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CMSC 671 Fall 2001

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CMSC 671Fall 2001

Class #25-26 – Tuesday, November 27 / Thursday, November 29

- Neural networks
- Bayesian learning

Machine Learning: Neural and Bayesian

Chapter 19

Some material adapted from lecture notes by Lise Getoor and Ron Parr

- Brain function (thought) occurs as the result of the firing of neurons
- Neurons connect to each other through synapses, which propagate action potential (electrical impulses) by releasing neurotransmitters
- Synapses can be excitatory (potential-increasing) or inhibitory (potential-decreasing), and have varying activation thresholds
- Learning occurs as a result of the synapses’ plasticicity: They exhibit long-term changes in connection strength
- There are about 1011 neurons and about 1014 synapses in the human brain

- Different areas of the brain have different functions
- Some areas seem to have the same function in all humans (e.g., Broca’s region); the overall layout is generally consistent
- Some areas are more plastic, and vary in their function; also, the lower-level structure and function vary greatly

- We don’t know how different functions are “assigned” or acquired
- Partly the result of the physical layout / connection to inputs (sensors) and outputs (effectors)
- Partly the result of experience (learning)

- We really don’t understand how this neural structure leads to what we perceive as “consciousness” or “thought”
- Our neural networks are not nearly as complex or intricate as the actual brain structure

- Computers are way faster than neurons…
- But there are a lot more neurons than we can reasonably model in modern digital computers, and they all fire in parallel
- Neural networks are designed to be massively parallel
- The brain is effectively a billion times faster

- Neural networks are made up of nodes or units, connected by links
- Each link has an associated weight and activation level
- Each node has an input function (typically summing over weighted inputs), an activation function, and an output

Output units

Hidden units

Input units

- Input units are set by some exterior function (think of these as sensors), which causes their output links to be activated at the specified level
- Working forward through the network, the input function of each unit is applied to compute the input value
- Usually this is just the weighted sum of the activation on the links feeding into this node

- The activation function transforms this input function into a final value
- Typically this is a nonlinear function, often a sigmoid function corresponding to the “threshold” of that node

- Backpropagation
- Cascade correlation: adding hidden units

Take it away, Chih-Yun!

Next up: Sohel

B

E

A

C

- Given training set
- Find B that best matches D
- model selection
- parameter estimation

Inducer

Data D

- Assume known structure
- Goal: estimate BN parameters q
- entries in local probability models, P(X | Parents(X))

- A parameterization q is good if it is likely to generate the observed data:
- Maximum Likelihood Estimation (MLE) Principle: Choose q* so as to maximize L

i.i.d. samples

- The likelihood decomposes according to the structure of the network
→ we get a separate estimation task for each parameter

- The MLE (maximum likelihood estimate) solution:
- for each value x of a node X
- and each instantiation u of Parents(X)
- Just need to collect the counts for every combination of parents and children observed in the data
- MLE is equivalent to an assumption of a uniform prior over parameter values

sufficient statistics

Moon-phase

- Why are the counts sufficient?

Light-level

Earthquake

Burglary

Alarm

Goal: Select the best network structure, given the data

Input:

- Training data
- Scoring function
Output:

- A network that maximizes the score

Same key property: Decomposability

Score(structure) = Si Score(family of Xi)

- Bayesian: prior over parameters and structure
- get balance between model complexity and fit to data as a byproduct

- Score (G:D) = log P(G|D) log [P(D|G) P(G)]
- Marginal likelihood just comes from our parameter estimates
- Prior on structure can be any measure we want; typically a function of the network complexity

Marginal likelihood

Prior

DeleteEA

AddEC

B

E

Δscore(C)

Δscore(A)

A

B

E

C

A

C

B

E

B

E

A

A

ReverseEA

Δscore(A)

C

C

DeleteEA

DeleteEA

AddEC

B

E

Δscore(C)

Δscore(A)

Δscore(A)

A

B

E

B

E

C

A

A

C

C

B

E

A

To recompute scores,

only need to re-score families

that changed in the last move

ReverseEA

Δscore(A)

C

- Known structure, fully observable: only need to do parameter estimation
- Unknown structure, fully observable: do heuristic search through structure space, then parameter estimation
- Known structure, missing values: use expectation maximization (EM) to estimate parameters
- Known structure, hidden variables: apply adaptive probabilistic network (APN) techniques
- Unknown structure, hidden variables: too hard to solve!

- Suppose that in some cases, we observe earthquake, alarm, light-level, and moon-phase, but not burglary
- Should we throw that data away??
- Idea: Guess the missing valuesbased on the other data

Moon-phase

Light-level

Earthquake

Burglary

Alarm

- Guess probabilities for nodes with missing values (e.g., based on other observations)
- Compute the probability distribution over the missing values, given our guess
- Update the probabilities based on the guessed values
- Repeat until convergence

- Suppose we have observed Earthquake and Alarm but not Burglary for an observation on November 27
- We estimate the CPTs based on the rest of the data
- We then estimate P(Burglary) for November 27 from those CPTs
- Now we recompute the CPTs as if that estimated value had been observed
- Repeat until convergence!

Earthquake

Burglary

Alarm