1. Probabilistic Influence & d-separation
Representation
Probabilistic
Graphical
Models
Bayesian Networks
Probabilistic Influence & d-separation

2. When can X influence Y given evidence about Z
Intelligence
Difficulty
Grade
Letter
SAT
pairs

3. When can X influence Y given evidence about Z
Intelligence
Difficulty
Grade
Letter
SAT
triples

4. When can X influence Y given evidence about Z
Intelligence
Difficulty
Grade
Letter
SAT
longer trails

5. Active Trails
A trail X1 ─ … ─ Xn is active given Z if:

6. d-separation
Definition: X and Y are d-separated given evidence Z if

7. Can Flow ≠ Must Flow
Degenerate dependency

8. Can Flow ≠ Must Flow
XOR example

9. Summary
Active trail in a graph G  influence might flow in any distribution P that factorizes over G
If a trail is active, influence might still not flow in a specific P that factorizes over G
Active trail is necessary but not sufficient for probabilistic influence to flow
If two nodes are d-separated, they have no active trails, and influence cannot flow in any P

10. END END END

14. Suppose q is at a local minimum of a function
Suppose q is at a local minimum of a function. What will one iteration of gradient descent do?
Leave q unchanged.
Change q in a random direction.
Move q towards the global minimum of J(q).
Decrease q.

15. Consider the weight update:
Which of these is a correct vectorized implementation?

16. Fig. A corresponds to a=0.01, Fig. B to a=0.1, Fig. C to a=1.