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

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

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

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

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

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

Can Flow ≠ Must Flow Degenerate dependency

Can Flow ≠ Must Flow XOR example

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

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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.

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

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