Deterministic decision nodes and random utility nodes

[jarrod-dalton]

(You might want to find something on which you can direct your frustration before reading on! Or at least a quiet place to lament collaborating with someone who doesn't quite have stuff figured out. In my defense, I'm not sure the rest of the statistical community has worked through these issues!)

The thought just occurred to me that I may have placed an unnecessary restriction on decision nodes. Currently, we restrict decision nodes to be random variables. So, the probability of making a decision of do this vs. do that for node D depends on the values of node D's parents.

The implication of this is that we can look at decisions as exhibiting a distribution across a population of decision makers, and that distribution depends on the parents of D.

What if a high-level decision maker sets policy so that they uniformly occur? These would be decisions like:

• always give antibiotics if there is a positive throat culture
• (blackjack decision net:) always take another card if the point total is <16

I can wrap my head around the first example. That would require some modifications to some of the downstream functions, but basically allowing for the possibility of a decision node to be deterministic would let us model that situation. But notice the second example represents a strategy that involves a sequence of decision nodes.

I've seen some literature describing sequential decisions in networks. That literature breaks networks down into the following structure:

Net = (X0 -> D1 -> X1 -> D2 -> X2 -> ... -> Dn -> Xn)

where X* and D* are subnetworks including random/deterministic nodes and decision nodes, respectively. (I guess utility nodes just do their thing and hang out as leaf nodes wherever they want. But I'm not 100% on this.) Maybe we can use that construct somehow.

For that matter, I think we may have also placed an unnecessary restriction on utility nodes to be deterministic? What if you wanted to represent utility as a belief distribution instead of a point-mass (single number)? In the below figure, the mean of this distribution is 0.75 (blue line) and the mode is around 0.94 (red line). Neither seem to be as good as using the whole belief density of utility.