Written response (with optional code) Question: You are given the following Bayesian network structure (no conditional probability tables provided): Rain → WetGrass ← Sprinkler This means: - Rain a ects whether the grass is wet. - Sprinkler also a ects whether the grass is wet. Task: Without using any software, answer the following:
1. (Short answer ) What independence assumptions are implied by this network structure?
The structure implies that Rain and Sprinkler are independent causes of the grass being wet. There is no arrow or path directly linking Rain to Sprinkler, so the model treats them as marginally independent. This kind of two separate arrows pointing into the same child node indicates that Rain and Sprinkler remain independent when we have no evidence, but they can become dependent if we observe their common effect WetGrass. In addition, WetGrass has no other parents, so once Rain and Sprinkler are known, WetGrass is conditionally independent of any other variables in the network. The key is that the sprinkler turning on is not influenced by the weather, and the weather does not affect whether someone turns on the sprinkler.
2. (True/False + Justify) Is `Rain ⊥ Sprinkler` (Rain independent of Sprinkler)?
Yes, Rain is independent of Sprinkler (Rain ⫫ Sprinkler) before observing any evidence. Because there is no direct link between them, the joint probability is:
P(Rain, Sprinkler, WetGrass) = P(Rain) × P(Sprinkler) × P(WetGrass | Rain, Sprinkler).
From this, we see that Rain and Sprinkler have independent priors (P(Rain, Sprinkler) = P(Rain) × P(Sprinkler)). The event it rains does not make it more or less likely that The sprinkler is on under the assumptions of this network.
3. Suppose you observe that the grass is wet. How does this a ect the relationship between Rain and Sprinkler? Is there explaining away? Briefly explain your reasoning.
Once we observe that the grass is wet, Rain and Sprinkler become dependent. This is a case of explaining away. If we already know the grass is wet, and then we discover it rained, that discovery explains the wet grass so strongly that it lowers the probability that the sprinkler was on. Likewise, if we find out the sprinkler was on, we become less likely to believe that it rained because the sprinkler alone could account for the wet grass.
In Bayesian network, observing the common effect induces dependence between its independent causes Rain and Sprinkler. Hence, Rain and Sprinkle are negatively correlated when we condition on WetGrass one cause explains away the effect, making the other cause less needed.
4. Sketch a di erent Bayesian network (3–4 nodes) where two causes independently a ect an outcome, and observing the outcome creates dependence between the causes. Explain why.
We imagine an network with two causes, Smart and Sporty, which both influence a shared outcome, Admitted. Initially, Smart and Sporty are independent. But once we observe that a student was admitted to college, these two causes become dependent due to explaining away.
If a student is admitted, the outcome is true, and we discover they are extremely Smart, that explains the admission, so we lower our belief that they must also be Sporty to have been admitted. Conversely, if they are known to be very Sporty, we reduce the likelihood they got in purely because they were Smart. Thus, observing the outcome Admitted induces a negative correlation between Smart and Sporty even though they were uncorrelated before. This illustrates the same structure effect as with Rain, Sprinkler, and WetGrass.
In summary, observing a common effect forces the two causes to be inferred in a correlated manner, even though they started as independent parameters in our probability model.