1. CS 416 Artificial Intelligence
Lecture 24
Statistical Learning
Chapter 20

2. AI: Creating rational agents
The pursuit of autonomous, rational, agents
It’s all about search
Varying amounts of model information
tree searching (informed/uninformed)
simulated annealing
value/policy iteration
Searching for an explanation of observations
Used to develop a model

3. Searching for explanation of observations
If I can explain observations…
can I predict the future?
Can I explain why ten coin tosses are 6 H and 4 T?
Can I predict the 11th coin toss

4. Running example: Candy
Surprise Candy
Comes in two flavors
cherry (yum)
lime (yuk)
All candy is wrapped in same opaque wrapper
Candy is packaged in large bags containing five different allocations of cherry and lime

5. Statistics
Given a bag of candy, what distribution of flavors will it have?
Let H be the random variable corresponding to your hypothesis
H1 = all cherry, H2 = all lime, H3 = 50/50 cherry/lime
As you open pieces of candy, let each observation of data: D1, D2, D3, … be either cherry or lime
D1 = cherry, D2 = cherry, D3 = lime, …
Predict the flavor of the next piece of candy
If the data caused you to believe H1 was correct, you’d pick cherry

6. Bayesian Learning
Use available data to calculate the probability of each hypothesis and make a prediction
Because each hypothesis has an independent likelihood, we use all their relative likelihoods when making a prediction
Probabilistic inference using Bayes’ rule:
P(hi | d) = aP(d | hi) P(hi)
The probability of of hypothesis hi being active given you observed sequence d equals the probability of seeing data sequence d generated by hypothesis hi multiplied by the likelihood of hypothesis i being active
hypothesis
prior
likelihood

7. Prediction of an unknown quantity X
The likelihood of X happening given d has already happened is a function of how much each hypothesis predicts X can happen given d has happened
Even though a hypothesis has a high prediction that X will happen, this prediction will be discounted if the hypothesis itself is unlikely to be true given the observation of d

8. Details of Bayes’ rule All observations within d are independent
identically distributed
The probability of a hypothesis explaining a series of observations, d
is the product of explaining each component

9. Example Prior distribution across hypotheses Prediction
h1 = 100% cherry = 0.1
h2 = 75/25 cherry/lime = 0.2
h3 = 50/50 cherry/lime = 0.5
h4 = 25/75 cherry/lime = 0.2
h5 = 100% lime = 0.1
Prediction
P(d|h3) = (0.5)10

10. Example
Probabilities for each hypothesis starts at prior value <.1, .2, .4, .2, .1>
Probability of h3 hypothesis as 10 lime candies are observed
P(d|h3)*P(h3) = (0.5)10*(0.4)

11. Prediction of 11th candy
If we’ve observed 10 lime candies, is 11th lime?
Build weighted sum of each hypothesis’s prediction
Weighted sum can become expensive to compute
Instead use most probable hypothesis and ignore others
MAP: maximum a posteriori
from
hypothesis
from
observations

12. Overfitting Remember overfitting from NN discussion?
The number of hypotheses influences predictions
Too many hypotheses can lead to overfitting

13. Overfitting Example Say we’ve observed 3 cherry and 7 lime
Consider our 5 hypotheses from before
prediction is a weighted average of the 5
Consider having 11 hypotheses, one for each permutation
The 3/7 hypothesis will be 1 and all others will be 0

14. Learning with Data First talk about parameter learning
Let’s create a hypothesis for candies that says the probability a cherry is drawn is q, hq
If we unwrap N candies and c are cherry, what is q?
The (log) likelihood is:

15. Learning with Data We want to find q that maximizes log-likelihood
differentiate L with respect to q and set to 0
This solution process may not be easily computed and iterative and numerical methods may be used