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Probability and Anthropic Reasoning in Small, Large, and Infinite Universes

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I will argue that anthropic reasoning is unnecessary or misleading when the universe/multiverse is small enough that another observer with exactly your memories is unlikely to exist. Instead, one can evaluate theories or make predictions in the standard Bayesian way, based on the conditional probability of something unknown given all that you do know. Things are not so clear when the universe is large enough that all competing theories predict that an observer with your exact memories exists with probability close to one. I will discuss issues that arise in such large or infinite universes, such as "Boltzmann brains", and will argue that pending better understanding of these issues one should be hesitant to draw conclusions different from those that would apply to a small iverse.