Part I: Conditional Probabilities

The dataset below shows whether a decision will be made for having an outdoor picnic given different weather conditions. For example, the first line states that if the weather is sunny, hot, not windy and not rainy, then there will be an outdoor picnic.

1. Compute the conditional probability of having a picnic given that the weather is sunny. That is: p(Outdoor Picnic = Yes | Sunny=True). Notice that this can be done by
   1. Extracting all the rows where sunny=True; these are the first 5 rows in the dataset.
   2. Then out of these 5 rows, check how many have a class label of Yes.
   3. The conditional probability will be the count you found in step 2 divided by 5.
2. Similar to step 1, find the conditional probabilities:
   1. p(Outdoor Picnic = Yes | Windy=True)
   2. p(Outdoor Picnic = Yes | Rainy=True)
   3. p(Outdoor Picnic = Yes | Hot=True)
   4. p(Outdoor Picnic = Yes | Rainy=False)

Sunny	Windy	Rainy	Hot	Outdoor Picnic?
True	False	False	True	Yes
True	False	False	False	Yes
True	True	True	True	No
True	True	False	True	No
True	True	True	False	No
False	False	False	True	Yes
False	False	False	False	Yes
False	True	True	True	No
False	True	True	False	No

Part II: Evaluating Evidence using Bayes' Rule

Bayes' Rule states that:

p(HE)=p(EH).p(H)p(E)p(HE)=p(EH).p(H)p(E)

Where H is some hypothesis and E is some evidence that may support this hypothesis.

The textbook used an example from the medical field to illustrate the use of this rule. The example was about estimating the probability that a patient has measles given that they have red spots. In this case:

• The hypothesis H is "The Patient has Measles", and
• The evidence E is "The Patient has Red Spots"

According to Bayes' Rule, the probability that the patient has measles given that he has red spots p(H|E) is

• Proportional to the probability of a patient developing red spots when he has measles, denoted by p(E|H)
• Proportional to the probability of developing measles in general.
• Inversely proportional to the probability of developing red spots in general.

Notice that the probability that they patient has measles given that he has red spots p(H|E) is considered the diagnosis.

Given this information, let us explore how the three different values for p(E|H), p(H) and p(E) can affect the overall probability p(H|E). Read the cases below and for each case, explain whether the patient diagnosis with measles will become more ore less likely.

1. If the probability of developing red spots when a person has measles is 99%, how will this affect the diagnosis? On the other hand, let's assume hypothetically that only 1% of those who have measles develop read spots. In this case, how will that affect P(H|E)?
2. If the probability of contracting measles is extremely low in the general population, how will this affect the diagnosis? If this probability kept steadily increasing over the years, how will this affect the diagnosis? Likewise, if this probability kept steadily decreasing over the years, how will this affect the diagnosis?
3. If there are many conditions that are more prevalent than measles, and all lead to the development of red spots, how will this affect p(E), and consequently, how will this affect the diagnosis?