Past exam of the systems biology course of the University of Cambridge / 2024 / Paper 4 / Section I / 1.1 / Solution

Write the parameters as ${\pi }_0,{\pi }_1,{\mu }_0,{\mu }_1,{\sigma }_0,{\sigma }_1$, where ${\pi }_j$ is the fraction of molecules in component $j$, ${\mu }_j$ is its mean current, and ${\sigma }_j>0$ is its current standard deviation. The mixture weights satisfy ${\pi }_j\ge 0$ and ${\pi }_0+{\pi }_1=1$. Thus six displayed parameters represent five independent degrees of freedom.

Assigning component 1 to modified molecules makes ${\pi }_1$ the inferred modification fraction. That biological assignment needs a reference or other identifying information: exchanging both component labels leaves the Gaussian mixture distribution unchanged. In the illustrated multi-sample model, component locations can be shared while the sample-specific fractions differ; this is different from fitting unrelated means to every sample.