10 Questions
What is the estimated probability of rain, given 511 out of 1000 samples have Rain = true?
0.511
What is the main purpose of the rejection sampling algorithm in Bayesian networks?
To determine conditional probabilities
What is the relationship between the number of samples used in the estimate and the standard deviation of the error in each probability?
The standard deviation is inversely proportional to the square root of the number of samples
What is the main problem with rejection sampling in complex problems?
It rejects too many samples
What is the result of the rejection sampling algorithm in estimating P(X | e)?
A consistent estimate of the true probability
What is the primary weakness of rejection sampling?
It may take a long time if the event of interest is very rare.
What is the role of the world in the context of estimating conditional probabilities using rejection sampling?
It plays the role of the sample-generation algorithm.
What is the primary advantage of likelihood weighting over rejection sampling?
It avoids the inefficiency of rejection sampling by generating only relevant events.
What is likelihood weighting an instance of?
A particular instance of the general statistical technique of importance sampling.
What is the primary goal of likelihood weighting in the context of Bayesian networks?
To estimate conditional probabilities in Bayesian networks.
Study Notes
Estimating Probabilities in Bayesian Networks
- The estimated probability of rain, ˆP(Rain = true), is 0.511 when 511 out of 1000 samples from the sprinkler network have Rain = true.
Rejection Sampling
- Rejection sampling is a general method for producing samples from a hard-to-sample distribution given an easy-to-sample distribution.
- It can be used to compute conditional probabilities, P(X | e), by generating samples from the prior distribution and rejecting those that do not match the evidence.
- The algorithm returns an estimated distribution, ˆP(X | e), which is obtained by counting how often X = x occurs in the remaining samples.
- ˆP(X | e) is a consistent estimate of the true probability, P(X | e).
Example of Rejection Sampling
- To estimate P(Rain | Sprinkler = true) using 100 samples, 27 samples with Sprinkler = true are generated, and 8 of those have Rain = true.
- P(Rain | Sprinkler = true) ≈ NORMALIZE(8, 19) = (0.296, 0.704), which is an estimate of the true answer (0.3, 0.7).
Limitations of Rejection Sampling
- The biggest problem with rejection sampling is that it rejects many samples, making it inefficient for complex problems.
- The fraction of samples consistent with the evidence e drops exponentially as the number of evidence variables grows.
Likelihood Weighting
- Likelihood weighting avoids the inefficiency of rejection sampling by generating only events that are consistent with the evidence e.
- It is a particular instance of the general statistical technique of importance sampling, tailored for inference in Bayesian networks.
- Likelihood weighting is more efficient than rejection sampling, as it only generates relevant samples.
Quiz about rejection sampling in Bayesian networks, computing conditional probabilities and determining P(X | e).
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