That’s basically right in terms of the math, but I’ll walk through the intuitive story a bit more.
First, “edge values” or “variables on edges”. When we think of a DAG model in message-passing terms, each edge carries a message from parent to child. The content of that message is the value of the parent variable. That’s the “edge value” or “edge variable” which gets included in the Markov blanket.
Now, if we find a cut through the graph (i.e. a set of edges which we could remove to split the graph into two disconnected pieces), then the edges in that cut carry all the messages between the two pieces. No communication between the two pieces occurs, except via the messages carried by those edges. So, intuitively, we’d expect the two pieces to be independent conditional on the messages passed over the edges in the cut. That’s the intuitive story behind a Markov blanket.
The usual definition of a Markov blanket is equivalent, but formulating it in terms of vertices rather than edges is more awkward. For instance, when we talk about the Markov blanket of one variable, we usually say that the blanket consists of the variable’s parents, children and spouses. What an awkward definition! What’s up with the spouses? With the edges picture, it’s clear what’s going on: we’re drawing a blanket around the variable and its children. The edges which pass through that blanket carry values of parents, children and spouses.
That’s basically right in terms of the math, but I’ll walk through the intuitive story a bit more.
First, “edge values” or “variables on edges”. When we think of a DAG model in message-passing terms, each edge carries a message from parent to child. The content of that message is the value of the parent variable. That’s the “edge value” or “edge variable” which gets included in the Markov blanket.
Now, if we find a cut through the graph (i.e. a set of edges which we could remove to split the graph into two disconnected pieces), then the edges in that cut carry all the messages between the two pieces. No communication between the two pieces occurs, except via the messages carried by those edges. So, intuitively, we’d expect the two pieces to be independent conditional on the messages passed over the edges in the cut. That’s the intuitive story behind a Markov blanket.
The usual definition of a Markov blanket is equivalent, but formulating it in terms of vertices rather than edges is more awkward. For instance, when we talk about the Markov blanket of one variable, we usually say that the blanket consists of the variable’s parents, children and spouses. What an awkward definition! What’s up with the spouses? With the edges picture, it’s clear what’s going on: we’re drawing a blanket around the variable and its children. The edges which pass through that blanket carry values of parents, children and spouses.
Thanks, that’s very clear. I’m a convert to the edge-based definition.