Large Language Models (LLMs) have shown remarkable progress in various tasks, but complex reasoning remains a challenge. While they can often generate logical steps, LLMs sometimes struggle to synthesize information effectively, leading to incorrect conclusions. Think of it like having all the pieces of a puzzle but not knowing how to put them together. Existing methods like self-consistency (voting on multiple LLM responses) and progressive hint prompting (giving previous answers as hints) have shown promise, but they often underutilize the information contained within those responses. Researchers have developed a novel technique called Hint Marginalization (HM) to address this limitation. This method refines the reasoning process by iteratively building and updating a probability distribution over possible answers. Instead of simply taking the most frequent answer, HM considers the likelihood of each answer given previous hints. It's like starting with a rough sketch and progressively adding details based on the hints, ultimately revealing a clearer picture. HM works by first generating an initial set of answers using standard prompting techniques. Then, in subsequent rounds, it presents unique answers from the previous round as hints to the LLM. The new answers generated are weighted by the probability of the hints they were based on, and this information is used to update the overall probability distribution of answers. This process is repeated iteratively, refining the distribution and converging towards the most probable answer. Experiments on various arithmetic reasoning benchmarks, including AddSub, MultiArith, SingleEQ, SVAMP, GSM8K, and AQuA, demonstrate that HM significantly improves LLM reasoning performance compared to existing methods. Using models like GPT-3.5 Turbo, GPT-4 Turbo, and GPT-4o-mini, HM consistently achieved higher accuracy while using a comparable number of LLM calls. This suggests that HM makes more efficient use of the LLM's resources by strategically incorporating hints into the reasoning process. While HM has shown impressive results, there are still areas for future improvement. The current implementation primarily focuses on arithmetic reasoning, and the hint phrasing might not be ideal for other types of problems. Further research into more general hint structures and dynamic allocation of LLM calls to different hints could further enhance the effectiveness of this promising technique. This approach offers a valuable step towards unlocking the full potential of LLMs for complex reasoning tasks, paving the way for more robust and reliable AI systems.