We study identification and estimation of a structural model of bargaining with optimism where players have heterogeneous beliefs about the final resolution of a dispute if they fail to reach an agreement. We show the distribution of the players' beliefs and the stochas- tic bargaining surplus are nonparametrically identified from the probability of reaching an agreement and the distribution of transfers in the final resolution of the dispute. We use a Maximum Simulated Likelihood approach to estimate the beliefs of doctors and patients in medical malpractice disputes in Florida during the 1980s and 1990s. We find strong evidence that beliefs for both parties vary with the severity of the injury and the qualification of the doctor named in the lawsuit, even though these characteristics are statistically insignificant in explaining whether the court rules in favor of the plaintiff or of the defendant. We also quantify the reduction in settlement amounts that would result from the introduction of a (counterfactual) policy that imposes caps on the total compensation for plaintiffs.