Inverse reinforcement learning (IRL) addresses the problem of recovering a task description given a demonstration of the optimal policy used to solve such a task. The optimal policy is usually provided by an expert or teacher, making IRL specially suitable for the problem of apprenticeship learning. The task description is encoded in the form of a reward function of a Markov decision process (MDP). Several algorithms have been proposed to find the reward function corresponding to a set of demonstrations. One of the algorithms that has provided best results in different applications is a gradient method to optimize a policy squared error criterion. On a parallel line of research, other authors have presented recently a gradient approximation of the maximum likelihood estimate of the reward signal. In general, both approaches approximate the gradient estimate and the criteria at different stages to make the algorithm tractable and efficient. In this work, we provide a detailed description of the different methods to highlight differences in terms of reward estimation, policy similarity and computational costs. We also provide experimental results to evaluate the differences in performance of the methods.