Who will win the 2027 French presidential election?

Bayesian poll-aggregation forecast. This is work in progress, I would love your feedback.

How to read this forecast

All panels are interactive: hover over the plots to read more information.
This model uses polls to simulate the upcoming French presidential election 1,000 times.
The panel Who wins the presidency? reports the share of simulations won by each candidate.
The panel First-round vote intention over time shows the polls used in the model, which tell us where candidates appear to stand today.
The panel First-round in 1,000 simulated elections displays the first-round scores in 1,000 simulations.
The last panel, Most likely runoff match-ups, shows the simulated runoff results in the most frequent pairings coming out of the simulated first rounds.
Note 1: past French elections tell me how uncertain the estimates should become between now and election day.
Note 2 candidate withdrawals are scenarios, not just predictions. They layer an explicit assumption about where a departing candidate's voters go on top of the poll-driven forecast.

Who wins the presidency?

Each dot is one simulated election, placed in the row of the candidate who won it — so the width of each cloud is that candidate's chance of winning. Hover a dot to see that simulation's first-round result and the runoff it produced. ()

First-round vote intention over time

Lines are the model's estimate; shaded bands are the 80% credible interval, widening toward election day. Dots are individual polls, each shown at the candidate's reported share (including alternatives not in the selected line-up).

First round in 1,000 simulated elections

Each dot is one simulated first-round result. The two candidates with the highest scores advance to the runoff, so candidates clustered near the top are fighting for the second qualifying spot.

Most likely runoff match-ups

The runoffs that occur most often across simulations. Each dot is one simulated runoff, placed by the RN candidate's second-round share and coloured by the winner; the dashed line is the 50% threshold. Hover a dot for that runoff's result.

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.

Latent support scale The vote shares range from 0 to 1 by definition, and add up to 1. This is hard to model directly: for instance, if you know that a candidate's vote share increases, it mechanically means that other candidates' vote shares decrease, but which ones? I instead estimate an unconstrained "latent support" number (latent because it is not observed) for each candidate, ranging from -∞ to ∞. I then turn the set of latent supports for each candidate back into shares using a "softmax" function. I can use Normal priors on this unconstrained latent scale. I explain how I estimate β_c,w in the next section.

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:

β_c,w = β_c,w−1 + σ ε_c,w,
With priors ε_c,w ∼ Normal(0, 1) and β_c,1 ∼ Normal(0, 1.5²).

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.

Pollster effect Pollsters may have persistent leans: one firm consistently giving a particular candidate a point or two higher because of its methods and weighting. Pollster effects use a hierarchical scale δ_p,c ∼ Normal(0, σ_house²) and are soft-centred (with ∑_p δ_p,c ≈ 0), so β tracks the average pollster.

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.

η_i,c = β_{c,w_i} + δ_{p_i,c}

Softmax function The softmax function (a generalization of the logistic function) turns a vector of unbounded latent numbers into positive shares that sum up to 1.

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

θ_i,c = e^η_i,c ∑_{c′∈S_i} e^η_i,c′ , c ∈ S_i

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:

y_i ∼ Multinomial(ñ_i, θ_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:

ñ_i = n_i d_eff · m_i

Design effect & effective sample size A real poll varies more than a simple random sample of the same size because of their quotas, weighting approaches, and more. I divide the sample size by a design effect d_eff ≈ 2.0 that measures how much more variation there is to obtain an effective sample size. I also divide the sample size by m_i, the number of line-ups tested in one survey i. Five hypothetical line-ups from a single sample therefore are not counted as five independent polls. FiveThirtyEight also scales each poll's effective sample size by pollster quality (ratings). Because ñ_i is generally not a whole number, the per-candidate effective counts ñ_i θ_i,c are rounded to integers for the multinomial likelihood.

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":

Var (poll error at horizon h) ≈ a² + c h,

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.

What the error is measured on I fit a² + c h to the squared error of individual polls (each poll's candidate share minus the eventual result), which comes down to one fit per cycle. So the intercept a² is the residual spread of a single poll on election day: it absorbs ordinary sampling noise and pollster effects, not only the irreducible aggregate miss.

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.

Forecast-horizon spread Two elements widen the forecast toward election day: opinion is still drifting (the random-walk term, growing at a speed of √time), and even final polls miss by a bit. To reflect this, I use a candidate-specific election-day polling error (the a term, which does not shrink no matter how many polls are averaged). I apply this miss to each candidate independently (see section on limitations at the end). Measuring both from past cycles is how the model knows to be wide now and tighter in April. The Economist model adds an explicit election-day polling-error term for exactly this reason, and Nate Silver likewise widens his intervals when polls are sparse or inconsistent, so the model isn't overconfident.

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.

β̃_c = β_c,w^∗ + s_rw √(W−w^∗) ζ_c + s_err υ_c,
ζ_c, υ_c ∼ Normal(0,1).

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.

Removing a candidate is a scenario, not a renormalisation For the default field, and for swapping one bloc nominee for another, the line-up share is exactly the softmax above. But when a candidate is withdrawn, their support is not simply left to renormalise proportionally over the rest (which is all the softmax denominator would do). It is reallocated to the remaining candidates using estimated head-to-head transfers where the data support them, and otherwise by inverse-distance weighting on the left–right axis with an abstention outlet (as described in the Parameters & assumptions section). This is why a withdrawal is flagged on the page as an explicit assumption rather than a poll-driven prediction.

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

p = logit⁻¹(s_A − s_B),
k_A ∼ BetaBinomial(ñ, pφ, (1−p)φ),

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

Overdispersion Overdispersion happens when the data has more variance than a given model would predict. Modelling the count k_A with a beta-binomial (with concentration parameter φ) lets runoff polls be more spread out than a standard binomial distribution would imply. This is standard for survey counts.

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.

p_X = logit⁻¹(s_X − s_Y + σ₂ ξ),
ξ ∼ Normal(0,1),
X wins ⇔ p_X > ½.

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.
Softmax independence. Two candidates never polled together are placed on a common scale only through shared opponents. For true substitutes (e.g., the two RN leaders, or Attal/Philippe for the centre), the softmax would overstate their combined support. This is why I don't allow both to be included in the picker.
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

quantity	symbol	value
election-day poll error (standard deviation)	a	≈ 3 pts (central; drawn per sim)
opinion drift (random walk)	√c	0.19 pt/day
pollster house-effect sd	σ_house	0.0107 pt
design effect	d_eff	2.0
weekly RW step (fixed, smoothing)	σ	0.045 (latent)
share→latent conversion	κ	≈ 5.9