D17 - 4.6 Bayesian Analysis of Learning In this section we...

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4.6 Bayesian Analysis of Learning In this section we will briefly review the Bayesian approach to learning. Its motivation does not come from bounds on the generalisation and so the section may seem out of place in this chapter. Though Bayesian analysis can be used to estimate generalisation, in this section we only cover that part of the theory needed to motivate the learning strategy. The pac style of analysis we have considered in the earlier sections of this chapter has focused on finding an error bound that will hold with high probability. That approach can be seen as conservative in that it attempts to find a bound on the error probability that will hold with high confidence. The Bayesian approach in contrast attempts to choose output values that are most likely based on the observed training values. This can result in a function not actually in the initial set of hypotheses. Alternatively if we restrict ourselves to choosing a function from the set, it can motivate an optimal choice. It therefore is attempting to make the best possible choice based on the available data. In order to make such a desirable calculation tractable a number of assumptions need to be made including the existence of a prior distribution over the set of hypotheses and a (Gaussian) noise model.
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