C.2 The probability density functionBayesâ€™ Theorem states that:
Since we are only seeking to maximise the quantity on the left, and the denominator, termed the Bayesian evidence, is independent of , we can neglect it and replace the equality sign with a proportionality sign. Furthermore, if we assume a uniform prior, that is, we assume that we have no prior knowledge to bias us towards certain more favoured values of , then is also a constant which can be neglected. We conclude that maximising is equivalent to maximising . Since we are assuming to be Gaussiandistributed observations of the true function , this latter probability can be written as a product of Gaussian distributions:
The product in this equation can be converted into a more computationally workable sum by taking the logarithm of both sides. Since logarithms are monotonically increasing functions, maximising a probability is equivalent to maximising its logarithm. We may write the logarithm of as:
where is some constant which does not affect the maximisation process. It is this quantity, the familiar sumofsquareresiduals, that we numerically maximise to find our bestfitting set of parameters, which I shall refer to from here on as .
