell2017; Hoffman and Gelman Reference Hoffman and Gelman2014).

With this posterior, the final predictive distribution of future election outcomes with new $\{\mathbf {f}_i^*(0)\}, \{\mathbf {z}_j^*\}$ will be defined by (13)–(16) marginalized by the posteriors:

(17) $$ \begin{align} &p\big(\mathbf{y}_j\mid\{\mathbf{z}^*_j\}, \{f_i(0)^*\}, \{\mathbf{y}_j \},\{\mathbf z_j\}, \{f_i(0)\}\big) \nonumber \\ &\quad =\int p(\mathbf{y}_j\mid \{\mathbf{z}^*_j\}, \{f_i(0)^*\}, \boldsymbol \Theta)p(\boldsymbol \Theta \mid \{\mathbf{y}_j \},\{\mathbf z_j\}, \{f_i(0)\} )d\boldsymbol{\Theta} \end{align} $$

A final issue is how to handle the dynamic nature of our forecasting task. While we have the complete set of polls for elections in our training set, when making real-time forecasts we have only the polls up to the current date. Training the model on the complete set of polls (all the way up to Election Day) is likely to lead to higher weight being assigned to polling data and poor predictive performance at remote time horizons. For instance, the coefficients for the Dirichlet regression component in the election model may put too much confidence on the polling. As noted above, this same issue applies to hyperparameter selection for the candidate-level model.

Figure 3

Posterior predictive densities for Sen. John McCain in the 2016 Arizona Election at different time horizons. The vertical line indicates the final vote share. Note that the posterior narrows noticeably as the time horizon shrinks.

To address this concern, we train the complete model at various time horizons denoted by $\tau $ . For any threshold, we discard all data where $|t|<\tau $ . Thus, when $\tau =28$ , we ignore all polls in the training data closer than 28 days to the election. This again helps calibrate the model for the levels of accuracy we can expect at various horizons. Table F.1 in Supplementary Appendix F shows the summaries for the posteriors of the model parameters at horizons ranging from $\tau =0$ to $\tau =56$ (8 weeks before the election). As expected, the $\theta $ parameter associated with $f_i(0)$ increases as Election Day approaches while the fundamental parameter become relatively less important.

Figure 3 shows the posterior prediction for Senator John McCain in 2016 for various time horizons. Note that the outcome (marked with the vertical blue line) is near the center of the posterior for all horizons, but that the prediction becomes more concentrated as Election Day nears. This reflects both more certainty in $f_i(0)$ and changing weights in the Dirichlet regression.

4.1 Data and Evaluation Criteria

We obtained opinion polls and election outcomes of all senate election races from 1992 to 2018 from www.fivethirtyeight.com and from CNN. On average 16 polls were conducted for each race, although some races such as the 2016 Florida election had over 80. Most of the surveys are conducted 2 weeks to 4 months prior to election, with a median number of respondents of 635. Over 470 entities have conducted these polls, but several active pollsters collectively contribute over half of them, including Rasmussen Reports, Mason-Dixon Polling, Public Policy Polling, SurveyUSA, YouGov, SurveyMonkey, Quinnipiac, and Zogby Interactive. To guarantee credibility, we eliminated polls sponsored by parties or candidates since unfavorable polls from these sources are not released.

We also acquired the partisan voting indices in every election cycle for each election state. The Cook PVI is a measurement of how strongly a U.S. congressional district or state leans toward the Democratic or Republican Party, compared to the nation as a whole. For example, PVI for California in 2018 is 10.76, indicating a strong preference for Democratic candidates, while PVI for the pro-republican state Texas in 2018 is $-7.02$ . For each candidate, we coded partisan affiliation and past experience (whether or not they held office). Where not provided in the CNN data, we coded these manually using ballotpedia.com.

To evaluate performance, we examine both the forecasting precision and the validity of our model. Hence, we consider the following measures: the averaged root-mean-squared-error (RMSE) between the expectation of the Dirichlet posterior samples and actual vote shares, the prediction accuracy of winners for election defined by higher winning probability (we calculate the winning probability of each candidate as the proportion of samples with the highest vote shares in Dirichlet posteriors), the coverage rate of actual vote shares in 95% credible intervals for Dirichlet posteriors, the averaged multinomial predictive likelihood and the averaged log-scaled Dirichlet predictive likelihood (LL). RMSE and prediction accuracy focus on the precision of the forecasting ability, while coverage rate focuses on the validity of the claimed credible intervals. The two likelihood measures serve as out-of-sample evaluation criteria for both vote share (Dirichlet) and final outcome (multinomial) that also reflect the uncertainty in the full posterior.

4.2 Baselines

We compare the performance of our combined GP and Dirichlet regression (GP+DR) model against three benchmarks. First, we compare our model to the dynamic Bayesian model in Linzer (Reference Linzer2013). This model was developed for predicting state-level results in presidential elections, but we adjusted it for predicting Senate races. Intuitively, this model is a dynamic Bayesian random walk (BRW) model similar to the nonlinear component in the model described above except that latent public opinion is assumed to be a random walk. We use the same informative prior for $\{{a}\}$ as used above and use the tuning parameters and basic estimation procedures as described in Linzer (Reference Linzer2013).Footnote ¹⁹

Second, we consider a baseline Dirichlet Regression model that uses a Bayesian linear regression model to forecast voter preferences. We refer the second baseline as LM+DR. To ensure a fair comparison, we choose the same priors for the linear coefficients as those in GP priors. We also chose the $\sigma ^2$ hyperparameter using the same cross-validation approach described above. This model is, in essence, the same as what we describe above without allowing for deviations from linearity. Finally, we examine the performance of the GP model in isolation excluding the Dirichlet regression portion of the hierarchy. Note that while we frame these as competitors to our favored model, both of these baselines are also novel.