# Forecasting elections in multiparty systems: a Bayesian approach combining polls and fundamentals

Uses the backward random-walk approach taken in Linzer2013 and Strauss2007, and includes two models that are fully integrated:

• The fundamentals-based model that provides a forecast for each party's results way ahead of the election;
• The measurement model that estimates the current level of party support (poll agregator).

$$\mathbf{y}_{ct} \sim \operatorname{Multinomial}(\mathbf{\pi_{ct}^{*}}, N_{ct})$$

The latent underlying support is normalized to sum to $$1$$ using a log-transform:

\begin{align*} \mathbf{\pi}_{ct} &= \left(\log\left(\frac{\pi^{*}_{ct1}}{\pi^{*}_{ctP}}\right), \dots, \log \left(\frac{\pi^{*}_{ct(P-1)}}{\pi^{*}_{ctP}}\right)\right)\\ &= \alpha_{t} + \delta_c \end{align*}

Q: Why not using a $$\operatorname{Softmax}$$ function instead?

And $$\alpha_t = \alpha_{t+1} + \omega_t$$ where $$\omega_t \sim \operatorname{Normal}(0, W)$$, and the $$W_{i,j}$$ covary.

The latent state of party support $$\alpha_T$$ on election day (with predictors $$x_E$$ that day) is given by

$$P(\alpha_{T}|x_{E}) = \operatorname{alr}\left(\int_{\theta}P(v_{E}|\theta, x_{E})P(\theta|V,X)\mathrm{d}\theta\right)$$

In other words, the prediction of the fundamental model and the intentions coimputed by the poll aggregator should somehow coincide. Joint posterior distribution of both the fundamentals model and the dynamics poll modelL

\begin{align*} \mathcal{L}_{poll} &= P(Y|\alpha, \delta, W)\,\left(\prod_{t=1}^{T} P(\mathbf{\alpha}_{t}|\mathbf{\alpha}_{t+1}, W)\right)\,P(W)\,P(\delta)\\ \mathcal{L}_{fund} &= P(V|\theta, X)\, P(\theta)\\ \mathcal{L} &= \mathcal{L}_{fund}\,\mathcal{L}_{poll}\,P(\alpha_T|x_E) \end{align*}