Instructions: Hand in the sheet to your instructor by the end of lab. If you don’t finish in lab, hand in on Gradescope by 9 AM the Wednesday of the next lab.
Give an example of time series data you’re interested in and explain why:
b) Why would assuming I.I.D. be problematic for your time series of interest?
a) Explain in your own words why the Markov assumption helps with the curse of dimensionality:
b) Give an example of a real-world time series that would be trend stationary but not weakly stationary:
c) Why is it useful that Gaussians are closed under linear operations for time series modeling? (Hint: forecasting.)
Graphical models and factorization:
a) Draw a directed graphical model where X_3 depends on both X_1 and X_2, and X_4 depends only on X_3:
b) Write the factorization of P(X_1, X_2, X_3, X_4) based on your graph:
c) If each variable is binary, how many parameters do we need with your factorization vs. without any assumptions?
Bayes Nets & the Green Party (courtesy of CS 188 at UC Berkeley :).
In a parallel universe the Green Party is running for presidency. Whether a Green Party President is elected (G) will have an effect on whether marijuana is legalized (M), which then influences whether the budget is balanced (B), and whether class attendance increases (C). Armed with the power of probability, the analysts model the situation with the Bayes Net below.
Here are the conditional probability tables (CPTs) for the factors in this graphical model:
P(G)
| P(G) | +g | -g |
|---|---|---|
| P(G) | 0.1 | 0.9 |
P(M|G)
| P(M|G) | P(+m | G) | P(-m | G) |
|---|---|---|
| +g | 0.667 | 0.333 |
| -g | 0.25 | 0.75 |
P(B|M)
| P(B|M) | P(+b | M) | P(-b | M) |
|---|---|---|
| +m | 0.4 | 0.6 |
| -m | 0.2 | 0.8 |
P(C|M)
| P(C|M) | P(+c | M) | P(-c | M) |
|---|---|---|
| +m | 0.25 | 0.75 |
| -m | 0.5 | 0.5 |
The full joint distribution is given below. Fill in the missing values.
| G | M | B | C | P(G, M, B, C) |
|---|---|---|---|---|
| + | + | + | + | 1/150 |
| + | + | + | - | ___ |
| + | + | - | + | 1/100 |
| + | + | - | - | ___ |
| + | - | + | + | 1/300 |
| + | - | + | - | 1/300 |
| + | - | - | + | ___ |
| + | - | - | - | 1/75 |
| - | + | + | + | ___ |
| - | + | + | - | 27/400 |
| - | + | - | + | ___ |
| - | + | - | - | 81/800 |
| - | - | + | + | 27/400 |
| - | - | + | - | 27/400 |
| - | - | - | + | ___ |
| - | - | - | - | 27/100 |
Now, add a node S to the Bayes net that reflects the possibility that a new scientific study could influence the probability that marijuana is legalized. Assume that the study does not directly influence B or C. Draw the new Bayes net below. Which CPT or CPT’s need to be modified?