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Machine Learning and Review Reading: C. 18 Bayesian Approach Each observed training example can incrementally decrease or increase probability of hypothesis instead of eliminate an hypothesis Prior knowledge can be combined with observed data to determine hypothesis

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bayesian approach
Bayesian Approach
  • Each observed training example can incrementally decrease or increase probability of hypothesis instead of eliminate an hypothesis
  • Prior knowledge can be combined with observed data to determine hypothesis
  • Bayesian methods can accommodate hypotheses that make probabilistic predictions
  • New instances can be classified by combining the predictions of multiple hypotheses, weighted by their probabilities
applying bayes theorem
Applying Bayes Theorem
  • Best hypothesis = most probable hypothesis
    • Maximum a posteriori (MAP) hypothesis
  • Variables
      • h = hypothesis
      • D = data
  • Prior probability
      • h: P(h)
      • training data observed: P(D)
  • P(D|h) = probability of observing data D given some world where hypothesis holds
  • Bayes theorem:
      • P(h|D) = P(D|h)*P(h) P(D)
defining the map hypothesis
Defining the MAP hypothesis
  • hMAP=argmax P(h|D) hεH
  • hMAP=argmax P(D|h)*P(h) hεH P(D)

(Using Bayes Theorem)

  • hMAP=argmax P(D|h)*P(h) hεH (P(D) is a constant independent of h)
  • hMAP=argmax P(D|h) hεH(when we can make the assumption that each hypothesis h is equally probable)
bayes optimal classifier
Bayes Optimal Classifier
  • The most probable classification of the new instance by combining the predictions of all hypotheses weighted by their posterior probabilities
      • Possible classifications: vjεV
      • Argmax ∑ P(vj|hi)P(hi|D)vjεVhiεH
example
Example
  • V = {p, n}
  • P(h1|D)=.4 P(p|h1)=0 P(n,h1)=1
  • P(h2|D)=.3 P(p|h2)=1 P(n,h2)=0
  • P(h3|D)=.3 P(p|h3)=1 P(n,h3)=0
      • ∑ P(n|hi)P(hi|D) = .4hiεH
      • ∑ P(p|hi)P(hi|D) = .6

hiεH

      • Argmax ∑ P(vj|hi)P(hi|D) = p

vjε{p,n}hiεH

properties of bayesian approach
Properties of Bayesian Approach
  • Bayesian learning is optimal
  • Easy to estimate P(h) by counting in training data
  • Estimating P(D|h) not feasible
  • Why?
na ve bayes
Naïve Bayes
  • Assume independence of attributes
    • D = a1,a2,…an
    • P(a1,a2,…an|vj)=∏P(ai|vj)i
  • Substitute into VMAP formula
    • VNB=argmax P(vj)∏P(ai|vj) vjV i
estimating probabilities
Estimating Probabilities
  • What happens when the number of data elements is small?
  • Suppose true P(S-length=high|verginica)=.05
  • There are only 2 instances with C=Verginica
  • We estimate probability by nc/n or #S-length|Verginica/C-Verginica
  • #S-length|Verginica must = 0
  • Then, instead of .05 we use estimated probability of 0
  • Two problems
      • Biased underestimate of probability
      • This probability term will dominate
instead
Instead
  • Use priors as well
  • nc+mp n+m
        • Where p = prior estimate
        • M is a constant called the equivalent sample size
          • Determines how heavily to weight p relative to observed data
          • Typical method: assume a uniform prior
benefits of na ve bayes
Benefits of Naïve Bayes
  • Practical
  • As effective and in some cases, more so, than other machine learners
review for midterm
Review for Midterm
  • Concepts you should know
  • Search algorithms
      • Depth-first, breadth-first, iterative deepening, A*, greedy, hill-climbing, beam
  • Constraint propagation
  • Game playing
  • Bayesian Nets
  • A little on machine learning
midterm format
Midterm format
  • Multiple choice
  • Short answer questions
  • Problem solving
  • Essay
  • An example midterm will be posted under links
concepts
Concepts
  • Any words in yellow or light blue or pink on slides
uninformed search
Uninformed Search
  • Depth-first
  • Breadth-first
  • Iterative Deepening
formulating problems as search
Formulating Problems as Search

Given an initial state and a goal, find the sequence of actions leading through a sequence of states to the final goal state.

Terms:

  • Successor function: given action and state, returns {action, successors}
  • State space: the set of all states reachable from the initial state
  • Path: a sequence of states connected by actions
  • Goal test: is a given state the goal state?
  • Path cost: function assigning a numeric cost to each path
  • Solution: a path from initial state to goal state
breadth first
Breadth first
  • OPEN = start node; CLOSED = empty
  • While OPEN is not empty do
      • Remove leftmost state from OPEN, call it X
      • If X = goal state, return success
      • Put X on CLOSED
      • SUCCESSORS = Successor function (X)
      • Remove any successors on OPEN or CLOSED
      • Put remaining successors on right end of OPEN
  • End while
depth first
Depth-first
  • OPEN = start node; CLOSED = empty
  • While OPEN is not empty do
      • Remove leftmost state from OPEN, call it X
      • If X = goal state, return success
      • Put X on CLOSED
      • SUCCESSORS = Successor function (X)
      • Remove any successors on OPEN or CLOSED
      • Put remaining successors on left end of OPEN
  • End while
can we combine benefits of both
Can we combine benefits of both?
  • Depth limited
      • Select some limit in depth to explore the problem using DFS
      • How do we select the limit?
  • Iterative deepening
      • DFS with depth 1
      • DFS with depth 2 up to depth d
complexity analysis
Complexity Analysis
  • Completeness: is the algorithm guaranteed to find a solution when there is one?
  • Optimality: Does the strategy find the optimal solution?
  • Time: How long does it take to find a solution?
  • Space: How much memory is needed to perform the search?

Is this notion of completeness the same as completeness in logic?

cost variables
Cost variables
  • Time: number of nodes generated
  • Space: maximum number of nodes stored in memory
  • Branching factor: b
      • Maximum number of successors of any node
  • Depth: d
      • Depth of shallowest goal node
  • Path length: m
      • Maximum length of any path in the state space
informed search
Informed Search
  • Best-first
  • A*
  • Greedy
  • Hill climbing
  • Variants
      • Randomness, Simulated annealing, Local beam search,
  • Online search will not be on midterm
greedy search
Greedy Search
  • OPEN = start node; CLOSED = empty
  • While OPEN is not empty do
      • Remove leftmost state from OPEN, call it X
      • If X = goal state, return success
      • Put X on CLOSED
      • SUCCESSORS = Successor function (X)
      • Remove any successors on OPEN or CLOSED
      • Compute heuristic function for each node
      • Put remaining successors on either end of OPEN
      • Sort nodes on OPEN by value of heuristic function
  • End while
a search
A* Search
  • Try to expand node that is on least cost path to goal
  • Evaluation function = f(n)
      • f(n)=g(n)+h(n)
      • h(n) is heuristic function: cost from node to goal
      • g(n) is cost from initial state to node
  • f(n) is the estimated cost of cheapest solution that passes through n
  • If h(n) is an underestimate of true cost to goal
      • A* is complete
      • A* is optimal
      • A* is optimally efficient: no other algorithm using h(n) is guaranteed to expand fewer states
admissable heuristics
Admissable heuristics
  • A heuristic that never overestimates the cost to the goal
  • h1 and h2 are admissable heuristics
  • Consistency: the estimated cost of reaching the goal from n is no greater than the step cost of getting to n’ plus estimated cost to goal from n’
      • h(n) <=c(n,a,n’)+h(n’)
local search algorithms
Local Search Algorithms
  • Operate using a single current state
  • Move only to neighbors of the state
  • Paths followed by search are not retained
  • Iterative improvement
      • Keep a single current state and try to improve it
problems for hill climbing
Problems for hill climbing

When the higher the heuristic function the better: maxima (objective fns); when the lower the function the better: minima (cost fns)

  • Local maxima: A local maximum is a peak that is higher than each of its neighboring states, but lower than the global maximum
  • Ridges: a sequence of local maxima
  • Plateaux: an area of the state space landscape where the evaluation function is flat
some solutions
Some solutions
  • Stochastic hill-climbing
      • Chose at random from among the uphill moves
  • First-choice hill climbing
      • Generates successors randomly until one is generated that is better than current state
  • Random-restart hill climbing
      • Keep restarting from randomly generated initial states, stopping when goal is found
  • Simulated annealing
      • Generate a random move. Accept if improvement. Otherwise accept with continually decreasing probability.
  • Local beam search
      • Keep track of k states rather than just 1
csp algorithm
CSP algorithm

Depth-first search often used

  • Initial state: the empty assignment {}; all variables are unassigned
  • Successor fn: assign a value to any variable, provided no conflicts w/constraints
      • All CSP search algorithms generate successors by considering possible assignments for only a single variable at each node in the search tree
  • Goal test: the current assignment is complete
  • Path cost: a constant cost for every step
local search
Local search
  • Complete-state formulation
      • Every state is a compete assignment that might or might not satisfy the constraints
  • Hill-climbing methods are appropriate
general purpose methods for efficient implementation
General purpose methods for efficient implementation
  • Which variable should be assigned next?
  • in what order should its values be tried?
  • Can we detect inevitable failure early?
  • Can we take advantage of problem structure?
order
Order
  • Choose the most constrained variable first
      • The variable with the fewest remaining values
      • Minimum Remaining Values (MRV) heuristic
  • What if there are >1?
      • Tie breaker: Most constraining variable
      • Choose the variable with the most constraints on remaining variables
order on value choice
Order on value choice
  • Given a variable, chose the least constraining value
      • The value that rules out the fewest values in the remaining variables
forward checking
Forward Checking
  • Keep track of remaining legal values for unassigned variables
  • Terminate search when any variable has no legal values
game playing
Game Playing
  • Minimax
  • Alpha-beta pruning
  • Evaluation function (what is the difference between a cost function, a utility function, a heuristic function, an evaluation function?)
bayesian nets
Bayesian nets
  • Example problem
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