# Metropolis–Hastings Sampling

The work introduces Parallel Power Tempering (PPT), which adapts classical parallel tempering (replica exchange) to sequence-level power-sharpened LLM sampling: multiple replicas at different sharpening levels run in parallel so that low-power replicas explore diverse reasoning trajectories while high-power chains exploit higher-likelihood responses, with swaps carrying promising traces to the sharpest output rung, and a fixed-horizon construction removes a structural truncation bias identified in early-stopped power samplers; across MATH500, GPQA, HumanEval, GSM8K, AIME 24&25 and LiveCodeBench v5 on Qwen3-4B, Qwen3-8B and Qwen3.5-9B, PPT attains best or tied-best accuracy in all model-benchmark settings, outperforming Power Sampling, PowerSMC and GRPO, and on Qwen3.

The work introduces Parallel Power Tempering (PPT), which adapts classical parallel tempering (replica exchange) to sequence-level power-sharpened LLM sampling: multiple replicas at different sharpening levels run in parallel so that low-power replicas explore diverse reasoning trajectories while high-power chains exploit higher-likelihood responses, with swaps carrying promising traces to the sharpest output rung, and a fixed-horizon construction removes a structural truncation bias identified in early-stopped power samplers; across MATH500, GPQA, HumanEval, GSM8K, AIME 24&25 and LiveCodeBench v5 on Qwen3-4B, Qwen3-8B and Qwen3.5-9B, PPT attains best or tied-best accuracy in all model-benchmark settings, outperforming Power Sampling, PowerSMC and GRPO, and on Qwen3.