Seventh Class

Toolbox so far

Prospects for automating statistical reasoning

Bayes nets

Dempster/Shafer

Kyburg (his seventieth birthday is friday!)

Today's readings.  <>, (!), (?), caps are mine.

	From Kyburg, Epistemology and Inference (collected papers)

Lecture:

	Toolbox:

	RULE-BASED:
		logics
			horne/EXPERT SYSTEM SHELL/prolog
				MEDICAL INFORMATICS GROUP
			fuzzy
			defeasible/nonmonotonic/taxonomical trees/inheritance
				LOUI
			theorem-proving
			procedurally encoded knowledge
		decision calculi
			MEU (maximal expected utility)
				SANDHOLM
			mean-risk
			Allais (applicable theory?)
			Prospect theory (full theory?)
			arguing for decisions/practical reasoning
				LOUI
		probability calculi
			applied probability modeling
				KAMBEROVA
			bayes nets
			maximum entropy (objective bayes)

	SEARCH-BASED:
		game-based
			heuristic adversarial
				CHESS
			admissible adversarial
		objective-based
			heuristic
				simulated annealing
					MARK FRANKLIN
				gradient methods
				Monte Carlo methods
				heuristic modeling for exact methods
					RODIN
			optimization:  SSM381
				linear programming
				combinatorial optimization
				dynamic programming
				Kuhn-Tucker
				optimizing models of search
					SANDHOLM
		situation-based
			planning
			reacting
				game theory
					SANDHOLM
		constraint-satisfaction
			SALLY GOLDMAN

	LEARNING-BASED:
		computational regression techniques (coming to curve-fit)
			genetic algorithms
			neural nets
				feed-forward (standard)
					backprop learning
						KIMURA
						PNP GROUP
				recurrent (almost standard)
					KALMAN-KWASNY
				symmetric (hopfield -- constraint sat)
					PINKAS
				boltzmann (probabilistic simulation)
		statistical regression techniques (optimal curve-fit)
			linear
			non-linear
		bayesian classification
			SHANNON (med school)
		non-bayesian non-classical statistical
			dempster/shafer
			kyburg with clopper-pearson intervals/prob arguments
				LOUI
		decision trees	
			ID3
			Q4.5
				SANDHOLM
		theoretical results
			SALLY GOLDMAN


	Bayes nets:

		use conditional independence to reduce the storage size
		of joint probability distribution.  note that this is
		an excellent example of the tradeoff between representation
		and reasoning (space and time).

		cond indep:  Pr(A | B,C) = Pr(A | B)

		so we say that B screens A from C (ex. A = dies from lung 
		cancer, B = smoker for ten years, C= cool teen)

		actually, it must be that 

			Pr(A=ai | B=bk, C=ck) = Pr(A=ai | B=bk)

		for all values that A, B, and C can take on!

		give example of net and representational savings.

		For example, if with A (2 values), B (3 values), C (4 values),
		D (5 values) and a linear topology:

			A <-- B <-- C <-- D,

		joint is 2x3x4x5 = 120.  bayes net requires
		2x3 + 3x4 + 4x5 = 38 values.

		in general combinatorial savings.

		is this knowledge really available?  yes, in well-understood
		domains, or more precisely, those for which there is a causal
		or quasi-causal theory of variables' interaction.

Statistical tool box (SSM325/326):

		order statistics

		time series analysis

		linear regression (l.s.e)

		neyman-pearson theory

			hypothesis testing (reject H0 --> accept H1?)

			confidence interval estimation (width vs. confidence)

			u.m.p.m.l. estimation (power vs. likelihood)

		neyman-pearson had competitors:  fisher, good, lindley;
		we have entrenched np-theory in the social sciences.
		this is history, not philosophy or mathematics.

		but np-theory doesn't solve the reference class problem:
		sampling theory can address sample size and the significance
		of subpopulation, but cannot say among a combinatorial number
		of possible sampling classes which to use.

		also, the narrowest intervals have to be justified (on power,
		under the assumption that each estimate have the required
		likelihood).  one could use all sorts of confidence curves 
		if one wanted merely to satisfy np theory.

		bernoulli example.

statisticians are mad:  it sometimes suffices to know how to come to know
	what you are doing even if you don't know what you are doing.

dempster-shafer

	1.  interval representation, which doesn't sum to 1.

		map example.

	2.  normalized orthogonal sum rule of combination.	

		example.

kyburg

	1.  interval representation, inherited from clopper-pearson

		(yes, arbitrary, but this is DESIGNED to fill-in
		the np-theory's gaps)

	2.  rule for choosing the reference class

		needs to be supplemented with XP classes in order
		to work well.

		disagreement of intervals is a cheap significance test.

	3.  implementation in rcstat

dempster vs. kyburg.

	when [.1, .1] vs. [.9, .9] and neither is subclass, kyburg 
	gets [0, 1] and dempster (under the usual interpretation)
	gets [.4,.6] (approximately).

	when [.7, .7] and [.8, .8] and neither is subclass, kyburg
	gets [.8, .9] (approximately), and dempster gets [.74, .76]
	(i am guessing).

	so the choice of calculus depends on how one views
	"combination" and "ignorance" -- is [.5, .5] how you 
	want to represent ignorance?  any bayesian would choose
	this.  but lots of people want [0, 1] to show ignorance.

	also, when two independent reports are "positive", do you want
	the system to become more positive, or to average the positive
	reports?

	the question really is what the designer wants the system to
	do, and how one frames problems.  no theory of induction is
	a magic box that can always tell you what to believe.  they
	are really just canonical forms of statistical argument
	for what one ought to believe.

Discussion:

	simple neomonopoly approaches

		accept loans that look good

			what is a good loan?

		don't let your balance drop too low

			how low is low?

		take a loan before buying when you have no cash

Problem:

	None this week.  Please program.