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Learning and discovery

Simon GrantRice University

&John Quiggin

Risk and Sustainable Management GroupSchools of Economics and Political

Science,University of Queensland

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Web sites

• RSMG http://www.uq.edu.au/economics/rsmg/index.htm

• Quiggin http://www.uq.edu.au/economics/johnquiggin

• WebLog http://johnquiggin.com

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Learning and discovery

Every solution of a problem raises new unsolved problems; the more so the deeper the original problem and the bolder its solution. The more we learn about the world, and the deeper our learning, the more conscious, specific, and articulate will be our knowledge of what we do not know, our knowledge of our ignorance. For this, indeed, is the main source of our ignorance - the fact that our knowledge can only be finite, while our ignorance must necessarily be infinite -Popper, 1963

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Unforeseen contingencies

Unknown unknowns (Rumsfeld)

Discovery in scientific research

Problems of decision theoryGeneralizations of EU within state-act framework

Ambiguity and uncertainty

Important contribution but don’t deal with crucial problem

State space cannot be an exhaustive description

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Aims of this paper

• Provide a formal representation of the process of discovery

• Determine conditions under which standard Bayesian learning theory is applicable

• Consider implications of continuing discovery for decision theory

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Research and discovery

Example of electron beam experimentResearcher decides whether to undertake experiment to test atomic hypothesis

Act of Nature may produce unanticipated possibility

Gamma ray emission

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Decision Tree for Electron Beam Experiment without Unconsidered Possibility of Gamma Ray Emission

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Tree for Electron Beam Experiment that Includes Unconsidered Possibility of Gamma Ray Emission

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Nodes and trees

Tree structure Nodes, occurrences, histories, events, instants,

Decision nodes represent acts of the individual or nature

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Propositions

• Sentences in a formal language

• Bounded rationality means that not all sentences are expressible

• Decision propositions correspond to nodes or events

• Modal logic of tense derived from tree structure

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Knowledge and modal logic

Modal logic of knowledge

Individual knows proposition p at node n if p is true at all nodes considered possible at n

Derived operators for “knowing whether p”, and “considers p”

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The tree structure

• Lattice of trees

• More refined tree implies larger set of expressible propositions

• Maximal tree represents external or unboundedly rational viewpoint

• A subjective tree associated with each node in the maximal tree

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Propositions and the tree structure

A proposition is expressible in a given tree structure if the set of nodes at which it is true in the maximal structure corresponds to a set of nodes in the given tree structure

The more refined the tree structure, the larger the set of expressible propositions

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A Coarsening of the Tree in Figure 2

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Lattice construction

• Mappings from more refined to less refined trees, and inverse correspondences

• These induce awareness operators

• Need to show that such mappings commute

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Direct (and Indirect) Mapping from tree ’’ to coarser tree (via ’).

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Commuting Information Correspondences for tree

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Dynamics of learning and discovery

Move from a node to its successor yields new, more refined subjective tree structure

Newly expressible propositions are said to be discovered

World is characterised by continuing discovery

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Awareness of unconsidered propositions

The crucial innovation in this paper

We use an existence operator to generate statements of the form ‘There exist propositions p that I have not considered’

Hence, individuals can reason about future discoveries

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Research example

• Investigator can entertain proposition ‘If I undertake action A, I will discover new possibilities’

• This is more plausible if action A is ‘conduct experiment’ than if action A is ‘do not conduct experiment’

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Criminal investigation

• ‘Person A is a suspect’

• There may exist evidence that, if discovered would imply guilt of Person A

• The nature of this evidence may not be known

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Contracting under ambiguity (Grant, Kline & Quiggin)

• Risk sharing contract• A pays if card is black, B pays if it is white

• Unconsidered ‘grey area’ leads to a dispute

• Contracting may or may not be optimal

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Discovery and knowledge

• Proposition ‘I will be more aware in the future than I am now’

• True if world is characterised by continuing discovery

• Propositions of this kind can not be known (in modal-logic sense) to be true

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Developments

• Probabilities

• Consistency conditions for Bayesian learning

• Ambiguity and multiple priors

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Potential applications

• Theory of research and discovery

• Precautionary principle

• Entrepreneurship

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Concluding comments

• The more we know, the more we discover our own ignorance

• Analysis here both formalises and exemplifies this point

• Given a formal way of describing discovery, can recognise great gaps in our understanding of this and related processes