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The main difference is that Bayesian incorporate subjective probability judgment prior to data collection as part of its input, but frequentist does not. This means that when it comes to making decision, frequentist relies on various rule of thumb like maximum likelihood principle, while Bayesian can compute the expected utility precisely. This has actual mathematical consequences, as Bayesian inference using what seems like a neutral subjective probability prior to data collection will produce different outcome from frequentist probability.