圓 1.t'sof the right shape P(Buys?-true Cos whose lcat Why the probit? 1)全T2+33 Continuous child variables > given Cost should be a soft"threshold Discrete variable w/continuous parents Summary shape to probit but much lnger ta Sigmoid (or git)distribution alo used in neural networks: Discrete variable contd.Continuous child variables 0 5 10 0 5 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Cost Harvest P(Cost|Harvest,Subsidy?=true) All-continuous network with LG distributions ⇒ full joint distribution is a multivariate Gaussian Discrete+continuous LG network is a conditional Gaussian network i.e., a multivariate Gaussian over all continuous variables for each combination of discrete variable values Chapter 14.1–3 25 Discrete variable w/ continuous parents Probability of Buys? given Cost should be a “soft” threshold: 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 P(Buys?=false|Cost=c) Cost c Probit distribution uses integral of Gaussian: Φ(x) = R −∞ x N(0, 1)(x)dx P(Buys? = true | Cost = c) = Φ((−c + µ)/σ) Chapter 14.1–3 26 Why the probit? 1. It’s sort of the right shape 2. Can view as hard threshold whose location is subject to noise Buys? Cost Cost Noise Chapter 14.1–3 27 Discrete variable contd. Sigmoid (or logit) distribution also used in neural networks: P(Buys? = true | Cost = c) = 1 1 + exp(−2 −c+µ σ ) Sigmoid has similar shape to probit but much longer tails: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 P(Buys?=false|Cost=c) Cost c Chapter 14.1–3 28 Summary Bayes nets provide a natural representation for (causally induced) conditional independence Topology + CPTs = compact representation of joint distribution Generally easy for (non)experts to construct Canonical distributions (e.g., noisy-OR) = compact representation of CPTs Continuous variables ⇒ parameterized distributions (e.g., linear Chapter Gaussian) 14.1–3 29