Exercise: Objects and Attributes

There are \(M\) objects, each with a number of attributes or labels. There are \(L\) possible labels and each object is associated with some subset of these labels. The indicator variables
\[
\alpha_{ml} = \left\{\begin{array}{cl}
1 & \textrm{if label } l \textrm{ is associated with object } m \\
0 & \textrm{otherwise}
\end{array}\right.
\]
define the association between objects and labels. The true values of the \(\alpha_{ml}\) are unknown. To infer the values of the \(\alpha_{ml}\), we ask \(N\) people to indicate which labels they believe are associated with each object. The outcomes of these trials are defined by another set of indicator variables,
\[
\beta_{nml} = \left\{\begin{array}{cl}
1 & \textrm{if person } n \textrm{ associated label } l \textrm{ with object } m \\
0 & \textrm{otherwise}
\end{array}\right.
\]
Now, people can make mistakes and we need to take two types of mistakes into account, namely false positives and false negatives. A false positive is when a person associates a label with an object, while the truth is that the label is not associated with it — i.e. \(\beta_{nml} = 1\) and \(\alpha_{ml} = 0\). A false negative is the opposite scenario, \(\beta_{nml} = 0\) and \(\alpha_{ml} = 1\). We will assume that each person makes each type of error with some fixed but unknown probability,
\begin{eqnarray}
e_n^{\textrm{pos}} &=& P(\beta_{nml}=1 | \alpha_{ml}=0) \quad\textrm{ and}\\
e_n^{\textrm{neg}} &=& P(\beta_{nml}=0 | \alpha_{ml}=1)
\end{eqnarray}

Questions

1. Come up with an appropriate prior over \(\left\{\alpha_{ml}\right\}\) and derive the posterior after observing \(\left\{\beta_{nml}\right\}\). You will also need priors over \(\left\{e^{\textrm{pos}}_n\right\}\) and \(\left\{e^{\textrm{neg}}_n\right\}\) and should derive their posteriors too.
2. Use the data file (LINK: \(N\approx20\), \(M\approx100\), \(L\approx100\)) and infer posteriors over \(\alpha_{ml}\), \(\left\{e^{\textrm{pos}}_n\right\}\) and \(\left\{e^{\textrm{neg}}_n\right\}\). Visualise the posteriors over \(\left\{e^{\textrm{pos}}_n\right\}\) and \(\left\{e^{\textrm{neg}}_n\right\}\) and note your observations. Visualise the posterior over \(\left\{\alpha_{ml}\right\}\) for each object.
3. Comment on whether there are enough data in the file or whether more measurements should be made.