This paper lays the foundation for a general theory of weakly admissible tests, where a test consists of both an experiment (a $\sigma$-field of information) and a decision function based on the experiment. A test is weakly admissible unless there is another test with as good a decision function, which uses no more information, and is strictly better in one of these respects. Weakly admissible tests are completely characterized. In the case where the experiment consists of a sequential sampling procedure, a method for constructing such tests is given.