Utility Nodes
A Utility Node allows you to assign utilities to combinations of states of its parent nodes. Utilities generally reflect value judgments of outcomes that are represented by the states of the parent nodes.
- As an alternative to directly assigning utilities in the Node Editor, you can also use equations to compute the utilities using arbitrary functions.
- While the assignment of values in a Utility Node is always deterministic, its parent nodes can certainly remain uncertain in the distribution of their states.
- On the basis of its parent nodes' distributions, the Utility Node computes the expected value of the utility.
- You can also use Utility Nodes in conjunction with Decision Nodes and Policy Learning.
Example
Single Utility Node
In the following, simple example, the Utility Node U1 evaluates the states of its parent nodes:
- Party Venue, with states Inside and Outside
- Weather, with states Rain and Sunshine
The situation is straightforward. We are trying to trade off the downside risk of a party in the rain (the worst outcome) with the upside of a beautiful event outdoors in the sunshine (the best outcome). At the same time, we can also quantify our potential "regret" in a situation like having decided to stay indoors for fear of rain, and then the weather turns out to be nice after all.
Note that the values in this Utility Node are assigned arbitrarily. Furthermore, there is no predefined scale, so you could define the worst-to-best range as 0 to 1, or -1,000 to 1,000, for instance. The only important convention is that higher numerical values of utilities mean a more desirable or better outcome.
Utility Nodes can only have inbound arcs, which is the reason why those arcs appear as Fixed Arcs, i.e., they cannot be inverted.
In Validation Mode, we can now evaluate the expected utility of our potential decisions regarding the venue. If the weather forecast suggests a 40% chance of rain, the evaluation will yield the following:
The purple Utility Monitors show that the expected utility of an "outdoor" decision is 60, whereas an "indoor" decision would only have a utility of 54.
Multiple Utility Nodes
Real-world problems are typically more complicated, though. Therefore, we can use multiple Utility Nodes to encode the "costs" and "benefits" of outcomes.
In our party planning example, we now add a second Utility Node U2 that reflects the additional setup cost of hosting the party outside as opposed to the default indoor setup. Note that we could have included this additional cost in the Utility Node U1. However, it is practical to have separate utility nodes so that any assumptions can be represented explicitly as opposed to "baking them in" with other values.
Given that setup cost is typically a specific dollar amount, we can assign this cost directly as a negative utility value. Here, we associate a utility of -25 (=cost of $25) with the outdoor setup and a cost of 0 for staying indoors.
At this point, we need to bear in mind that defining U1 was a purely arbitrary judgment in assigning utilities to certain outcomes. Now, however, we have an actual dollar amount. Given the currently-specified utilities, our model infers, for example, that we "value" an indoor event in nice weather $10 less than the same indoor event while it's raining. That places a dollar value on the regret, which we need to verify. Whenever we introduce an actual monetary cost for one of the utilities, we need to make sure that our "value system" between all Utility Nodes is in alignment.
Once we evaluate our potential decisions again in Validation Mode, BayesiaLab sums up all utilities and shows the (same) sum in each of the Monitors of both Utility Nodes. Note that we do not need to draw any additional arcs to sum up the utilities.
So, for the "indoor" decision, we now have a total utility of 54, as shown on the bottom bars in the Monitors of U1 and U2. The top bars in the Monitors of U1 and U2 show their individual utilities of 54 and 0, respectively.
For the "outdoor" decision, we obtain a total utility of 35, which is the sum of the U1 utility of 54 and the U2 utility of -25.
Based on this new assessment of utilities (54 > 35), we would choose to remain indoors for our event.