If this project survived, I will post codes for the "non-experts" problem.

• Use PCA instead of linear algebra.
• Forget to normalize $\theta$ at the beginning. However, normalized $\widehat{\theta}$ and $\widetilde\theta$ is estimated later.
• $\alpha^{(r)}$ should be different for each annotator. However, I forgot it and let the intercept be the same because the logistic regression by default has only one intercept.
  • However, at present I still failed to handle it beautifully. It seems that I should set some constraints on the intercept $\alpha^{(r)}$. My requirements are not support by standard logistic codes offered by sklearn. I should write my own codes.
• The ultimate $\widetilde\theta$ should has norm 1. However, after one-step update, it may not repect this restriction. Thus, a standardization is forced on $\widetilde\theta$.