Probabilistic reasoning in intelligent systems: networks of plausible inference
Probabilistic reasoning in intelligent systems: networks of plausible inference
Probabilistic reasoning in expert systems: theory and algorithms
Probabilistic reasoning in expert systems: theory and algorithms
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What sorts of observations could confirm the universal hypothesis that all ravens are black? Carl Hempel proposed a number of simple and plausible principles which had the odd ("paradoxical") result that not only do observations of black ravens confirm that hypothesis, but so too do observations of yellow suns, green seas and white shoes. Hempel's response to his own paradox was to call it a psychological illusion---i.e., white shoes do indeed confirm that all ravens are black. Karl Popper on the other hand needed no response: he claimed that no observation can confirm any general statement---there is no such thing as confirmation theory. Instead, we should be looking for severe tests of our theories, strong attempts to falsify them. Bayesian philosophers have (in a loose sense) followed the Popperian analysis of Hempel's paradox (while retaining confirmation theory): they have usually judged that observing a white shoe in a shoe store does not qualify as a severe test of the hypothesis and so, while providing Bayesian confirmation, does so to only a minute degree. This rationalizes our common intuition of non-confirmation. All of these responses to the paradox are demonstrably wrong---granting an ordinary Bayesian measure of confirmation. A proper Bayesian analysis reveals that observations of white shoes may provide the raven hypothesis any degree of confirmation whatsoever.