Who will win the 2027 French presidential election?

The forecast is a Bayesian poll-aggregation model in the tradition of Nate Silver's FiveThirtyEight and the Economist's 2020 model (Gelman, Morris & Heidemanns), which in turn build on Linzer (2013). I wanted to adapt these to France's two-round system. In this system, many candidates enter the first round and then, either (i) one candidate receives an absolute majority of the votes and is elected, or (ii) the two candidates who receive the highest number of votes in the first round advance to the runoff, which takes place two weeks later. My model has two stages: a first-round vote-share model and a runoff model. It is a Bayesian model, which requires setting priors. Some are deliberate regularisation choices (e.g. the fixed random-walk step and the runoff-strength prior); the key forecast-uncertainty parameters — how fast uncertainty grows toward election day, and the irreducible election-day poll error — are calibrated on the 2012, 2017, and 2022 elections. Three cycles is not a lot of history, so these are estimated with real uncertainty. Everything is fit in Stan. This page recomputes the forecast for the candidate line-up you select from the posterior draws.

This is work in progress and I appreciate your feedback. Credits (but no responsibility for my errors) are due to Ronald Neath, who taught the Bayesian Statistics course I took in the Fall of 2023. I used LLMs to think through part of the modeling, and for a lot of the dynamic visualization (I have no prior D3.js expertise). The data comes from Wikipedia.

1 · First round: latent support and random walk

Consider candidates c = 1,…,C over weeks w = 1,…,W (where W is the election week). β_c,w is the latent support of candidate c in week w, on an unbounded scale.

The latent support of each candidate "wanders" randomly from week to week. This is the "random walk" from Linzer (2013) commonly found in other forecasts. Each week, the random walk adds a value randomly drawn from a Normal distribution to the previous week's latent support of each candidate. Here the weekly step σ is fixed (this is a smoothing choice, it's not estimated from the data): it controls how much the latent line may move between polls while I estimate where the candidates stand today. This in-sample walk is distinct from the forecast-horizon uncertainty (how far opinion can still drift from the last poll to election day), which is governed by a separate, historically-calibrated random walk described in Section 3, and is what actually makes the bands widen into the future.

So the latent support of candidate c in week w is:

The overall level of the latent supports is not identified by the softmax, as adding the same constant to every β_c,w would leave all shares unchanged. Each week is therefore softly anchored with a sum-to-zero constraint: ∑_c β_c,w ∼ Normal(0, 0.1²).

2 · Estimation of the latent support

French polls test different line-ups: one poll may or may not include a given candidate. A poll i fielded in week w_i by pollster p_i reports shares only over the candidate subset S_i present in that hypothesis. I model it as the softmax over that subset S_i of the latent supports, plus a pollster effect δ_p,c.

For each candidate c in poll i in week w, I call η_i,c the sum of the latent support and the pollster effect.

Softmax turns the latent η_i,c into shares θ_i,c:

Each poll is then a draw from a Multinomial distribution with ñ_i effective respondents and the vector of probabilities θ_i for candidates c∈S_i:

Because I condition each poll on their restricted line-up, all polls provide conditional shares used to inform the same β. The sample sizes are deflated to an "effective" size ñ_i reflecting "design effects" and the fact that pollsters test multiple line-ups in the same polls:

3 · Calibration from history, and the forecast

By comparing polls to actual electoral results in the 2012, 2017, and 2022 presidential elections, I estimate how the (squared) error of polls grows with horizon h (time to the election). I use this to calibrate: (i) the parameter of a random walk making variance grow over time, and (ii) an "election-day miss":

where a is the election-day miss and √c the daily step of the random walk. I fit this per cycle, weighting each election equally. The three cycles genuinely disagree, with election-day misses of roughly 1, 3 and 5 points respectively. Rather than picking one, I treat 2027 as new: every simulation draws (a, √c) from a heavy-tailed predictive across the three cycles (central election-day error ≈ 3 points, drift ≈ 0.2 pt/day). This helps propagate the uncertainty from estimating the election-day miss using three elections only.

For the forecast, I take the estimate of the latent state at the last poll week w^∗ and add (i) the random-walk drift to election day and (ii) the systematic election-day miss. That component does not shrink with more polls or as election day gets closer.

These elements of horizon spread are added to the latent support of the candidate β̃_c, before it is transformed into shares using the softmax function.

The share for a chosen line-up 𝒜 is the softmax over just those candidates, π_c = softmax_𝒜(β̃_c). The drift and election-day miss standard deviations are the historical quantities discussed above, but mapped from the "shares" scale to the "latent" scale by κ = 1⁄[π(1−π)] ≈ 5.9 (evaluated at a typical π ≈ 0.22), i.e. s_rw = √c · √7 · κ and s_err = a · κ.

4 · Runoff: Bradley–Terry model

I use a paired-comparison model (Bradley & Terry, 1952) that gives every candidate a “strength”, and in which the chance that A beats B depends on the difference in strengths. The model pools head-to-heads and predicts unpolled pairings by transitivity: if A beats C handily and B only narrowly, we can estimate A vs. B. For each candidate's latent runoff strength s_c, a head-to-head poll of A vs. B gives A's expected share as

with s_c ∼ Normal(0, 1.5²) (with a soft sum-to-zero) and over-dispersion φ ∼ Gamma(2, 0.05).

Bradley–Terry pools sparse matchup polls and predicts any pairing through shared opponents. For a runoff between the first-round top two X, Y, I draw each strength from its full posterior and add a horizon inflation σ₂ = √(s_err² + s_rw,day² H) to get X's second-round share. This is a single Normal shock on the strength difference s_X − s_Y — not two independent per-candidate shocks, which would widen the gap by a further √2.

X wins the runoff if that share is larger than 0.5.

Because each candidate has a single runoff strength, a poll of Philippe vs. Bardella sharpens s_Philippe, which in turn tightens the predicted Philippe vs. Le Pen result. Information works through the fact that Le Pen and Bardella have a shared opponent. This is the Bradley–Terry transitivity. This is different from the first round, in which Le Pen and Bardella have separate latent support with no party-level pooling. They are comparable only because they face the same opponents.

5 · What the dots are

Each dot in the plots represents one simulated election. For each election, the simulation draws β̃, s, a latent support and runoff strength for each candidate, from the corresponding posterior distributions. After adding the random walk drift, the election-day miss, and the runoff noise, it then calculates the first-round shares π, takes the top two, calculates their runoff share p_X and awards the win to the candidate that clears 0.5. (No first-round candidate comes anywhere near an outright majority, so the simulator always proceeds to a runoff. The direct first-round win mentioned at the top is not separately implemented.) Repeating over the posterior generates the posterior predictive distributions displayed. The trajectory bands are week-by-week quantiles of π, and each dot in the "who wins" and "1,000 simulations" plots is one such draw.

6 · Assumptions and limitations

• Shares are only estimated among expressed votes. Turnout and indecision are not included in the model. The reported shares are renormalised over modelled candidates: undecideds are assumed to split like decideds.
• The line-up is a scenario. I don't make any prediction about primary results and the final line-up.
• Independent election-day miss. The election-day miss is applied to each candidate independently in latent space, before the softmax. It is not added as a correlated national factor. When I check against history, I find that at the current horizon the model's candidate-gap errors are slightly narrower than in the 2012, 2017, and 2022 record.
• Runoff reuses first-round error calibration. The runoff horizon shock σ₂ reuses the election-day miss and drift calibrated from first-round history. Applying them to the two-candidate runoff logit is an additional assumption, not something Bradley–Terry implies.
• Calibration over only three elections. The forecast-uncertainty parameters rest on just three elections, I try to propagate their uncertainty but three elections provide a thin historical basis.
• Polls-only. I use no economic fundamentals, incumbency, approval ratings, or other political variables.

Calibrated quantities