1 / 31

Bayes Net Learning

Bayes Net Learning. Oliver Schulte Machine Learning 726. Learning Bayes Nets. Structure Learning Example: Sleep Disorder Network. Source: Development of Bayesian Network models for obstructive sleep apnea syndrome assessment Fouron , Anne Gisèle . (2006) . M.Sc. Thesis, SFU. .

rob
Télécharger la présentation

Bayes Net Learning

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Bayes Net Learning Oliver Schulte Machine Learning 726

  2. Learning Bayes Nets

  3. Structure Learning Example: Sleep Disorder Network Source: Development of Bayesian Network models for obstructive sleep apnea syndrome assessmentFouron, Anne Gisèle. (2006) . M.Sc. Thesis, SFU.

  4. Parameter Learning Scenarios • Complete data (today). • Later: Missing data (EM).

  5. The Parameter Learning Problem • Input: a data table XNxD. • One column per node (random variable) • One row per instance. • How to fill in Bayes net parameters? Humidity PlayTennis

  6. Start Small: Single Node • What would you choose? Humidity • How about P(Humidity = high) = 50%?

  7. Parameters for Two Nodes Humidity PlayTennis • Is θ as in single node model? • How about θ1=3/7? • How about θ2=6/7?

  8. Maximum Likelihood Estimation

  9. MLE • An important general principle: Choose parameter values that maximize the likelihood of the data. • Intuition: Explain the data as well as possible. • Recall from Bayes’ theorem that the likelihood isP(data|parameters) = P(D|θ).

  10. Finding the Maximum Likelihood Solution: Single Node Humidity independent identically distributed data! iid • Write down • In example, P(D|θ)= θ7(1-θ)7. • Maximize θfor this function.

  11. Solving the Equation

  12. Finding the Maximum Likelihood Solution: Two Nodes • In a Bayes net, can maximize each parameter separately. • Fix a parent condition  single node problem.

  13. Finding the Maximum Likelihood Solution: Single Node, >2 possible values. • Lagrange Multipliers

  14. Problems With MLE • The 0/0 problem: what if there are no data for a given parent-child configuration? • Single point estimate: does not quantify uncertainty. • Is 6/10 the same as 6000/10000? • [show Bayes net with playtennis as child, three parents.

  15. Classical Statistics and MLE • To quantify uncertainty, specify confidence interval. • For the 0/0 problem, use data smoothing.

  16. Bayesian Parameter Learning

  17. Parameter Probabilities • Intuition: Quantity uncertainty about parameter values by assigning a prior probability to parameter values. • Not based on data. • [give Russell and Norvig example]

  18. Bayesian Prediction/Inference • What probability does the Bayesian assign to PlayTennis = true? • I.e., how should we bet on PlayTennis = true? • Answer: • Make a prediction for each parameter value. • Average the predictions using the prior as weights. [Russell and Norvig Example]

  19. Mean • Bayesian prediction can be seen as the expected value of a probability distribution P. • Aka average or mean of P. • Notation: E(P), mu.

  20. Variance • Define • Variance of a parameter estimate = uncertainty. • Decreases with learning.

  21. Continuous priors • Probabilities usually range over a continuous interval. • Then probabilities of probabilities are probabilities of continuous variables. • Probability of continuous variables = probability density function. • p(x) behaves like probability of discrete value, but with integrals replacing sum. • E.g. [integral over 01 = 1]. • Exercise: Find the p.d.f. of the uniform distribution over an interval [a,b].

  22. Bayesian Prediction With P.D.F.s

  23. Bayesian Learning

  24. Bayesian Updating • Update prior using Bayes’ theorem. • Exercise: Find the posterior of the uniform distribution given 10 heads, 20 tails.

  25. The Laplace Correction • Start with uniform prior: the probability of Playtennis could be any value in [0,1], with equal prior probability. • Suppose I have observed n data points. • Find posterior distribution. • Predict probability of heads using posterior distribution. • Integral: • Solved by Laplace in A.D. x!

  26. Parametrized Priors • Motivation: Suppose I don’t want a uniform prior. • Smooth with m>0. • Express prior knowledge. • Use parameters for the prior distribution. • Called hyperparameters. • Chosen so that updating the prior is easy.

  27. Beta Distribution: Definition

  28. Beta Distribution: Examples

  29. Updating the Beta Distribution

  30. Conjugate Prior for non-binary variables • Dirichlet distribution: generalizes Beta distribution for variables with >2 values.

  31. Summary • Maximum likelihood: general parameter estimation method. • Choose parameters that make the data as likely as possible. • For Bayes net parameters: MLE = match sample frequency.Typical result! • Problems: • not defined for 0/0 situation. • doesn’t quantity uncertainty in estimate. • Bayesian approach: • Assume prior probability for parameters; prior has hyperparameters. • E.g., beta distribution. • Problems: • prior choice not based on data. • inferences (averaging) can be hard to compute.

More Related