2. Bayes Decision Theory
Prof. A.L. Yuille
Stat 231. Fall 2004.
Decisions with Uncertainty
• Bayes Decision Theory is a theory for how
to make decisions in the presence of
uncertainty.
• Input data x.
• Salmon y= +1, Sea Bass y=-1.
• Learn decision rule: f(x) taking values
Decision Rule for Fish.
• Classify fish as
Salmon or Sea Bass
by decision rule f(x).
Basic Ingredients.
• Assume there are probability distributions
for generating the data.
• P(x|y=1) and P(x|y=-1).
• Loss function L(f(x),y) specifies the loss of
making decision f(x) when true state is y.
• Distribution P(y). Prior probability on y.
• Joint Distribution P(x,y) = P(x|y) P(y).
Minimize the Risk
• The risk of a decision rule f(x) is:
• Bayes Decision Rule f*(x):
• The Bayes Risk:
•
Minimize the Risk.
• Write P(x,y) = P(y|x) P(x).
• Then we can write the Risk as:
• The best decision for input x is f*(x):
Bayes Rule.
• Posterior distribution P(y|x):
• Likelihood function P(x|y)
• Prior P(y).
• Bayes Rule has been controversial (historically)
because of the Prior P(y) (subjective?).
• But in Bayes Decision Theory, everything starts
from the joint distribution P(x,y).
Risk.
• The Risk is based on averaging over all
possible x & y. Average Loss.
• Alternatively, can try to minimize the worst
risk over x & y. Minimax Criterion.
• This course uses the Risk, or average
loss.
Generative & Discriminative.
• Generative methods aim to determine probability
models P(x|y) & P(y).
• Discriminative methods aim directly at estimating
the decision rule f(x).
• Vapnik argues for Discriminative Methods: Don’t
solve a harder problem than you need to. Only
care about the probabilities near the decision
boundaries.
Discriminant Functions.
• For two category case the Bayes decision rule
depends on the discriminant function:
• The Bayes decision rule is of form:
• Where T is a threshold, which is determined by
the loss function.
Two-State Case
• Detect “target” or “non-target”.
• Let loss function pay a penalty of 1 for misclassification,
0 otherwise.
• Risk becomes Error. Bayes Risk becomes Bayes Error.
• Error is the sum of false positives F+ (non- targets
classified as targets) and false negatives F- (targets
classified as non-targets).
Gaussian Example: 1
• Is a bright light flashing?
• n is no. photons emitted by dim or bright light.
8. Gaussian Example: 2
•
are Gaussians with
•
means and s.d. .
Bayes decision rule selects “dim” if
•
Errors:
;
Example: Multidimensional
Gaussian Distributions.
• Suppose the two classes have Gaussian
distributions for P(x|y).
• Different means
but same covariance
• The discriminant function is a plane:
• Alternatively, seek a planar decision rule without
attempting to model the distributions.
• Only care about the data near the decision
boundary.
Generative vrs. Discriminant.
• The Generative approach will attempt to
estimate the Gaussian distributions from
data – and then derive the decision rule.
• The Discriminant approach will seek to
estimate the decision rule directly by
learning the discriminant plane.
• In practice, we will not know the form of the
distributions of the form of the discriminant.
Gaussian.
• Gaussian Case with unequal covariance.
Discriminative Models & Features.
• In practice, the Discriminative methods are usually
defined based on features extracted from the data. (E.g.
length and brightness of fish).
• Calculate features z=h(x).
• Bayes Decision Theory says that this throws away
information.
• Restrict to a sub-class of possible decision rules – those
that can be expressed in terms of features z=h(x).
Bayes Decision Rule and Learning.
• Bayes Decision Theory assumes that we know,
or can learn, the distributions P(x|y).
• This is often not practical, or extremely difficult.
• In real problems, you have a set of classified
data
• You can attempt to learn P(x|y=+1) & P(x|y=-1)
from these (next few lectures).
• Parametric & Non-parametric approaches.
• Question: when do you have enough data to
learn these probabilities accurately?
• Depends on the complexity of the model.
Machine Learning.
• Replace Risk by Empirical Risk
• How does minimizing the empirical risk relate to
minimizing the true risk?
• Key Issue: When can we generalize? Be
confident that the decision rule we have learnt
on the training data will yield good results on
unseen data?
Machine Learning
• Vapnik’s theory gives a mathematically elegant
way of answering these issues.
• It assumes that the data is sampled from an
unknown distribution.
• Vapnik’s theory gives bounds for when we can
generalize.
• Unfortunately these bounds are very
conservative.
• In practice, train on part of dataset and test on
other part(s).
Extensions to Multiple Classes
Conceptually straightforward – see Duda, Hart & Stork.
The decision partitionsf the feature space into k subspaces
  ik1 i
i   j   , i  j
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3
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1
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