Sam and the Rain
Sam is an n^{\text{th}}-year graduate student at MIT. Because Sam just wants to finish his thesis and get out of here, he has shut himself in his office until he finishes. His office has no windows, so he cannot directly observe the weather outside; however, he wants to know the weather so he has something to talk about when his mother calls.
Sam decides to guess at the weather using probabilistic state estimation. He uses a binary random variable R to represent whether it is raining on a given day. R takes value 1 if it is raining, and 0 if it is not raining. Knowing nothing about the weather, Sam decides to start with a uniform prior:
The Professor
Sam also notices that his supervisor, the esteemed Professor Alyssa P. Hacker, carries an umbrella on some days, and wears a hat on some days.Let U_{Prof} be a binary random variable which takes value 1 if the Professor is carrying an umbrella, and 0 otherwise. Similarly, let H_{Prof} be a binary random variable that takes value 1 if the Professor is wearing a hat, and 0 otherwise. Sam builds up the following model, a conditional joint distribution \Pr(U_{Prof},H_{Prof}|R):
| \Pr(U_{Prof},H_{Prof}|R=1) | ||
| H_{Prof}=1 | H_{Prof}=0 | |
| U_{Prof}=1 | 0.32 | 0.48 |
| U_{Prof}=0 | 0.08 | 0.12 |
| \Pr(U_{Prof},H_{Prof}|R=0) | ||
| H_{Prof}=1 | H_{Prof}=0 | |
| U_{Prof}=1 | 0.03 | 0.07 |
| U_{Prof}=0 | 0.27 | 0.63 |
A Question of Magnitude
Sam wants to use this model to update his belief based on what he sees about Professor Hacker's attire.Assuming Sam starts from a uniform belief state as shown above, which observation (U_{Prof},H_{Prof}) would cause the biggest change in his belief? Enter your answer as a Python tuple of two numbers, with the first representing the value of U_{Prof} and the second representing the value of H_{Prof}:
Which observation (U_{Prof},H_{Prof}) would cause the smallest change in his belief?
Transitions
Sam decides to expand on his model by including a transition model. In this model, if it rained on day n, then it will rain on day n+1 with probability 0.7. If it did not rain on day n, then it will rain on day n+1 with probability 0.5.
At the start of day 0, Sam decides that the probability it is raining is 0.5:
Professor Hacker walks in and Sam notices she is wearing a hat, but can't tell whether she is carrying an umbrella. What is his updated belief after observing H_{Prof} = 1? Enter your answers as decimal numbers accurate to within 10^{-4}.
After a day passes, what is his updated belief state?
Raindrops Keep Falling on My Fred
Passing by Sam's office one day, you notice that he is working on another observation model based on his officemate Fred using a variable U_{Fred}, which is analogous to U_{Prof} in the previous sections.
Sam is asleep on his stack of papers, and so you can only see half of his observation model. Scrawled next to a small puddle of drool, you see:
You also find what appears to be his work for updating his belief based on a single observation. Starting with the belief state shown below and observing U_{Fred} = 1, you see that Sam updated his belief state as follows:
| \Pr(R=0) | \Pr(R=1) | |
| **Prior** | \frac{5}{6} | \frac{1}{6} |
| **Updated** | \frac{5}{13} | \frac{8}{13} |
Given this evidence, solve for the following quantities, entering your answers as decimal numbers accurate to within 10^{-4}.