Recognition: 2 theorem links
· Lean TheoremReasoning with Sampling: Your Base Model is Smarter Than You Think
Pith reviewed 2026-05-17 03:11 UTC · model grok-4.3
The pith
A simple iterative sampling algorithm using only a base model's likelihoods can elicit reasoning performance that nearly matches or exceeds reinforcement learning on tasks like math and coding.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper shows that an MCMC-style iterative sampler operating on a base model's likelihoods generates substantially better reasoning trajectories on single-shot tasks than standard sampling, achieving results that nearly match and sometimes exceed those obtained from reinforcement learning post-training while preserving greater output diversity.
What carries the argument
An iterative sampling procedure inspired by Markov chain Monte Carlo that repeatedly uses the base model's own likelihoods to reshape and refine reasoning outputs without any external guidance.
If this is right
- Reasoning gains on math, coding, and science questions become available at inference time without any model updates.
- Generated solutions maintain higher diversity across repeated draws than is typical after reinforcement learning.
- The same procedure works on multiple base models and across tasks that do not require hand-curated data.
- No external verifier or reward model is needed to obtain the reported performance lifts.
Where Pith is reading between the lines
- If the pattern holds, reinforcement learning may largely be surfacing and amplifying capabilities already latent in base models rather than installing entirely new ones.
- The approach could be tested on open-ended or non-verifiable tasks where traditional RL is harder to apply.
- Combining the sampler with other inference-time techniques such as self-consistency or tree search might produce further additive gains.
- Model developers could focus more on base pretraining quality if sampling methods prove sufficient for many reasoning needs.
Load-bearing premise
The base model's likelihoods contain enough useful signal to let a simple iterative sampler turn them into higher-quality reasoning paths without training or an external verifier.
What would settle it
Running the iterative sampler on a base model for the MATH500 benchmark and finding no gain or a clear drop in accuracy compared with ordinary sampling would falsify the central claim.
read the original abstract
Frontier reasoning models have exhibited incredible capabilities across a wide array of disciplines, driven by posttraining large language models (LLMs) with reinforcement learning (RL). However, despite the widespread success of this paradigm, much of the literature has been devoted to disentangling truly novel behaviors that emerge during RL but are not present in the base models. In our work, we approach this question from a different angle, instead asking whether comparable reasoning capabilites can be elicited from base models at inference time by pure sampling, without any additional training. Inspired by Markov chain Monte Carlo (MCMC) techniques for sampling from sharpened distributions, we propose a simple iterative sampling algorithm leveraging the base models' own likelihoods. Over different base models, we show that our algorithm offers substantial boosts in reasoning that nearly match and even outperform those from RL on a wide variety of single-shot tasks, including MATH500, HumanEval, and GPQA. Moreover, our sampler avoids the collapse in diversity over multiple samples that is characteristic of RL-posttraining. Crucially, our method does not require training, curated datasets, or a verifier, suggesting broad applicability beyond easily verifiable domains.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes an MCMC-inspired iterative sampling procedure that reshapes reasoning trajectories at inference time by using only the base LLM's token-level likelihoods for acceptance decisions. The central claim is that this training-free method produces substantial accuracy gains on single-shot reasoning benchmarks (MATH500, HumanEval, GPQA) that nearly match or exceed those obtained from RL post-training, while preserving sample diversity and requiring neither curated data nor an external verifier.
Significance. If the empirical results and the underlying assumption about likelihood signal hold, the work would be significant: it would demonstrate that advanced reasoning behaviors are already latent in base models and can be elicited by inference-time sampling rather than expensive RL, while avoiding the diversity collapse typical of RL-tuned models. The absence of a verifier requirement would also broaden applicability to domains without easy verification.
major comments (2)
- [§3] §3 (Algorithm description): The acceptance step is defined solely via likelihood ratios drawn from the base model. On MATH500, HumanEval, and GPQA, however, correct solutions are characteristically longer and contain lower-probability token sequences than fluent but incorrect answers; an acceptance rule based only on the model's own likelihoods therefore risks preferentially retaining incorrect trajectories. The manuscript must supply either a direct correlation analysis between likelihood and correctness or an ablation that isolates the effect of the acceptance criterion to establish that the likelihood signal is aligned with accuracy rather than anti-aligned.
- [§4 and §5] §4 and §5 (Experimental results): The abstract states that the sampler 'nearly match[es] and even outperform[s]' RL on MATH500, HumanEval, and GPQA, yet the reported tables and figures lack error bars, statistical significance tests, and controls for prompt length, temperature, and number of MCMC steps. Without these, it is impossible to determine whether the claimed gains are robust or whether they could be explained by differences in effective compute or sampling budget.
minor comments (2)
- The pseudocode for the iterative sampler is referenced but not shown in a single, self-contained listing; adding an explicit algorithm box would improve reproducibility.
- Figure captions should explicitly state the number of samples drawn per method and the exact temperature schedule used for the base-model baseline.
Simulated Author's Rebuttal
We thank the referee for their constructive comments, which have helped us improve the clarity and rigor of our work. We address each major comment below and outline the revisions we will make to the manuscript.
read point-by-point responses
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Referee: [§3] §3 (Algorithm description): The acceptance step is defined solely via likelihood ratios drawn from the base model. On MATH500, HumanEval, and GPQA, however, correct solutions are characteristically longer and contain lower-probability token sequences than fluent but incorrect answers; an acceptance rule based only on the model's own likelihoods therefore risks preferentially retaining incorrect trajectories. The manuscript must supply either a direct correlation analysis between likelihood and correctness or an ablation that isolates the effect of the acceptance criterion to establish that the likelihood signal is aligned with accuracy rather than anti-aligned.
Authors: We appreciate this insightful observation regarding the potential misalignment between likelihood and correctness. While it is true that correct solutions can be longer and thus lower probability under the base model, our iterative sampling procedure is designed to explore multiple trajectories and accept based on relative likelihoods, which empirically leads to better reasoning performance. To directly address the referee's request, we will include in the revised version a correlation analysis plotting the model's likelihood against solution correctness on the benchmarks, as well as an ablation study comparing the full sampler to a version without the acceptance criterion (i.e., pure sampling). This will demonstrate that the acceptance step contributes positively to accuracy. revision: yes
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Referee: [§4 and §5] §4 and §5 (Experimental results): The abstract states that the sampler 'nearly match[es] and even outperform[s]' RL on MATH500, HumanEval, and GPQA, yet the reported tables and figures lack error bars, statistical significance tests, and controls for prompt length, temperature, and number of MCMC steps. Without these, it is impossible to determine whether the claimed gains are robust or whether they could be explained by differences in effective compute or sampling budget.
Authors: We agree that the experimental section would benefit from greater statistical detail to substantiate the robustness of our results. In the revised manuscript, we will add error bars to all reported metrics, computed over multiple independent runs with different random seeds. We will also include statistical significance tests (e.g., paired t-tests) comparing our sampler to the baselines. Additionally, we will provide controls by varying prompt lengths, temperatures, and the number of MCMC steps, and report performance as a function of these parameters to rule out explanations based solely on increased sampling budget or compute. revision: yes
Circularity Check
No significant circularity; method and claims are independently specified
full rationale
The paper defines a straightforward iterative MCMC-style sampler that directly consumes the base model's token likelihoods as the sole source of signal, without any parameter fitting to the evaluation tasks, without renaming an empirical pattern as a derivation, and without load-bearing self-citations or uniqueness theorems. The algorithm is presented as a direct application of existing sampling ideas to LLM logits; the reported gains on MATH500, HumanEval, and GPQA are obtained by running the sampler on external benchmarks rather than by construction from the same quantities used to define the procedure. No equation or step reduces the claimed improvement to a tautological restatement of the input likelihoods or to prior work by the same authors.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Base LLMs contain latent reasoning capabilities that can be surfaced by reshaping their output distribution through repeated sampling from their own likelihoods.
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Inspired by Markov chain Monte Carlo (MCMC) techniques for sampling from sharpened distributions, we propose a simple iterative sampling algorithm leveraging the base models' own likelihoods... Metropolis-Hastings... A(x,x_i) = min{1, p^α(x) q(x_i|x) / p^α(x_i) q(x|x_i)}
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IndisputableMonolith/Foundation/AlphaCoordinateFixation.leanJ_uniquely_calibrated_via_higher_derivative unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
sampling from p^α encourages sampling tokens which have fewer but higher likelihood future paths
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Forward citations
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discussion (0)
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