Probabilistic Inference Tasks

Tasks are each with respect to a graphical model \(\mathcal{M} = < X, D, F>\), where:

  • \(X = \{ X_1, X_2, ..., X_N \}\) is the set of the model’s variables
  • \(D = \{ D_{X_1}, D_{X_2}, ..., D_{X_N} \}\) is the set of discrete domains for each variable
  • \(F = \{ f_1, f_2, ..., f_M \}\) is the set of the model’s functions

\(X\) can be further partitioned into two sets, evidence variables \(E\) and the rest \(X'= X \setminus E\).

PR :   computing the partition function (ie. normalizing constant) \(\begin{align*} PR(E=e) = \sum_{X'} \prod_{F} f(x',e) \end{align*}\)

MAR :   computing the marginal probability distribution over all variables given evidence \(\begin{align*} MAR(X_i|E=e) = \frac{ \sum_{X'' = X' \setminus X_i} \prod_{F} f(x'',e) }{ PR(E=e) } \end{align*}\)

MPE :   computing the most likely assignment to all variables given evidence \(\begin{align*} MPE(E=e) = \arg \max_{X'} \prod_{F} f(x',e) \end{align*}\)

MMAP :   computing the most likely assignment to the query variables, \(X_M \subset X'\) after marginalizing out when marganlizing the remaining variables \(X_S = X' \setminus X_M\).

\[\begin{align*} MMAP(E=e) = \arg \max_{X_M} \sum_{X_S} \prod_{F} f(x_M, x_S, e) \end{align*}\]

All inference tasks will be given one CPU and 8GB of ram and will be tested in three different time categories:

  • 20 seconds
  • 20 minutes
  • 1 hour

Multi-Label Classification Task

MLC (new!) :   multi-label classification of a subset of variables of given a model and observed evidence (through learning, inference, or any other method of your choice).