We build our activity model based on the Relational Markov
Network (RMN) framework [Taskar et al., 2002]. RMNs describe
specific relations between objects using clique templates
specified by SQL queries: each query C selects the
relevant objects and their attributes, and specifies a potential
function, or clique potential, C, on the possible values of
these attributes. Intuitively, the clique potentials measure the
“compatibility” between values of the attributes. Clique potentials
are usually defined as a log-linear combinations of
feature functions, i.e., C(vC) = exp{wTC
· fC(vC)}, where vC are the attributes selected in the query, fC() is a feature
vector for C, and wTC
is the transpose of the corresponding
weight vector. For instance, a feature could be the number of
different homes defined using aggregations.
To perform inference, an RMN is unrolled into a Markov
network, in which the nodes correspond to the attributes of
objects. The connections among the nodes are built by applying
the SQL templates to the data; each template C can result
in several cliques, which share the same feature weights.
Standard inference algorithms, such as belief propagation and
MCMC, can be used to estimate the conditional distribution
of hidden variables given all the observations.