Shortcomings of modern views of statistical inference have had negative effects on the image of statistics, whether through students, clients or the press. Here, I question the underlying foundations of modern inference, including the existence of 'true' models, the need for probability, whether frequentist or Bayesian, to make inference statements, the assumed continuity of observed data, the ideal of large samples and the need for procedures to be insensitive to assumptions. In the context of exploratory inferences, I consider how much can be done by using minimal assumptions related to interpreting a likelihood function. Questions addressed include the appropriate probabilistic basis of models, ways of calibrating likelihoods involving differing numbers of parameters, the roles of model selection and model checking, the precision of parameter estimates, the use of prior empirical information and the relationship of these to sample size. I compare this direct likelihood approach with classical Bayesian and frequentist methods in analysing the evolution of cases of acquired immune deficiency syndrome in the presence of reporting delays.
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