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# Machine Learning and Review - PowerPoint PPT Presentation

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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### Machine Learning and Review

• 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

• 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)

• 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)

• 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

• 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

• Bayesian learning is optimal

• Easy to estimate P(h) by counting in training data

• Estimating P(D|h) not feasible

• Why?

• 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

VNB=argmax P(vj)∏P(ai|vj) vjV

• 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

• 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

• Practical

• As effective and in some cases, more so, than other machine learners

• 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

• Multiple choice

• Problem solving

• Essay

• An example midterm will be posted under links

• Any words in yellow or light blue or pink on slides

• Depth-first

• Iterative Deepening

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

• 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

• 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

• 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

• 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?

• 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

• Best-first

• A*

• Greedy

• Hill climbing

• Variants

• Randomness, Simulated annealing, Local beam search,

• Online search will not be on midterm

• 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

• 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

• 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’)

• 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

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

• 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

• 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

• Complete-state formulation

• Every state is a compete assignment that might or might not satisfy the constraints

• Hill-climbing methods are appropriate

• 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?

• 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

• Given a variable, chose the least constraining value

• The value that rules out the fewest values in the remaining variables

• Keep track of remaining legal values for unassigned variables

• Terminate search when any variable has no legal values

• Minimax

• Alpha-beta pruning

• Evaluation function (what is the difference between a cost function, a utility function, a heuristic function, an evaluation function?)

• Example problem