Another Bias Correction Method John Ashburner

By integrating over k, we obtain the likelihood of yi:

P(yi|,,,) =kP(yi,k|k,k,k,) =kP(yi|k,k,k,) P(k|k)

Likelihood of the entire dataset is derived by assuming independence:

P(y|,,,) = i P(yi|,,,)

• = i{k i() (2k2)-1/2 exp {-(yi-k /i())2/(2k2)} }

Likelihood is maximised when the following is minimised:

E = -log{P(y|,,,)}

= -i log{k i() (2k2)-1/2 exp {-(yi-k /i())2/(2k2)} }

Non-parametric Generalisation

• Consider a simple mixture model, where the means and variances are known, and therefore fixed by the model. Also assume the means are equally spaced, and the variances are identical. The only unknown parameters are the mixing proportions (). By representing the density of the kth Gaussian at yi by k(yi), the negative log-likelihood of the model is:

•