NOTE: This assignment, like others in this class, is due at a particular time. This means that if you are even a minute late, you lose 20%. If you are worried about potentially being late, turn in your homework ahead of time. Do this by submitting it to me during office hours or by sliding it under my office door. Do not send assignments to me through email or leave them in my departmental mail box.
Unfortunately, few things in the real world are known with absolute certainty. This causes several types of difficulties for reasoning about the real world. Two difficulties are considered here.
When we discussed propositional and predicate logic, we needed to know things with certainty. As a simple example, we considered the propositions "it is raining" and "the sidewalk is wet." We related these using the connective ⇒ (implies) to indicate that "if it is raining, then the sidewalk is wet." We acknowledged that this might not be absolutely certain because something might prevent the sidewalk from being wet when it is raining (maybe someone has covered the sidewalk with a tarp to keep it dry) but to make any progress in these logics, we needed to ignore exceptional cases such as these.
List and explain a way that we can handle such exceptional cases using probabilistic reasoning. Note that it is not sufficient to simply suggest that we will list all exceptional cases and assign them small probability values - listing all of them is, in many cases, just as unrealistic as ignoring them.
If we know that "if it is raining, then the sidewalk is wet" and we know that "the sidewalk is wet" the conclusion that "it is raining" is known as the fallacy of affirming the consequent. After all, it might not be raining - maybe the sprinklers are running instead.
Does the fallacy of affirming the consequent exist when we are using probabilistic reasoning? That is, in our example, if I know that "the sidewalk is wet", can I conclude anything about the chance that "it is raining"? Explain your answer.
Consider again the wet sidewalk.
Rain could make the sidewalk wet. So could running sprinklers. Describe at least two other events that could make the sidewalk wet.
You might believe that the sidewalk is probably wet if you see people slipping while they walk on it. Describe at least two other indicators that might allow you probabilistically infer that the sidewalk is wet.
Pick one of the events that could cause the sidewalk to be wet. (You can choose rain, running sprinklers, or one of the others you described.) List the event you picked and describe at least two indicators that might allow you to probabilistically infer that this event is happening.
Construct a Bayes Net for the wet sidewalk domain that you described above. While the exact values you fill into the conditional probability tables (CPTs) are up to you, you should choose a causal topology for the network and probability values for the CPTs that are reasonable and do not violate that basic concepts behind probability theory and Bayes Nets.
Construct a table to show the full joint probability distribution for any three of the variables you encoded in the Bayes Net you constructed above.