A Bayesian Treatment of Self-Consistency Voting in Language Model Reasoning

1. Introduction

Self-consistency [Wang et al. 2022] has emerged as one of the simplest and most effective inference-time techniques for chain-of-thought reasoning: sample $K$ rationales independently from the same prompt and return the modal final answer. Despite its widespread use, self-consistency provides only a point estimate. There is no principled way to express the difference in confidence between, say, an 8/10 majority and a 4/10 plurality, even though their reliability characteristics differ markedly.

In this paper we recast self-consistency voting as posterior inference and derive a small set of closed-form formulas that turn vote counts into calibrated probabilities and credible intervals.

2. Generative Model

For a fixed prompt $x$, suppose the model induces a distribution $\boldsymbol{\theta} = (\theta_1, \dots, \theta_M)$ over $M$ candidate final answers, where $\theta_m$ is the probability that an independent sample produces answer $m$. We do not observe $\boldsymbol{\theta}$; we observe vote counts $\boldsymbol{n} = (n_1, \dots, n_M)$ from $K = \sum_m n_m$ independent samples.

We place a Dirichlet prior $\boldsymbol{\theta} \sim \mathrm{Dir}(\boldsymbol{\alpha})$ with weakly-informative concentration $\boldsymbol{\alpha} = \alpha_0 \mathbf{1}$. By conjugacy,

$$\boldsymbol{\theta} \mid \boldsymbol{n} \sim \mathrm{Dir}(\boldsymbol{\alpha} + \boldsymbol{n}).$$

The posterior probability that answer $m^*$ is correct, if we identify correctness with mode in the population sense, is

$$\Pr[m^* = \arg\max_m \theta_m \mid \boldsymbol{n}] = \int_{\Delta^{M-1}} \mathbb{1}[\theta_{m^*} = \max_m \theta_m] , p(\boldsymbol{\theta} \mid \boldsymbol{n}) , d\boldsymbol{\theta}.$$

This integral admits a tractable Monte-Carlo estimate, but for the binary-relevant case (top-1 versus the rest) the marginal is a Beta:

$$\theta_{m^$$} \mid \boldsymbol{n} \sim \mathrm{Beta}(\alpha_0 + n_{m^}, , (M-1)\alpha_0 + K - n_{m^*}).

3. Calibration

The naive baseline reports confidence $n_{m^*} / K$. Our posterior-mean estimate is

$$\hat{c}$${\text{Bayes}} = \frac{\alpha_0 + n{m^*}}{M \alpha_0 + K}

with a $1 - \delta$ credible interval from the Beta quantile function.

We choose $\alpha_0$ by maximum marginal likelihood on a held-out calibration set per benchmark; values cluster between 0.4 and 1.1 across our domains.

from scipy.stats import beta

def bayes_confidence(n_top, K, M, alpha0=0.7):
    a = alpha0 + n_top
    b = (M - 1) * alpha0 + K - n_top
    mean = a / (a + b)
    lo, hi = beta.ppf([0.025, 0.975], a, b)
    return mean, (lo, hi)

4. Experimental Setup

We evaluate on six benchmarks: GSM8K, MATH, MMLU-pro, ARC-Challenge, BIG-Bench-Hard, and a private code-debugging set. For each we sample $K = 16$ rationales per problem from a 70B open-weights model.

We report Expected Calibration Error (ECE), Brier score, and the area under the selective-prediction curve (AURC).

5. Results

The Bayesian estimate reduces ECE by a factor of 3-5x consistently. The improvement is most pronounced when the vote count is small or when $M$ (the empirical answer cardinality) is large.

5.1 Selective Prediction

Using the Bayesian confidence as a deferral score, we obtain a coverage-accuracy curve dominating the vote-fraction baseline at every coverage level above 0.5. At a deferral budget of 18% of inputs, accuracy on accepted inputs rises from 64.0 to 73.6 on MATH.

6. Discussion

The key insight is that vote fractions are not probabilities — they are maximum-likelihood estimates with noise that depends on $K$ and on $M$. Adding a small Dirichlet pseudocount produces a shrinkage estimator that calibrates well even at the small $K$ commonly used in production.

A philosophical caveat: we identified correctness with population mode, which is not always the same as ground-truth correctness. When the model has a systematic bias toward a wrong answer, Bayes confidence will be high and wrong. We see this rarely in practice (less than 4% of GSM8K errors) but it bounds achievable calibration.

7. Limitations

The approach assumes independent samples. Sequential decoding with shared prefix or KV-cache shortcuts may induce correlation; in such cases $K$ should be replaced by an effective sample size. We have not investigated this empirically.

We also assume a flat prior on $\boldsymbol{\theta}$. Hierarchical priors that pool across problems within a benchmark may yield further gains and remain future work.

8. Conclusion

A two-line modification turns self-consistency into a calibrated, uncertainty-aware procedure. We hope this encourages reporting of full confidence distributions rather than point answers, particularly in settings where downstream policy depends on confidence.

References

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