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Instead of forcing researchers to decide on weightings that may be idiosyncratic, BayesFold uses Bayes' Theorem to objectively combine results from multiple sources of data. Bayes' Theorem provides a unique, optimal method of updating beliefs in the light of new data:
In other words, the probability of a hypothesis H given some new data D is equal to the prior probability of the hypothesis before the new data were observed, multiplied by the conditional probability of observing the data if the hypothesis were true, divided by the unconditional probability of observing the data when the hypothesis is unknown (or chosen at random). In keeping with common sense, a hypothesis becomes more likely when an outcome that it predicts to be frequent is actually observed, especially when the alternative hypotheses predict that the outcome is rare. For example, if you walk outside and get wet, it is reasonable to believe that it is raining, since getting wet is far more probable when it is raining than when it is not. However, there could be alternative explanations: for example, the water could have fallen from a passing plane. Since both the 'rain' and 'plane' hypotheses make equivalent predictions about subsequent wetness, the decision about which to believe must take into account the prior probabilities of the hypotheses (rain is a much more familiar experience), or additional sources of information (such as the presence or absence of clouds or engine noise). Bayes' theorem formalizes this type of common-sense reasoning.
When considering several hypotheses, Bayes' Theorem provides the probability that, given the evidence, each of the hypotheses is the true hypothesis. This gives the probability that the hypothesis is true (rather than the probability that some null hypothesis is false and should therefore be rejected). If there are multiple hypotheses under consideration, Bayes' Theorem can give the probability, given the evidence, that each of the hypotheses is the true hypothesis. Bayesian methods contrast with the more familiar 'frequentist' type of hypothesis testing, in which a single null hypothesis is chosen and then accepted or rejected according to the probability that the results would be as surprising as those observed if the null hypothesis were true.
When considering several hypotheses, Bayes' Theorem provides the probability that, given the evidence, each of the hypotheses is the true hypothesis. In practice, Pr(D), the probability of the data, is usually unknown. However, if an exhaustive list of possible hypotheses is known in advance, then Pr(D) can be calculated by a technique called marginalization. This technique takes into account the probability of observing D under each H, weighted by the probability of H, as follows:
Bayes' Theorem can be used with multiple types of data Di, by calculating posterior probabilities Pr(H|Di) for the first type of data, using these posteriors as priors for the next type of data, and so on until the last kind of data is reached. This method assumes that the different types of data are not correlated with each other.