We present Information Gain Fine-Tuning (IGFT), a novel approach for training medical conversational AI to conduct effective patient interviews and generate comprehensive History of Present Illness (HPI) without requiring pre-collected human conversations. IGFT combines online Group Relative Policy Optimization (GRPO) with information-theoretic rewards, enabling models to learn from self-generated conversations with simulated patients. Unlike existing approaches that rely on expensive expert-annotated conversations or static datasets, our online RL framework allows models to discover effective questioning strategies through exploration. Our key innovation is an information gain reward function that tracks which clinical entities such as symptoms, temporal patterns, and medical history, are revealed during conversation. Each question's reward is computed based on its expected information gain combined with GPT-4o-mini quality assessments across dimensions including clinical relevance, patient engagement, and specificity. This hybrid approach ensures models learn to ask targeted, clinically appropriate questions that efficiently gather diagnostic information. We fine-tune two models using LoRA: Llama-3.1-8B-Instruct and DeepSeek-R1-Distill-Qwen-7B (a reasoning-optimized model). Training exclusively on Avey data containing concise HPIs, we evaluate generalization to MIMIC data with longer, more elaborate HPIs. DeepSeek-R1-Distill-Qwen-7B (IGFT) achieves F1 scores of 0.408 on Avey (10.9% improvement over base) and 0.289 on MIMIC (12.9% improvement), while Llama-3.1-8B-Instruct (IGFT) reaches 0.384 and 0.336 respectively. Both models outperform OpenAI's model on MIMIC and surpass medical domain-specific baselines like HuatuoGPT and UltraMedical, which were optimized for single-turn medical QA rather than multi-turn conversations.