Informational Theories of Content and Mental Representation
Miguel Sebastian2018, Review of Philosophy and Psychology
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Can Informational Theories Account for Metarepresentation?
TopoiAn International Review of Philosophy, 2020
In this essay we discuss recent attempts to ana-lyse the notion of representation, as it is employed in cognitive science, in purely informational terms. In particular, we argue that recent informational theories cannot accommodate the existence of metarepresentations. Since metarepresentations play a central role in the explanation of many cognitive abilities, this is a serious shortcoming of these proposals.
Representation and content in some (actual) theories of perception
1988
Recent discussions in the philosophy of psychology have examined the use and legitimacy of such notions as “representation”, “content”, “computation”, and “inference” within a scientific psychology. While the resulting assessments have varied widely, ranging from outright rejection of some or all of these notions to full vindication of their use, there has been notable agreement on the considerations deemed relevant for making an assessment. The answer to the question of whether the notion of, say, representational content may be admitted into a scientific psychology has often been made to hinge upon whether the notion can be squared with our “ordinary” or “folk” style of psychological explanation, with its alleged commitment to the idiom of beliefs and desires. In this article, I proceed the other way around, starting with experimental psychology itself and asking whether such notions as the above play a legitimate role within a particular area of contemporary theory, the psychology of perception. My conclusion will be that especially the first three have a legitimate place in theories of perception, albeit one that differs from that ascribed to them in a significant portion of the philosophical literature. Along the way I develop a notion of noncognitive functional analysis, which separates the notion of contentful perceptual processing and perceptual representation from cognitive or conceptual content. (This paper has been reprinted, with stylistic revisions, in Perception and Cognition, Essays in the Philosophy of Psychology, Oxford, Clarendon Press, 2009, pp. 50-87.)
The Formats of Cognitive Representation: A Computational Account
«Philosophy of Science», 2023
Cognitive representations are typically analysed in terms of content, vehicle and format. While current work on formats appeals to intuitions about external representations, such as words and maps, in this paper we develop a computational view of formats that does not rely on intuitions. In our view, formats are individuated by the computational profiles of vehicles, i.e., the set of constraints that fix the computational transformations vehicles can undergo. The resulting picture is strongly pluralistic, it makes space for a variety of different formats, and is intimately tied to the computational approach to cognition in cognitive science and artificial intelligence.
Does Representation Need Reality? Rethinking Epistemological Issues In the Light of Recent Developments and Concepts In Cognitive Science
… Representation in the Cognitive Sciences-Does …
“Linguistic” vs.“Non-Linguistic” Representations: On the Nature of the Theoretical Constructs of the Cognitive Sciences
2008
Within the field of the cognitive sciences (CSs), frequently happened that some form of “linguistic” representations (symbols, propositional representations, and so on) have been opposed to representations of some different kind. In this paper I claim that such an opposition, in spite of its prima facie intuitive strength, cannot be stated in a precise and general way. The possible characterisations of a general notion of “linguistic”, or “symbolic” representation are either too permissive or too vague to turn out to be useful.
A Functional View Toward Mental Representations
Computer-Based Diagnostics and Systematic Analysis of Knowledge, 2009
Notes Toward a Structuralist Theory of Mental Representation
One of the principal tasks confronting cognitive science is to explain how cognitive systems are capable of representing their environment. It is a commonplace in the philosophy of mind that a theory of mental representation must be naturalistic, in the sense that it must explain mental representation without appealing to properties that are either non-physical or already representational. What is not quite so commonplace is the further injunction that such a theory must explain mental representation in a fashion consistent with its casual role in shaping behaviour. We call this the "causal constraint" on a theory of mental representation. Remarkably, even though the well-known causal, functional, and teleosemantic theories of mental representation are all naturalistic, they all violate the causal constraint. Despite their internal differences, these theories ultimately ground mental content in the behaviour of cognitive systems. And any theory that grounds mental content in behaviour is in principle unable to explain how mental representation is causally responsible for such behaviour. The task we undertake in this paper is to sketch the outlines of a naturalistic theory of mental representation that satisfies the causal constraint. We call this a structuralist theory of mental representation for reasons that will become apparent as we proceed.
Representation and mental representation
Philosophical Explorations, 2018
This paper engages critically with anti-representationalist arguments pressed by prominent enactivists and their allies. The arguments in question are meant to show that the “as-such” and “job-description” problems constitute insurmountable challenges to causal-informational theories of mental content. In response to these challenges, a positive account of what makes a physical or computational structure a mental representation is proposed; the positive account is inspired partly by Dretske’s views about content and partly by the role of mental representations in contemporary cognitive scientific modeling.
The concepts of representation and information in explanatory theories of human behavior
Focusing in experimental study of human behavior, this article discusses the concepts of information and mental representation aiming the integration of their biological, computational, and semantic aspects. Assuming that the objective of any communication process is ultimately to modify the receiver’s state, the term correlational information is proposed as a measure of how changes occurring in external world correlate with changes occurring inside an individual. Mental representations are conceptualized as a special case of information processing in which correlational information is received, recorded, but also modified by a complex emergent process of associating new elements. In humans, the acquisition of information and creation of mental representations occurs in a two-step process. First, a sufficiently complex brain structure is necessary to establishing internal states capable to co-vary with external events. Second, the validity or meaning of these representations must be gradually achieved by confronting them with the environment. This contextualization can be considered as part of the process of ascribing meaning to information and representations. The hypothesis introduced here is that the sophisticated psychological constructs classically associated with the concept of mental representation are essentially of the same nature of simple interactive behaviors. The capacity of generating elaborated mental phenomena like beliefs and desires emerges gradually during evolution and, in a given individual, by learning and social interaction.
■♥❢♦r♠❛t✐♦♥❛❧ ❚❤❡♦r✐❡s ♦❢ ❈♦♥t❡♥t ❛♥❞ ▼❡♥t❛❧
❘❡♣r❡s❡♥t❛t✐♦♥
▼❛r❝ ❆rt✐❣❛ ❛♥❞ ▼✳➪✳ ❙❡❜❛st✐á♥
✭❢♦rt❤❝♦♠✐♥❣✱
❘❡✈✐❡✇ ♦❢ P❤✐❧♦s♦♣❤② ❛♥❞ Ps②❝❤♦❧♦❣② ✮
❆❜str❛❝t
■♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ♦❢ s❡♠❛♥t✐❝ ❝♦♥t❡♥t ❤❛✈❡ ❜❡❡♥ r❡❝❡♥t❧② ❣❛✐♥✐♥❣
♣r♦♠✐♥❡♥❝❡ ✐♥ t❤❡ ❞❡❜❛t❡ ♦♥ t❤❡ ♥♦t✐♦♥ ♦❢ ♠❡♥t❛❧ r❡♣r❡s❡♥t❛t✐♦♥✳ ■♥ t❤✐s
♣❛♣❡r ✇❡ ❡①❛♠✐♥❡ ♥❡✇✲✇❛✈❡ ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ✇❤✐❝❤ ❤❛✈❡ ❛ s♣❡❝✐❛❧
❢♦❝✉s ♦♥ ❝♦❣♥✐t✐✈❡ s❝✐❡♥❝❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❛r❣✉❡ t❤❛t t❤❡s❡ t❤❡♦r✐❡s ❢❛❝❡
❢♦✉r ✐♠♣♦rt❛♥t ❞✐✣❝✉❧t✐❡s✿ t❤❡② ❞♦ ♥♦t ❢✉❧❧② s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❡rr♦r✱
❢❛❧❧ ♣r❡② t♦ t❤❡ ✇r♦♥❣ ❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠✱ ❤❛✈❡ s❡r✐♦✉s ❞✐✣❝✉❧✲
t✐❡s ❛❝❝♦✉♥t✐♥❣ ❢♦r ❛♠❜✐❣✉♦✉s ❛♥❞ r❡❞✉♥❞❛♥t r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ❢❛✐❧ t♦
❞❡❧✐✈❡r ❛ ♠❡t❛s❡♠❛♥t✐❝ t❤❡♦r② ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❛r❣✉❡
t❤❛t t❤❡s❡ ❞✐✣❝✉❧t✐❡s ❞❡r✐✈❡ ❢r♦♠ t❤❡✐r ❡①❝❧✉s✐✈❡ r❡❧✐❛♥❝❡ ♦♥ t❤❡ ♥♦t✐♦♥
♦❢ ✐♥❢♦r♠❛t✐♦♥✱ s♦ ✇❡ s✉❣❣❡st t❤❛t ♣✉r❡ ✐♥❢♦r♠❛t✐♦♥❛❧ ❛❝❝♦✉♥ts s❤♦✉❧❞ ❜❡
❝♦♠♣❧❡♠❡♥t❡❞ ✇✐t❤ ✭♦r ♣❡r❤❛♣s s✉❜st✐t✉t❡❞ ❜②✮ ❢✉♥❝t✐♦♥❛❧ ❛♣♣r♦❛❝❤❡s✳
✶ ■♥tr♦❞✉❝t✐♦♥
❘❡♣r❡s❡♥t❛t✐♦♥s ❛r❡ ♣✉③③❧✐♥❣ ❡♥t✐t✐❡s✳ ▼♦r❡ t❤❛♥ ✜❢t② ②❡❛rs ❛❢t❡r t❤❡ ❝♦❣♥✐t✐✈❡
r❡✈♦❧✉t✐♦♥ ❛♥❞ ❛t ❛ t✐♠❡ ✐♥ ✇❤✐❝❤ r❡♣r❡s❡♥t❛t✐♦♥s ❛r❡ ✇✐❞❡❧② ❛ttr✐❜✉t❡❞ ✐♥ ❝♦❣♥✐✲
t✐✈❡ s❝✐❡♥❝❡✱ ❢✉♥❞❛♠❡♥t❛❧ q✉❡st✐♦♥s ❛❜♦✉t t❤❡✐r ♥❛t✉r❡ st✐❧❧ r❡♠❛✐♥ ✉♥❛♥s✇❡r❡❞✳
❋♦r ♦♥❡ t❤✐♥❣✱ ✐t ✐s ♥♦t ♦❜✈✐♦✉s ✇❤❛t ♠❛❦❡s ✐t t❤❡ ❝❛s❡ t❤❛t ❝❡rt❛✐♥ st❛t❡s ❛r❡
r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ s♦♠❡ ❛r❡ ♥♦t✳ ❋♦r ❛♥♦t❤❡r✱ ✇❡ ❧❛❝❦ ❛ s❛t✐s❢❛❝t♦r② ❛❝❝♦✉♥t ♦❢
✇❤❛t ❞❡t❡r♠✐♥❡s r❡♣r❡s❡♥t❛t✐♦♥❛❧ ❝♦♥t❡♥t✳ ◆♦✇✱ ❣✐✈❡♥ t❤❛t ❝♦❣♥✐t✐✈❡ s❝✐❡♥t✐st
s②st❡♠❛t✐❝❛❧❧② ❛ttr✐❜✉t❡ r❡♣r❡s❡♥t❛t✐♦♥s✱ ✐t ✐s ♥♦t ✉♥r❡❛s♦♥❛❜❧❡ t♦ s✉♣♣♦s❡ t❤❛t
t❤❡② ♠✐❣❤t ❜❡ ✐♠♣❧✐❝✐t❧② ❛ss✉♠✐♥❣ ❛ s❡t ♦❢ ✐♥t✉✐t✐✈❡ ❝♦♥❞✐t✐♦♥s t❤❛t ❛r❡ s✉✣✲
❝✐❡♥t ♦r ❡✈❡♥ ♥❡❝❡ss❛r② ❢♦r ❛ st❛t❡ t♦ q✉❛❧✐❢② ❛s ❛ r❡♣r❡s❡♥t❛t✐♦♥✳ ❋♦❧❧♦✇✐♥❣ t❤✐s
✐♥t✉✐t✐♦♥✱ r❡❝❡♥t❧② s♦♠❡ ♣❤✐❧♦s♦♣❤❡rs ❛♥❞ ♣s②❝❤♦❧♦❣✐sts ❤❛✈❡ tr✐❡❞ t♦ ✉♥r❛✈❡❧
t❤✐s ✐♥t✉✐t✐✈❡ ♠❡t❤♦❞♦❧♦❣② ❛♥❞ ❞❡✈❡❧♦♣ ✐t ✐♥t♦ ❛ ❢✉❧❧✲❜❧♦✇♥ ♥❛t✉r❛❧✐st✐❝ t❤❡♦r②
♦❢ r❡♣r❡s❡♥t❛t✐♦♥✳ ❙✐♥❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ♣❧❛②s ❛♥ ❡ss❡♥t✐❛❧ r♦❧❡✱ ✇❡
✇✐❧❧ ❝❛❧❧ t❤❡♠ ❵❙❝✐❡♥t✐✜❝❛❧❧② ●✉✐❞❡❞ ■♥❢♦r♠❛t✐♦♥❛❧ ❚❤❡♦r✐❡s✬ ✭❵❙●■❚✬✱ ❢♦r s❤♦rt✮✳
■♥ t❤✐s ❡ss❛② ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❝r✐t✐❝❛❧❧② ❛ss❡ss ❙●■❚✳ ■♥ ❛ ♥✉ts❤❡❧❧✱ ✇❡ ✇✐❧❧
❛r❣✉❡ t❤❛t✱ ❡✈❡♥ ✐❢ s♦♠❡ ❙●■❚ ♠✐❣❤t ❝❛♣t✉r❡ ❝❡♥tr❛❧ ❛ss✉♠♣t✐♦♥s ✐♥ ❝✉rr❡♥t s❝✐✲
❡♥t✐✜❝ ♣r❛❝t✐❝❡✱ t❤❡② ❢❛✐❧ t♦ s❛t✐s❢❛❝t♦r✐❧② ❡①♣❧❛✐♥ t❤❡ ♥❛t✉r❡ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥s
❛♥❞ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ❝♦♥t❡♥t✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ❢♦✉r ♦❜❥❡❝t✐♦♥s ✇❡ ✇✐❧❧ ❞❡✲
✈❡❧♦♣ ✐s t❤❛t ❙●■❚ ❞♦ ♥♦t ❛❝❝♦✉♥t ❢♦r s♦♠❡ ❝❛s❡s ♦❢ ❡rr♦r✱ ❢❛❧❧ ♣r❡② t♦ t❤❡ ✇r♦♥❣
✶
❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠✱ ❤❛✈❡ s❡r✐♦✉s ❞✐✣❝✉❧t✐❡s ❛❝❝♦✉♥t✐♥❣ ❢♦r ❛♠❜✐❣✉♦✉s
❛♥❞ r❡❞✉♥❞❛♥t r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ❢❛✐❧ t♦ ❞❡❧✐✈❡r ❛ ♠❡t❛s❡♠❛♥t✐❝ t❤❡♦r② ♦❢ r❡♣✲
r❡s❡♥t❛t✐♦♥✳ ❍❡♥❝❡✱ ❛❧t❤♦✉❣❤ t❤❡s❡ ❛❝❝♦✉♥ts ♠✐❣❤t ❢❛✐t❤❢✉❧❧② ❡♠❜♦❞② ✭❛t ❧❡❛st
s♦♠❡ ♦❢✮ t❤❡ ✐♥t✉✐t✐✈❡ str❛t❡❣✐❡s ❡♠♣❧♦②❡❞ ✐♥ ♥❡✉r♦s❝✐❡♥❝❡ ✐♥ ♦r❞❡r t♦ ❛ttr✐❜✉t❡
r❡♣r❡s❡♥t❛t✐♦♥s✱ ✇❡ ✇✐❧❧ ❛r❣✉❡ t❤❛t t❤❡② ❢❛✐❧ ❛s t❤❡♦r✐❡s ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✳
✷ ❘❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ■♥❢♦r♠❛t✐♦♥
■♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ❤❛✈❡ ❛ ❧♦♥❣ ❤✐st♦r②✳ ❖♥❡ ♦❢ t❤❡ ✜rst ❛♥❞ ❜❡tt❡r ❦♥♦✇♥
✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ✇❛s ❉r❡ts❦❡✬s ✭✶✾✽✶❛✮✱ ✇❤♦ tr✐❡❞ t♦ ❛♥❛❧②③❡ s❡♠❛♥t✐❝ ❝♦♥✲
t❡♥t ❜② ❛♣♣❡❛❧✐♥❣ t♦ ✐♥❢♦r♠❛t✐♦♥❛❧ ❝♦♥t❡♥t ❛♥❞ ❞❡✜♥❡❞ ✐♥❢♦r♠❛t✐♦♥❛❧ ❝♦♥t❡♥t ✐♥
t❡r♠s ♦❢ ♣r♦❜❛❜✐❧✐t② r❡❧❛t✐♦♥s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❛❝❝♦r❞✐♥❣ t♦ ❤✐s ❛♣♣r♦❛❝❤ ❛ st❛t❡
❘ ❝❛rr✐❡s ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❛♥♦t❤❡r st❛t❡ ❙ ✐✛ ❣✐✈❡♥ ❝❡rt❛✐♥ ❜❛❝❦❣r♦✉♥❞ ❝♦♥❞✐✲
t✐♦♥s P (S | R) = 1✱ ❜✉t P (S) < 1 ❣✐✈❡♥ ❜❛❝❦❣r♦✉♥❞ ❝♦♥❞✐t✐♦♥s ❛❧♦♥❡✳ ❲❤✐❧❡ t❤❡
✐❞❡❛ ♦❢ ❡①♣❧❛✐♥✐♥❣ s❡♠❛♥t✐❝ ♣r♦♣❡rt✐❡s ✐♥ t❡r♠s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✇❛s r❡✈♦❧✉t✐♦♥✲
❛r② ❛♥❞ ✈❡r② ✐♥✢✉❡♥t✐❛❧✱ t❤❡r❡ ✇❡r❡ ❛t ❧❡❛st t✇♦ ❞❡❡♣ ♣r♦❜❧❡♠s ✇✐t❤ ❉r❡ts❦❡✬s
♣r♦♣♦s❛❧ t❤❛t ❝❛✉s❡❞ ❛ ❝♦♥t✐♥✉♦✉s ❧♦ss ♦❢ s✉♣♣♦rt ❢♦r ✐♥❢♦r♠❛t✐♦♥❛❧ ❛❝❝♦✉♥ts✳
❋✐rst ♦❢ ❛❧❧✱ ✐♥ t❤❡ ♥❛t✉r❛❧ ✇♦r❧❞ ✐t ✐s ❡①tr❡♠❡❧② ❞✐✣❝✉❧t ✭✐❢ ♥♦t ✐♠♣♦ss✐❜❧❡✮ t♦
✜♥❞ t✇♦ ❞✐✛❡r❡♥t st❛t❡s s✉❝❤ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♦♥❡ ♦❢ t❤❡♠ ♠❛❦❡s t❤❡ ♦t❤❡r
❝❡rt❛✐♥ ✭❡✈❡♥ ✐❢ ❝❡rt❛✐♥ ❜❛❝❦❣r♦✉♥❞ ❝♦♥❞✐t✐♦♥s ❛r❡ ❛ss✉♠❡❞✮✳ ❙❡❝♦♥❞❧②✱ ✐t ✇❛s
✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❢❛❝t t❤❛t r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡s ❝❛♥ ♠✐sr❡♣r❡s❡♥t✳ ❖♥
❉r❡ts❦❡✬s ❛♣♣r♦❛❝❤✱ ❛ ❜r❛✐♥ st❛t❡ ❝❛♥ r❡♣r❡s❡♥t ❛ st❛t❡ ♦❢ ❛✛❛✐rs ♦♥❧② ✐❢ ❜♦t❤
♦❜t❛✐♥✱ s♦ ❛ t②♣✐❝❛❧ ❝❛s❡ ♦❢ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥ ✭✇❤✐❝❤ ✉s✉❛❧❧② ✐♥✈♦❧✈❡s ❛♥ ❡①✐st✲
✐♥❣ st❛t❡ r❡♣r❡s❡♥t✐♥❣ ❛ ♥♦♥✲❡①✐st✐♥❣ ♦♥❡✮ ✐s r❡♥❞❡r❡❞ ✐♠♣♦ss✐❜❧❡✳✶ ❚❤❡s❡ ❛♥❞
♦t❤❡r ❞✐✣❝✉❧t✐❡s ❧❡❞ ♠♦st ♣❡♦♣❧❡ t♦ t❤✐♥❦ t❤❛t ❛ ♣✉r❡❧② ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r② ♦❢
❝♦♥t❡♥t ✇❛s ✉♥✇♦r❦❛❜❧❡✳
❘❡❝❡♥t❧②✱ ❤♦✇❡✈❡r✱ ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ❛r❡ r❡✈✐✈✐♥❣✳ ❚♦ ❛✈♦✐❞ t❤❡s❡ ♣r♦❜✲
❧❡♠s✱ ❙●■❚ ❞❡✜♥❡ ❛♥❞ ✉s❡ t❤❡ ♥♦t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✐♥ ❞✐✛❡r❡♥t ✇❛②s✳ ❚❤❡
❝♦♠♠♦♥ t❤r❡❛❞ ✐s t❤❡ r❡❥❡❝t✐♦♥ ♦❢ t❤❡ r❡q✉✐r❡♠❡♥t t❤❛t P (S | R) = 1✱ ✇❤✐❝❤
✇❛s t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ t✇♦ ♠❛✐♥ ♦❜❥❡❝t✐♦♥s t♦ ❉r❡ts❦❡✬s ❛♣♣r♦❛❝❤✳ ■♥st❡❛❞✱ ♠❛♥②
♦❢ t❤❡♠ r❡❧② ♦♥ t❤❡ ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ♣r♦❜❛❜✐❧✐t✐❡s✳ ❆❝❝♦r❞✐♥❣ t♦ ❙●■❚ ✇❤❛t
✐s r❡❧❡✈❛♥t ✐s ♥♦t ❤♦✇ ♠✉❝❤ ❛ s✐❣♥❛❧ r❛✐s❡s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥♦t❤❡r st❛t❡✱ ❜✉t
✇❤❡t❤❡r ✐t r❛✐s❡s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥♦t❤❡r st❛t❡ ♠♦r❡ t❤❛♥ ❛♥② ♦t❤❡r r❡♣r❡✲
s❡♥t❛t✐♦♥ ❞♦❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡s❡ ✈✐❡✇s ❢♦❝✉s ♦♥ t❤❡ ❞✐st✐♥❝t✐✈❡ st❛t✐st✐❝❛❧
❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ✐ts r❡❢❡r❡♥t✳
❊❧✐❛s♠✐t❤ ✭✷✵✵✵✱ ✷✵✵✺❜✱ ✷✵✵✺❛✱ ✷✵✶✸✮✱ ❢♦r ✐♥st❛♥❝❡✱ ❤❛s ❞❡❢❡♥❞❡❞ ❛♥ ✐♥❢♦r✲
♠❛t✐♦♥❛❧ t❤❡♦r② ❜❛s❡❞ ♦♥ t❤✐s ✐♥t✉✐t✐♦♥✳ ❆❝❝♦r❞✐♥❣ t♦ ❤✐♠✿
❚❤❡ s❡t ♦❢ ❡✈❡♥ts r❡❧❡✈❛♥t t♦ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❝♦♥t❡♥t ♦❢ ♥❡✉r❛❧ r❡✲
s♣♦♥s❡s ✐s t❤❡ ❝❛✉s❛❧❧② r❡❧❛t❡❞ s❡t t❤❛t ❤❛s t❤❡ ❤✐❣❤❡st st❛t✐st✐❝❛❧ ❞❡✲
♣❡♥❞❡♥❝❡ ✇✐t❤ t❤❡ ♥❡✉r❛❧ st❛t❡s ✉♥❞❡r ❛❧❧ st✐♠✉❧✉s ❝♦♥❞✐t✐♦♥s ✭❊❧✐❛✲
s♠✐t❤✱ ✷✵✵✵✱ ♣✳ ✸✹✮✳
✶ ❖❢ ❝♦✉rs❡✱ ❉r❡ts❦❡ ✭✶✾✽✶❛✮ ✇❛s ✇❡❧❧ ❛✇❛r❡ ♦❢ t❤❡s❡ ♣r♦❜❧❡♠s✱ ❛♥❞ ❤❡ tr✐❡❞ t♦ s♦❧✈❡ t❤❡♠
❜② ❞✐st✐♥❣✉✐s❤✐♥❣ ❛ ❧❡❛r♥✐♥❣ ♣❡r✐♦❞ ✭✐♥ ✇❤✐❝❤ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥ ✐s ✐♠♣♦ss✐❜❧❡✮ ❢r♦♠ ❛ ♣♦st✲
❧❡❛r♥✐♥❣ ♣❡r✐♦❞✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ✐t ✐s ✇✐❞❡❧② ❛❣r❡❡❞ t❤❛t t❤✐s ♣r♦♣♦s❛❧ st✐❧❧ ❢❛❝❡s ❞❛✉♥t✐♥❣
♣r♦❜❧❡♠s✳ ❋♦r ♦♥❡ t❤✐♥❣✱ ✐t s❡❡♠s t❤❛t t❤❡ s❛♠❡ ❞✐✣❝✉❧t✐❡s r❡❛♣♣❡❛r ❛t t❤❡ ❧❡❛r♥✐♥❣ ♣❡r✐♦❞✳
✷
❊❧✐❛s♠✐t❤ ♣✉ts ❢♦r✇❛r❞ t✇♦ ❝♦♥❞✐t✐♦♥s ❢♦r ❛ st❛t❡ ❘ t♦ r❡♣r❡s❡♥t ❙✿ ❘ r❡♣r❡s❡♥ts
❙ ✐✛ t❤❡r❡ ✐s ❛ ❝❛✉s❛❧ ❧✐♥❦ ❛♥❞ ❛ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❘ ❛♥❞
❙✳
✷ ❇✉t ❤♦✇ ❞♦❡s ❊❧✐❛s♠✐t❤ ✐♥t❡r♣r❡t t❤❡ ♥♦t✐♦♥ ♦❢ ❵st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡✬ ❄
❍❡ ❝❧❛✐♠s t❤❛t ✏❛ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❡✈❡♥ts ♠❡❛♥s t❤❛t t❤❡
♦❝❝✉rr❡♥❝❡ ♦❢ ♦♥❡ ❡✈❡♥t ❝❤❛♥❣❡s ✭❡✐t❤❡r ✐♥❝r❡❛s✐♥❣ ♦r ❞❡❝r❡❛s✐♥❣✮ t❤❡ ♣r♦❜❛❜✐❧✐t②
♦❢ t❤❡ ♦❝❝✉rr❡♥❝❡ ♦❢ t❤❡ ♦t❤❡r ❡✈❡♥t✑ ✭❊❧✐❛s♠✐t❤✱ ✷✵✵✵✱ ♣✳ ✻✾✮✳ ❚❤❡r❡ ✐s ❛ ❤✐❣❤
st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ st❛t❡s ✇❤❡♥ t❤❡ ♦❝❝✉rr❡♥❝❡ ♦❢ ♦♥❡ ♦❢ t❤❡♠
✐♥❝r❡❛s❡s ♦r ❞❡❝r❡❛s❡s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ♦t❤❡r✳
❚❤✉s✱ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢
❝♦❣♥✐t✐✈❡ s❝✐❡♥❝❡✱ ❊❧✐❛s♠✐t❤ ❤♦❧❞s t❤❛t ❝♦♥t❡♥t ✐s ✜①❡❞ ❜② t❤❡ ♣♦s✐t✐✈❡ st❛t✐st✐❝❛❧
❞❡♣❡♥❞❡♥❝❡ ♦❢ st✐♠✉❧✐ ♦♥ r❡s♣♦♥s❡s ❛♥❞ ❛❧s♦ ❜② t❤❡ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ♦❢
r❡s♣♦♥s❡s ♦♥ st✐♠✉❧✐ ✭✐✳❡✳
P (R | S)
❛♥❞
P (S | R)✮✳
❚❤❛t ✐s ✇❤❛t ❤❡ ❝❛❧❧s t❛❦✐♥❣
t❤❡ ❵♦❜s❡r✈❡r✬ ❛♥❞ t❤❡ ❵❛♥✐♠❛❧✬s ♣❡rs♣❡❝t✐✈❡✬✱ r❡s♣❡❝t✐✈❡❧② ✭❊❧✐❛s♠✐t❤✱ ✷✵✵✺❜✮✳ ❆
✸ t❤❡ ❡♥t✐t② ✇✐t❤ ✇❤✐❝❤ ✐t ❤❛s ❛ ❤✐❣❤❡r st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝②✳
st❛t❡ r❡♣r❡s❡♥ts
❆ ♣r❡❝✐s❡ s✉♠♠❛r② ❛♥❞ ❡❧❡❣❛♥t ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤✐s ✐❞❡❛ ❤❛s ❛❧s♦ ❜❡❡♥ ❞❡✲
❢❡♥❞❡❞ ❜② ❯s❤❡r ✭✷✵✵✶✮✳ ❯s❤❡r✱ ✇❤♦ ❡①♣❧✐❝✐t❧② ❜❛s❡s ❤✐s ❛♣♣r♦❛❝❤ ♦♥ ❙❤❛♥♥♦♥✬s
♥♦t✐♦♥ ♦❢ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥✱ ♣r♦♣♦s❡s t❤❛t ❘ r❡♣r❡s❡♥ts ❙ ✐✛ ✭✶✮ t❤❡ ♠✉t✉❛❧
✐♥❢♦r♠❛t✐♦♥ t❤❛t ❘ ❝❛rr✐❡s ❛❜♦✉t ❙ ✐s ❣r❡❛t❡r t❤❛t t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❘ ❝❛rr✐❡s
❛❜♦✉t ❛♥② ♦t❤❡r ❡♥t✐t② ❛♥❞ ✭✷✮ t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ t❤❛t ❙ ❝❛rr✐❡s ❛❜♦✉t ❘
✐s ❣r❡❛t❡r t❤❛♥ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❙ ❝❛rr✐❡s ❛❜♦✉t ❛♥② ♦t❤❡r r❡♣r❡s❡♥t❛t✐♦♥✳ ▼♦r❡
❢♦r♠❛❧❧②✿
■♥❢♦ Ri
r❡♣r❡s❡♥ts
Si
✐✛ ❢♦r ❛❧❧
✶✳
P (Ri | Si ) > P (Ri | Sj )
✷✳
P (Si | Ri ) > P (Si | Rj )✹
j 6= i
❚❤❡s❡ t✇♦ ❝♦♥❞✐t✐♦♥s ❛r❡ s✉♣♣♦s❡❞ t♦ ❝❛♣t✉r❡ t❤❡ t✇♦ ❞✐♠❡♥s✐♦♥s t❤❛t ❛r❡ r❡❧✲
❡✈❛♥t ❢♦r ❝♦♥t❡♥t ❞❡t❡r♠✐♥❛t✐♦♥✿ t❤❡ ❜❛❝❦✇❛r❞ ❛♥❞ ❢♦r✇❛r❞ ♣r♦❜❛❜✐❧✐t✐❡s ✭❝♦r✲
r❡s♣♦♥❞✐♥❣ t♦ ✇❤❛t ❊❧✐❛s♠✐t❤ ❝❛❧❧s ❵♦❜s❡r✈❡r✬ ❛♥❞ ❵❛♥✐♠❛❧ ♣❡rs♣❡❝t✐✈❡s✬✮✳ ❚❤❡
✜rst ❝♦♥❞✐t✐♦♥ ❝❧❛✐♠s t❤❛t ❛♠♦♥❣ ❛❧❧ t❤❡ ❡♥t✐t✐❡s t❤❛t ✐♥❝r❡❛s❡ t❤❡ ♣r♦❜❛❜✐❧✐t②
♦❢
Ri
♦❝❝✉rr✐♥❣✱
Si
✐s t❤❡ ♦♥❡ t❤❛t ✐♥❝r❡❛s❡s t❤✐s ♣r♦❜❛❜✐❧✐t② ♠♦st✳ ❙♦✱ ❢♦r ❡①✲
❛♠♣❧❡✱ ❛❧t❤♦✉❣❤ t❤❡ ❛❝t✐✈❛t✐♦♥ ♦❢ ❝❡rt❛✐♥ ♥❡✉r❛❧ ♣♦♣✉❧❛t✐♦♥✱ ✇❤✐❝❤ ✇❡ ❝❛♥ ❝❛❧❧
❵❉❖●✬✱ ❝❛♥ ❜❡ tr✐❣❣❡r❡❞✱ ✉♥❞❡r s♦♠❡ ❝✐r❝✉♠st❛♥❝❡s✱ ❜② ❞♦❣s✱ ✇♦❧❢s✱ ❢❛t ❝❛ts✱
✷ ◆♦t❡✱ ❤♦✇❡✈❡r✱ t❤❛t t❤❡ ❡❧❡♠❡♥t t❤❛t ✐s ❞♦✐♥❣ t❤❡ r❡❛❧ ✇♦r❦ ✐♥ ✜①✐♥❣ ❝♦♥t❡♥t ✐s t❤❡
st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ r❡❧❛t✐♦♥❀ t❤❡ ❝❛✉s❛❧ ❡❧❡♠❡♥t ✐s ♠❛✐♥❧② ✐♥tr♦❞✉❝❡❞ ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞
❝❡rt❛✐♥ ❝♦✉♥t❡r❡①❛♠♣❧❡s✱ s✉❝❤ ❛s ❝❛s❡s ✐♥✈♦❧✈✐♥❣ t✇♦ ♠❡♥t❛❧ st❛t❡s ✇✐t❤ ❛ ❝♦♠♠♦♥ ❝❛✉s❡
✭❊❧✐❛s♠✐t❤✱ ✷✵✵✵✱ ♣✳ ✺✾✮✳
✸ ❆ t❡r♠✐♥♦❧♦❣✐❝❛❧ ♥♦t❡✿ ✐♥ t❤✐s ❡ss❛②✱ ✇❡ ❝❛❧❧ ❵r❡♣r❡s❡♥t❛t✐♦♥❛❧ ❝♦♥t❡♥t✬ ✇❤❛t ❊❧✐❛s♠✐t❤
❝❛❧❧s ❵r❡❢❡r❡♥t✬ ✭❤❡ ❞✐st✐♥❣✉✐s❤❡s ❵r❡❢❡r❡♥t✬ ❢r♦♠ ❵❝♦♥t❡♥t✬ ✉♥❞❡rst♦♦❞ ✐♥ ❛ ❞✐✛❡r❡♥t s❡♥s❡✮✳
✹ ❚❤❡ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ❙❤❛♥♥♦♥✬s ♠❡❛s✉r❡ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❜❡❝♦♠❡s ❝❧❡❛r ♦♥❝❡ ✐t ✐s ♥♦t❡❞
t❤❛t ✶ ❛♥❞ ✷ ❛r❡ s✐♠♣❧✐✜❝❛t✐♦♥s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t✐❡s ✭❯s❤❡r✱ ✷✵✵✶✱ ♣✳ ✸✷✶✮✿
P (Ri |Sj )
i |Si )
✭❛✮ M I(Ri ; Si ) = log P P(R(R
> log P (R
= M I(Ri ; Sj )
i)
i)
P (Si |Rj )
i |Ri )
✭❜✮ M I(Ri ; Si ) = log P P(S(S
> log P (S
= M I(Rj ; Si )
i)
i)
❆s ❯s❤❡r ♥♦t❡s ✭♣✳✸✷✵✮✱ s✐♥❝❡ t❤❡ ❧♦❣❛r✐t❤♠ ✐s ❛ ♠♦♥♦t♦♥✐❝ ❢✉♥❝t✐♦♥✱ ❛♥❞ ✇❡ ♦♥❧② ♠❛❦❡ ✉s❡
♦❢ ♦r❞✐♥❛❧ r❡❧❛t✐♦♥s✱ ✇❡ ❝❛♥ r❡❧② ♦♥ ❡①♣✭▼■✮ t❤❛t ♣r♦✈✐❞❡s t❤❡ s❛♠❡ ❡①♣r❡ss✐♦♥ ❜✉t ✇✐t❤♦✉t
t❤❡ ❧♦❣❛r✐t❤♠✳ ❆s ✐♥ ✶ ❛♥❞ ✷ ✐♥ ■◆❋❖✳
✸
❡t❝✳✱ ❉❖● r❡♣r❡s❡♥ts ❞♦❣s ♣❛rt❧② ❜❡❝❛✉s❡ ❛♠♦♥❣ ❛❧❧ t❤❡ st✐♠✉❧✐ ❡❧✐❝✐t✐♥❣ ❉❖●✱
❞♦❣s ❛r❡ t❤❡ ❡♥t✐t✐❡s t❤❛t ❜❡tt❡r ♣r❡❞✐❝t t♦❦❡♥s ♦❢ ❉❖●✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡
✇✐❧❧ s♦♠❡t✐♠❡s ❡①♣r❡ss t❤✐s ✐❞❡❛ ❜② s❛②✐♥❣ t❤❛t ❉❖● ✬tr❛❝❦s✬ ❞♦❣s ❜❡tt❡r✳ ❚❤❡
s❡❝♦♥❞ r❡q✉✐r❡♠❡♥t ❞♦❡s ♥♦t ❝♦♠♣❛r❡ st✐♠✉❧✐ ❜✉t r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡s✳ ❚❤❡
✐❞❡❛ ✐s t❤❛t Ri r❡♣r❡s❡♥ts Si ♦♥❧② ✐❢ Ri ✐s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡ t❤❛t ♠♦st
✐♥❝r❡❛s❡s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ Si ❜❡✐♥❣ t❤❡ ❝❛s❡✳ ❍❡r❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ♠❛tt❡rs
✐s t❤❡ ❜❛❝❦✇❛r❞ ♣r♦❜❛❜✐❧✐t②✱ ✐✳❡✳ ❝♦♥❞✐t✐♦♥❛❧✐③❡❞ ♦♥ r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡s✳ ❋♦r
✐♥st❛♥❝❡✱ s✉♣♣♦s❡ t❤❛t t❤❡r❡ ✐s ❛ ♥❡✉r❛❧ ♣♦♣✉❧❛t✐♦♥ t❤❛t ✐s s❡♥s✐❜❧❡ t♦ ❛❧❧ ❦✐♥❞s
♦❢ ♠❛♠♠❛❧s ✭❝❛❧❧ ✐t ❵▼❆▼▼❆▲✬✮✳ ❊✈❡♥ ✐❢ ▼❆▼▼❆▲ ✐s s♦♠❡t✐♠❡s tr✐❣❣❡r❡❞ ❜②
❞♦❣s✱ t❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ❞❡t❡r♠✐♥❡s t❤❛t ❉❖●✱ ❜✉t ♥♦t ▼❆▼▼❆▲✱ r❡♣r❡✲
s❡♥ts ❞♦❣s✱ ❜❡❝❛✉s❡ ✐t ✐s ♠♦r❡ ♣r♦❜❛❜❧❡ t❤❛t t❤❡r❡ ✐s ❛ ❞♦❣ ✐❢ ❉❖● ✐s ✐♥st❛♥t✐❛t❡❞
t❤❛♥ ✐❢ ▼❆▼▼❆▲ ✐s t♦❦❡♥❡❞✿ P (Dog | DOG) > P (Dog | M AM M AL)✳ ■♥✲
❞❡❡❞✱ t❤✐s ✐♥❡q✉❛❧✐t② ❤♦❧❞s ❡✈❡♥ ✐❢ ♥♦ ♦t❤❡r st✐♠✉❧✉s ✐s ❛s ❡✣❝✐❡♥t ✐♥ tr✐❣❣❡r✐♥❣
▼❆▼▼❆▲ ❛s ❞♦❣s ❛r❡ ✐♥ t❤❡ ✐♥❞✐✈✐❞✉❛❧✬s ❡♥✈✐r♦♥♠❡♥t✳
❇♦t❤ ❊❧✐❛s♠✐t❤✬s ❛♥❞ ❯s❤❡r✬s ♣r♦♣♦s❛❧s tr② t♦ ♥❛t✉r❛❧✐③❡ r❡♣r❡s❡♥t❛t✐♦♥s ❜②
❛♣♣❡❛❧✐♥❣ t♦ ❤✐❣❤ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝② r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞
r❡♣r❡s❡♥t❛t❛✳ ❆ s❧✐❣❤t❧② ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤ ✐s s✉❣❣❡st❡❞ ❜② ❘✉♣❡rt ✭✶✾✾✾✮✳ ❆❧✲
t❤♦✉❣❤ ❤❡ ❛❧s♦ ❛♥❛❧②③❡s r❡♣r❡s❡♥t❛t✐♦♥s ✐♥ t❡r♠s ♦❢ ♣r♦❜❛❜✐❧✐t② r❡❧❛t✐♦♥s ❜❡t✇❡❡♥
❡♥t✐t✐❡s✱ ❤❡ ♦♥❧② ❝♦♥s✐❞❡rs ❢♦r✇❛r❞ ♣r♦❜❛❜✐❧✐t✐❡s ✭✐✳❡✳ ❝♦♥❞✐t✐♦♥❛❧✐③❡❞ ♦♥ ❡♥t✐t✐❡s
r❛t❤❡r t❤❛♥ ♦♥ st❛t❡s✮ ❛♥❞ ❤❡ r❡str✐❝ts ❤✐s ❛❝❝♦✉♥t t♦ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ♥❛t✲
✉r❛❧ ❦✐♥❞s✳ ❖♥ t❤✐s ❛❝❝♦✉♥t✱ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ❝♦♥t❡♥t ❡①❝❧✉s✐✈❡❧② ❞❡♣❡♥❞s ♦♥
✇❤❡t❤❡r ♠❡♠❜❡rs ♦❢ ❙ ❛r❡ ♠♦r❡ ❡✣❝✐❡♥t ✐♥ t❤❡✐r ❝❛✉s✐♥❣ ❘✺ t❤❛♥ ❛r❡ ♠❡♠❜❡rs
♦❢ ❛♥② ♦t❤❡r ❦✐♥❞✳ ❍✐s ❛❝❝♦✉♥t ❝❛♥ ❜❡ ❡✛❡❝t✐✈❡❧② ❝♦♥s✐❞❡r❡❞ ❛ ✈❡rs✐♦♥ ♦❢ ❙●■❚✱
✐♥ ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥ ✶ ♦❢ ❙●■❚ ✐s ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❢♦r ❛ st❛t❡ t♦ r❡♣r❡s❡♥t
❛♥♦t❤❡r st❛t❡✱ ♦♥❝❡ Si ❛♥❞ Sj ❛r❡ r❡str✐❝t❡❞ t♦ st❛t❡s ✐♥✈♦❧✈✐♥❣ ♥❛t✉r❛❧ ❦✐♥❞s✳
❙●■❚ ❤❛✈❡ ❝❡rt❛✐♥ ❢❡❛t✉r❡s t❤❛t ♠❛❦❡ t❤❡♠ ✇♦rt❤ ❝♦♥s✐❞❡r✐♥❣ ✐♥ ❞❡t❛✐❧✳
❋♦r ♦♥❡ t❤✐♥❣✱ t❤❡② s❡❡♠ t♦ s♦❧✈❡ t❤❡ t✇♦ ♠♦st ♣r❡ss✐♥❣ ♣r♦❜❧❡♠s ♦❢ ❉r❡ts❦❡✬s
❛♣♣r♦❛❝❤✱ ♥❛♠❡❧② t❤❡ ♣r♦❜❧❡♠ ♦❢ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ✐ts ❡♠♣✐r✐❝❛❧ ✐♠♣❧❛✉✲
s✐❜✐❧✐t②✳ ❋✐rst✱ s✐♥❝❡ t❤❡② r❡❥❡❝t ❉r❡ts❦❡✬s s✉❣❣❡st✐♦♥ t❤❛t t❤❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ t❤❡
r❡❢❡r❡♥t ❣✐✈❡♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❤❛s t♦ ❜❡ ♦♥❡✱ t❤❡s❡ t❤❡♦r✐❡s ♠❛❦❡ ✐t ♣♦ss✐❜❧❡
❢♦r ❛ st❛t❡ t♦ r❡♣r❡s❡♥t ❙ ✇❤❡♥ ❙ ✐s ♥♦t t❤❡ ❝❛s❡✳ ❘❡♣r❡s❡♥t❛t✐♦♥❛❧ r❡❧❛t✐♦♥s ❛r❡
❣r♦✉♥❞❡❞ ♦♥ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝✐❡s ❜❡t✇❡❡♥ ❡♥t✐t✐❡s✱ s♦ ✐♥ ❛ ❣✐✈❡♥ ♦❝❝❛s✐♦♥ ❛
r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡ ♠✐❣❤t ❜❡ ❝❛✉s❡❞ ❜② ❛♥ ❡♥t✐t② t❤❛t ✐s ♥♦t ✐♥ ✐ts ❡①t❡♥s✐♦♥
✭s❡❡ ❜❡❧♦✇✮✳ ❙❡❝♦♥❞❧②✱ ❙●■❚ ❛r❡ ❛❧s♦ ❢❛r ♠♦r❡ r❡❛❧✐st✐❝ t❤❛♥ ♣r❡✈✐♦✉s ♣r♦♣♦s❛❧s
✐♥ t❤✐s tr❛❞✐t✐♦♥✳ ■♥❞❡❡❞✱ ❛s t❤❡② ✐♥s✐st✐♥❣❧② ♣♦✐♥t ♦✉t✱ t❤❡s❡ ❛❝❝♦✉♥ts ♠✐❣❤t
❛❝t✉❛❧❧② ❝❛♣t✉r❡ t❤❡ ✇❛② ♥❡✉r♦s❝✐❡♥t✐sts r❡❛s♦♥ ✭❯s❤❡r✱ ✷✵✵✶✱ ♣✳ ✸✷✵✮✳✻ ❋♦r
✐♥st❛♥❝❡✱ ❢♦❧❧♦✇✐♥❣ ❍✉❜❡❧ ❛♥❞ ❲✐❡s❡❧✬s ✭✶✾✺✾✮ ♣✐♦♥❡❡r✐♥❣ ♠❡t❤♦❞♦❧♦❣②✱ ♠❛♥②
♥❡✉r♦s❝✐❡♥t✐sts ✐❞❡♥t✐❢② t❤❡ r❡❢❡r❡♥t ♦❢ ❛ ♥❡✉r♦♥❛❧ str✉❝t✉r❡ ✐♥ ❡❛r❧② ✈✐s✐♦♥ ✇✐t❤
✺ ❘✉♣❡rt ❞♦❡s ♥♦t ❢♦r♠✉❧❛t❡ ❤✐s ❛♣♣r♦❛❝❤ ✐♥ t❡r♠s ♦❢ r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡s✱ ❜✉t ❛s ❛♣♣❧②✐♥❣
t♦ ✬t❡r♠s ✐♥ ❛ ❧❛♥❣✉❛❣❡ ♦❢ t❤♦✉❣❤t✬✳
◆♦♥❡t❤❡❧❡ss✱ s✐♥❝❡ t❤❡ ❡①❛❝t ♥❛t✉r❡ ♦❢ t❤❡ ❡♥t✐t② t❤❛t
❞♦❡s t❤❡ r❡♣r❡s❡♥t✐♥❣ ✐s ✐rr❡❧❡✈❛♥t ❢♦r ♦✉r ❞✐s❝✉ss✐♦♥✱ ✇❡ ❞❡s❝r✐❜❡ ❛❧❧ ❛❝❝♦✉♥ts ❛s t❛❧❦✐♥❣ ❛❜♦✉t
st❛t❡s✳
✻ ❚❤❡
❢❛❝t t❤❛t
■♥❢♦
♠✐❣❤t ❝❛♣t✉r❡ t❤❡ ✐♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥s ♠❛❞❡ ❜② ♥❡✉r♦s❝✐❡♥t✐sts ✐♥
❡st❛❜❧✐s❤✐♥❣ ❤②♣♦t❤❡s❡s ❛❜♦✉t r❡♣r❡s❡♥t❛t✐♦♥❛❧ r❡❧❛t✐♦♥s ❞♦❡s ♥♦t ♠❡❛♥ t❤❛t ❙●■❚ ❛r❡ ♣✉r❡❧②
❞❡s❝r✐♣t✐✈❡ t❤❡♦r✐❡s✳ ❊❧✐❛s♠✐t❤ ✭✷✵✵✺❜✮✱ ❢♦r ❡①❛♠♣❧❡✱ ❤❛s ❝r✐t✐❝✐③❡❞ ♥❡✉r♦s❝✐❡♥t✐sts ❢♦r ❡①❝❡s✲
s✐✈❡❧② r❡❧②✐♥❣ ♦♥ ✇❤❛t ❤❡ ❝❛❧❧s ❵t❤❡ ♦❜s❡r✈❡r✬s ♣❡rs♣❡❝t✐✈❡✬ ✭✐✳❡✳
t❤❡ ❵❛♥✐♠❛❧✬s ♣❡rs♣❡❝t✐✈❡✬ ✭✐✳❡✳
P (S | R)✮✳
✹
P (R | S)✮ ❛♥❞ ❢♦r❣❡tt✐♥❣ ❛❜♦✉t
t❤❡ st✐♠✉❧✉s t❤❛t ✐s ♠♦r❡ ❧✐❦❡❧② t♦ ❡❧✐❝✐t ❛ str♦♥❣❡r r❡s♣♦♥s❡✳ ❆❧♦♥❣ t❤❡ s❛♠❡
❧✐♥❡s✱ ❛♥ ❛❞❞✐t✐♦♥❛❧ ✈✐rt✉❡ ♦❢ t❤❡s❡ ❛♣♣r♦❛❝❤❡s ✐s t❤❛t t❤❡② ♣r♦✈✐❞❡ ❛ ♣r❡❝✐s❡
♠❡t❤♦❞ ❢♦r ❞✐s❝♦✈❡r✐♥❣ t❤❡ ❝♦♥t❡♥t ♦❢ ♥❡✉r❛❧ ❡✈❡♥ts✳ ❚❤❡② ♠❛❦❡ ✈❡r② ❞❡t❡r♠✐✲
♥❛t❡ ♣r❡❞✐❝t✐♦♥s ❛❜♦✉t t❤❡ ❝♦♥t❡♥t ♦❢ r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡s✱ ✇❤✐❝❤ ♠✐❣❤t ❜❡
❡①tr❡♠❡❧② ✈❛❧✉❛❜❧❡ ✐♥ s❝✐❡♥t✐✜❝ ♣r♦❥❡❝ts ✭❊❧✐❛s♠✐t❤✱ ✷✵✵✵✱ ♣✳ ✼✶❀ ❙❦②r♠s✱ ✷✵✶✵❛✮✳
❆♥❞ t❤❡② ❛❧s♦ ❛tt❡♠♣t t♦ ❡①♣❧❛✐♥ ❛♥❞ ✈✐♥❞✐❝❛t❡ t❤❡ ♥❛t✉r❡ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥s
❛ttr✐❜✉t❡❞ ❜② ❝♦❣♥✐t✐✈❡ s❝✐❡♥t✐sts✳
❋♦r t❤❡s❡ ❛♥❞ ♦t❤❡r r❡❛s♦♥s✱ ✐♥ r❡❝❡♥t ②❡❛rs t❤❡r❡ ❤❛s ❜❡❡♥ ❛ ❣r♦✇✐♥❣ ❜♦❞② ♦❢
✐♥t❡r❡st ♦♥ ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ✭❙❦②r♠s✱ ✷✵✶✵❜❀ ❇✐r❝❤✱ ✷✵✶✹✮ ❛♥❞ ✐ts r❡❧❛t✐♦♥
t♦ ❝♦❣♥✐t✐✈❡ s❝✐❡♥❝❡ ✭❡✳❣✳ P✐❝❝✐♥✐♥✐ ❛♥❞ ❙❝❛r❛♥t✐♥♦✱ ✷✵✶✵❀ ❙t❡❣♠❛♥♥✱ ✷✵✶✸❀ ✷✵✶✺✮✳
■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ❤♦✇❡✈❡r✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛r❣✉❡ t❤❛t t❤❡s❡ t❤❡♦r✐❡s ❢❛❧❧ ♣r❡②
t♦ ✐♠♣♦rt❛♥t ❞✐✣❝✉❧t✐❡s✳ ❚❤❡s❡ ♦❜❥❡❝t✐♦♥s s✉❣❣❡st t❤❛t t❤❡ t♦♦❧s ❡♠♣❧♦②❡❞ ❜②
❙●■❚ ❛r❡ ♣r♦❜❛❜❧② ✐♥❛❞❡q✉❛t❡ ❢♦r ❛♥❛❧②③✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡s✳ ■❢ t❤❡s❡
❛r❣✉♠❡♥ts ❛r❡ ♦♥ t❤❡ r✐❣❤t tr❛❝❦✱ t❤❡ ♣r♦s♣❡❝ts ♦❢ ❙●■❚ ✇♦✉❧❞ ♥❡❡❞ t♦ ❜❡
s❡r✐♦✉s❧② r❡✈✐s❡❞✳
✸ Pr♦❜❧❡♠s ❢♦r ❙●■❚
❲❡ ✇✐❧❧ ♣r❡s❡♥t ❢♦✉r ♦❜❥❡❝t✐♦♥s ❛❣❛✐♥st ❙●■❚✿ t❤❡② s✉✛❡r ❢r♦♠ t❤❡ ♣r♦❜❧❡♠ ♦❢
❡rr♦r✱ ❞❡❧✐✈❡r ♣r♦①✐♠❛❧ ❝♦♥t❡♥ts✱ ❞♦ ♥♦t ❛❧❧♦✇ ❢♦r st❛t❡s ✇✐t❤ ❞✐s❥✉♥❝t✐✈❡ ❝♦♥t❡♥ts
♦r ♠✉❧t✐♣❧❡ st❛t❡s ✇✐t❤ t❤❡ s❛♠❡ ❝♦♥t❡♥t✱ ❛♥❞ ❢❛✐❧ t♦ ♣r♦✈✐❞❡ ❛ ♠❡t❛s❡♠❛♥t✐❝
❛❝❝♦✉♥t ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✳ ❖✉r ✜rst ❣♦❛❧ ✐s t♦ ❛r❣✉❡ t❤❛t ♥❡✇ ✐♥❢♦r♠❛t✐♦♥❛❧
t❤❡♦r✐❡s ❢❛❝❡ t❤❡s❡ ❞✐✣❝✉❧t✐❡s✳ ❙❡❝♦♥❞❧②✱ ✇❡ ✇✐❧❧ s❤♦✇ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s r♦♦t❡❞
✐♥ s♦♠❡ ❛s♣❡❝t ♦❢ t❤❡ ♥♦t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❛t r❡s✉❧t ✇♦✉❧❞ str♦♥❣❧② s✉❣❣❡st
t❤❛t ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ♥❡❡❞ t♦ ❜❡ s✉♣♣❧❡♠❡♥t❡❞ ✇✐t❤ ✭♦r ♣❡r❤❛♣s✱ r❡♣❧❛❝❡❞
❜②✮ ❛ t❤❡♦r② ✇✐t❤ ❛ ❞✐✛❡r❡♥t s❡t ♦❢ t♦♦❧s✳
✸✳✶
❊rr♦r
▲❡t ✉s ❜❡❣✐♥ ✇✐t❤ t❤❡ ✜rst ❞✐✣❝✉❧t② ♦❢ ❝❧❛ss✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s t❤❛t ❙●■❚
✇❡r❡ ❞❡s✐❣♥❡❞ t♦ s♦❧✈❡✿ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❡rr♦r✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❛r❣✉❡ t❤❛t
❙●■❚ ❢❛✐❧ t♦ ❢✉❧❧② ❛❞❞r❡ss t❤❡ ♣r♦❜❧❡♠ ♦❢ ❡rr♦r t❤❛t ❝❛✉s❡❞ t❤❡ ❞✐s♠✐ss❛❧ ♦❢
♣r❡✈✐♦✉s ❛♣♣r♦❛❝❤❡s ✐♥ t❤✐s tr❛❞✐t✐♦♥✳
❋✐rst ♦❢ ❛❧❧✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t ❙●■❚ ❝❛♥ ✐♥❞❡❡❞ ❛❝❝♦✉♥t ❢♦r s♦♠❡
❡rr♦r ✭s♦ ✐t ♣r♦✈✐❞❡s ❛ s✐❣♥✐✜❝❛♥t ✐♠♣r♦✈❡♠❡♥t ♦✈❡r ❉r❡ts❦❡✬s ❛♣♣r♦❛❝❤✮✳ ❚♦
❛❝❝♦♠♠♦❞❛t❡ s♦♠❡ ❝❛s❡s ♦❢ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥✱ ❛♥ ❛❝❝♦✉♥t ♦♥❧② ♥❡❡❞s t♦ ❜❡
❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝♦♥❞✐t✐♦♥s✿ ✭✶✮ ❘ r❡♣r❡s❡♥ts ❙ ❛♥❞ ✭✷✮ ¬S ✳
❚❤✐s ✐s ❝❧❡❛r❧② ♣♦ss✐❜❧❡ ♦♥ ❙●■❚✳ ■♥❞❡❡❞✱ ❛♥② ❛♣♣r♦❛❝❤ t❤❛t r❡q✉✐r❡s ❛ ❝♦♥✲
❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② ❜❡❧♦✇ ✶ ❛✉t♦♠❛t✐❝❛❧❧② ❧❡❛✈❡s r♦♦♠ ❢♦r s♦♠❡ ❝❛s❡s ♦❢ ❡rr♦r
✭❑r❡❛♠❡r✱ ✷✵✶✸✮✳ ❋♦r ✐♥st❛♥❝❡✱ ❛ t❤❡♦r② ❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ ❘ r❡♣r❡s❡♥t❡❞ ❙ ♦♥❧②
✐❢ P (R | S) ≥ 0.8✱ ✇♦✉❧❞ ❝❡rt❛✐♥❧② ❛❧❧♦✇ ❢♦r s♦♠❡ ❝❛s❡s ♦❢ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥✳
❍♦✇❡✈❡r✱ ✇❤❡r❡❛s ❙●■❚ ❝❛♥ ❛❝t✉❛❧❧② ❛❝❝♦✉♥t ❢♦r ♦❝❝❛s✐♦♥❛❧ ❡rr♦r ❛♥❞ ❡✈❡♥
❢♦r s♦♠❡ ❢♦r♠s ♦❢ ❝♦♠♠♦♥ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥ ✭❯s❤❡r✱ ✷✵✵✶✱ ♣✳ ✸✸✶✱ ❢♦r ✐♥st❛♥❝❡✱
♣r❡s❡♥ts ♦♥❡ s✉❝❤ ❝❛s❡✮✱ t❤❡r❡ ❛r❡ ♣❧❛✉s✐❜❧❡ ❝❛s❡s ♦❢ ❢r❡q✉❡♥t ♠✐sr❡♣r❡s❡♥t❛t✐♦♥
t❤❛t ❝❛♥♥♦t ❜❡ ❛❝❝♦♠♠♦❞❛t❡❞✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ✇✐❧❧ ❛r❣✉❡ t❤❛t t❤❡② ❝❛♥♥♦t
✺
❛❝❝♦✉♥t ❢♦r s♦♠❡ ❝❛s❡s ✐♥ ✇❤✐❝❤ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥ ✐s ♠♦r❡ ❢r❡q✉❡♥t t❤❛t ❛❝❝✉r❛t❡
r❡♣r❡s❡♥t❛t✐♦♥✳
▲❡t ✉s tr② t♦ s♣❡❧❧ ♦✉t t❤✐s ✐❞❡❛ ✐♥ ❞❡t❛✐❧ ❛♥❞ ✐❧❧✉str❛t❡ ✐t ✇✐t❤ s♦♠❡ ❡①✲
✼ ❘❡❝❛❧❧ t❤❛t✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✜rst ❝♦♥❞✐t✐♦♥ ♦❢ ■♥❢♦✱
Ri ✬s r❡♣r❡s❡♥✲
Si t❤❛t Ri tr❛❝❦s t❤❡ ❜❡st✱
✐✳❡✳ ❢♦r ❛♥② ♦t❤❡r st❛t❡ Sj ✱ P (Ri | Si ) > P (Ri | Sj )✳ ❚♦ ❝❤❛❧❧❡♥❣❡ t❤✐s
❝❧❛✐♠✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ s♦♠❡ ❡①❛♠♣❧❡s ✐♥ ✇❤✐❝❤ Ri r❡♣r❡s❡♥ts Si ❛♥❞✱ ♥♦♥✲
t❤❡❧❡ss✱ P (Ri | Si ) < P (Ri | Sj )✳ ❈♦♥s✐❞❡r t❤❡ ❝❛s❡ ♦❢ ♠✐♠✐❝r②❀ t❤❡ ❞r♦♥❡✢②
❛♠♣❧❡s✳
t✐♦♥❛❧ ❝♦♥t❡♥t ✐s ♣❛rt❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ st❛t❡
❊r✐st❛❧✐s t❡♥❛①✱ ❢♦r ❡①❛♠♣❧❡✱ ✐s ❛ ❇❛t❡s✐❛♥ ♠✐♠✐❝ ♦❢ t❤❡ ❤♦♥❡②❜❡❡ ❆♣✐s ♠❡❧❧✐❢✲
❡r❛ ✿ t❤❡ ❢♦r♠❡r ✐s ❛ ❞❡❢❡♥s❡❧❡ss ✐♥s❡❝t t❤❛t ❤❛s ❝♦♣✐❡❞ t❤❡ ❛♣♣❡❛r❛♥❝❡ ♦❢ ❤♦♥❡②
❜❡❡ ✭✐♥❝❧✉❞✐♥❣ ✐ts ✢✐❣❤t ✕ ●♦❧❞✐♥❣ ❡t ❛❧✳
✷✵✵✶✮ t♦ ❛✈♦✐❞ ❜❡✐♥❣ ♣r❡②❡❞ ✉♣♦♥✳
■❢ t❤❡ t♦❦❡♥✐♥❣ ♦❢ t❤❡ ♣r❡❞❛t♦r✬s ❜r❛✐♥ st❛t❡
Ri
❞❡♣❡♥❞s ✉♣♦♥ t❤❡ ❛♣♣❡❛r❛♥❝❡
♦❢ t❤❡ ♣r❡②✱ ❛♥❞ t❤❡ ❞r♦♥❡✢② ♠✐♠✐❝s s✉✣❝✐❡♥t❧② ✇❡❧❧ t❤❡ ❛♣♣❡❛r❛♥❝❡ ♦❢ t❤❡
❞❛♥❣❡r♦✉s ❜❡❡✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t
t❤✐s ♠✐❣❤t ❧❡❛❞ t♦ ❛ ❝❛s❡ ✐♥ ✇❤✐❝❤
P (Ri | bee) = P (Ri | dronef ly)✳
P (Ri | bee) < P (Ri | dronef ly)
■♥❞❡❡❞✱
❢♦r t❤❡
❢♦❧❧♦✇✐♥❣ r❡❛s♦♥✿ t❤❡ ✜t♥❡ss ♦❢ t❤❡ ♠✐♠✐❝❦❡❞ ♦r❣❛♥✐s♠ ❞❡❝r❡❛s❡s ✇❤❡♥ ♠✐♠✲
✐❝s ❡①♣❛♥❞ ✭❜❡❝❛✉s❡ ✇❤❡♥ ♦♥❧② ❢❡✇ ❜❡❡✲❧♦♦❦✐♥❣ ❛♥✐♠❛❧s ❛r❡ ❛❝t✉❛❧❧② ❜❡❡s✱ ✐t
♠✐❣❤t ♣❛② ♣r❡❞❛t♦rs t♦ r✐s❦ ❛♥❞ ♣r❡② t❤❡♠ ❛♥②✇❛② ✕ s❡❡ ❈❡❝❝❛r❡❧❧✐ ❛♥❞ ❈r♦③✐❡r✱
✷✵✵✼✮❀ t❤✉s✱ ❜❡❡s ❝♦✉❧❞ ❛❝t✉❛❧❧② ❡✈♦❧✈❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❧♦♦❦✐♥❣ ✏❧❡ss ❜❡❡✲❧✐❦❡✑✱
P (Ri | bee) < P (Ri | dronef ly)✳ ❚❤✐s ✐s ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡ ♦❢
P (Ri | Si ) < P (Ri | Sj )✱ ✇❤✐❝❤ s❤♦✉❧❞ ❜❡ tr♦✉❜❧✐♥❣ ❢♦r ■♥❢♦✱ s✐♥❝❡ ✐t s❡❡♠s t❤❛t
❛♥❞ ❛s ❛ r❡s✉❧t
♣r❡❞❛t♦rs r❡♣r❡s❡♥t ❜❡❡s✱ ❛❧t❤♦✉❣❤ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ tr❛❝❦s ❞r♦♥❡✢✐❡s ❜❡tt❡r✳
◆♦✇✱ t❤❡r❡ ✐s ❛ s✐♠✐❧❛r s❝❡♥❛r✐♦ t♦ s❤♦✇s t❤❛t
✐♥❢♦ ✐s ✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ s♦♠❡
❝❛s❡s ♦❢ s②st❡♠❛t✐❝ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥✳ ❋♦r♠❛❧❧②✱ s②st❡♠❛t✐❝ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥
∗
∗
♦❝❝✉rs ✇❤❡♥ R r❡♣r❡s❡♥ts S ✱ S 6= S ❛♥❞✱ ♥♦♥❡t❤❡❧❡ss✱ P (S | Ri ) < P (S | Ri )✳
■♥ ♣r✐♥❝✐♣❧❡✱ t❤✐s s✐t✉❛t✐♦♥ ✐s ❢✉❧❧② ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s st❛t❡❞
✐♥
■♥❢♦✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤❡ ♣r♦❜❧❡♠ ♣♦✐♥t❡❞ ♦✉t ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤ s✉❣✲
❣❡sts t❤❛t t❤❡r❡ ❛r❡ s♦♠❡ ❝❛s❡s ♦❢ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥ t❤❛t ❝❛♥♥♦t ❜❡ ❛❝❝♦♠♠♦✲
❞❛t❡❞✳ ❋♦r ✐♥st❛♥❝❡✱ s✉♣♣♦s❡ t❤❛t ❜♦t❤ ❜❡❡s ❛♥❞ ❞r♦♥❡✢✐❡s ❝❛✉s❡ t❤❡ ♣r❡❞❛t♦r✬s
P (Ri | bee) = P (Ri |
P (Ri | bee) = P (Ri | dronef ly) = 0.6✳ ❚❤✐s ✐s
❜r❛✐♥ st❛t❡ ✐♥ r♦✉❣❤❧② ❛ s✐♠✐❧❛r ♣r♦♣♦rt✐♦♥ ♦❢ ❝❛s❡s✱ ✐✳❡✳
dronef ly)✳
▲❡t ✉s s❛② t❤❛t
❛ ❢♦r✇❛r❞ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② ✭✐✳❡✳ ❝♦♥❞✐t✐♦♥❛❧✐③❡❞ ♦♥ st❛t❡s✮✱ ✇❤❡r❡❛s t❤❡
♣r♦❜❧❡♠ ♦❢ s②st❡♠❛t✐❝ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥ ✐♥✈♦❧✈❡s ❜❛❝❦✇❛r❞ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛✲
❜✐❧✐t✐❡s ✭✐✳❡✳❝♦♥❞✐t✐♦♥❛❧✐③❡❞ ♦♥ r❡♣r❡s❡♥t❛t✐♦♥s✮✳
❆❝❝♦r❞✐♥❣ t♦ ❇❛②❡s✬ t❤❡♦r❡♠✱
t♦ ❣❡t t❤❛t ✈❛❧✉❡ ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s✳ ❚❤✉s✱ s✉♣♣♦s❡
✇❡ ❦♥♦✇ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❜❡❡s ❛♥❞ ❞r♦♥❡❧✐❡s✱ ❛♥❞ ✐♠❛❣✐♥❡ t❤❛t
P (bee) = 0.4 ❛♥❞
P (bee | Ri ) < P (dronef ly | Ri )✳✽ ■♥ ♦t❤❡r
✇♦r❞s✱ t❤✐s ✐s ❛ s✐t✉❛t✐♦♥ ✐♥ ✇❤✐❝❤ Ri ✐♥t✉✐t✐✈❡❧② r❡♣r❡s❡♥ts ❜❡❡s✱ ❜✉t ♠♦st ♦❢ t❤❡
t❤❡ ❧❛tt❡r ❛r❡ s❧✐❣❤t❧② ♠♦r❡ ❝♦♠♠♦♥ t❤❛♥ t❤❡ ❢♦r♠❡r✱ ❡✳❣✳
P (dronef ly) = 0.6✳
✼ ❲❡
■♥ t❤❛t ❝❛s❡✱
✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ❛♥ ❛♥♦♥②♠♦✉s r❡✈✐❡✇❡r ❢♦r ♣r❡ss✐♥❣ ✉s ♦♥ t❤✐s ✐ss✉❡✳
✽ ❆❝❝♦r❞✐♥❣
P (dronef ly | Ri )
✇❡ ♥❡❡❞ t♦
P (R |S )(S )
i i
i
✳
P (Si | Ri ) =
P (Ri )
❦♥♦✇ P (Ri )✳ ❋♦rt✉♥❛t❡❧②✱ ❛s
t♦ ❇❛②❡s✬ r✉❧❡✱
❚❤✉s✱ t♦ ❞❡r✐✈❡
P (bee | Ri )
❛♥❞
❯s❤❡r r❡♠❛r❦s ✭s❡❡ ❢♦♦♥♦t❡ ✹✮✱ ✐♥
t❤❡ ♣r❡s❡♥t ❝♦♥t❡①t t❤✐s ✈❛❧✉❡ ✐s ♥♦t r❡q✉✐r❡❞✱ ❜❡❝❛✉s❡ ✇❡ ❛r❡ ♦♥❧② ✐♥t❡r❡st❡❞ ✐♥ ❝♦♠♣❛r✐♥❣
P (bee | Ri ) ❛♥❞ P (dronef ly | Ri )✱ ❛♥❞ ✐♥ ❜♦t❤ ❝❛s❡s t❤❡ ♥✉♠❡r❛t♦r ✐s t❤❡ s❛♠❡✱ ♥❛♠❡❧②
P (Ri )✳ ❚❤✉s✱ t❤❡ ❢❛❝t t❤❛t P (Ri | bee)P (bee) < P (Ri | dronef ly)P (dronef ly) ✐s ❡♥♦✉❣❤ ❢♦r
s❤♦✇✐♥❣ t❤❛t P (bee | Ri ) < P (dronef ly | Ri )✳
✻
t✐♠❡ ✐t ✐s t♦❦❡♥❡❞ ✇❤❡♥ t❤❡r❡ ✐s ❛ ❞r♦♥❡✢② ✭❛♥❞✱ ♦❜✈✐♦✉s❧②✱ bee 6= dronef ly ✮ s♦
✐t ✐s ❛ s②st❡♠❛t✐❝ ♠✐sr❡♣r❡s❡♥t❛t✐♦♥✳ ❍♦✇❡✈❡r✱ s✐♥❝❡ ❝♦♥❞✐t✐♦♥ ✶ ✐s ♥♦t s❛t✐s✜❡❞✱
✐t ✐s ❛ ❝❛s❡ ■♥❢♦ ❝❛♥♥♦t ❛❝❝♦♠♠♦❞❛t❡✳
■♥❞❡❡❞✱ ❛s ✇❡ s❡❡ ✐t✱ t❤❡ ❞✐✣❝✉❧t② ✐s ♥♦t ❥✉st t❤❛t ✐♥ ❝❡rt❛✐♥ s❝❡♥❛r✐♦s✱ s✉❝❤
❛s t❤❡ ♦♥❡ ❞❡♣✐❝t❡❞ ❛❜♦✈❡✱ ■♥❢♦ ❞❡❧✐✈❡rs t❤❡ ✇r♦♥❣ r❡s✉❧ts✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s
t❤❛t r❡♣r❡s❡♥t❛t✐♦♥❛❧ ❝♦♥t❡♥t ❤❡❛✈✐❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢
st❛t❡s✱ ❛♥❞ t❤❛t s❡❡♠s t♦ ❜❛s❡ ❝♦♥t❡♥t ♦♥ t❤❡ ✇r♦♥❣ ❦✐♥❞ ♦❢ ❝♦♥s✐❞❡r❛t✐♦♥s✳ ■♥
❣❡♥❡r❛❧✱ Ri ❝❛♥ ♦♥❧② r❡♣r❡s❡♥t Si ❝♦♥s✐st❡♥t ✇✐t❤ ❢❛❧s❡ ♣♦s✐t✐✈❡s ♦✉t♥✉♠❜❡r✐♥❣
tr✉❡ ♣♦s✐t✐✈❡s ✐❢ P (Si ) > ¬P (Si )✳ ❇❛s✐♥❣ ❝♦♥t❡♥t ♦♥ ❝♦♠♣❛r✐s♦♥s ❜❡t✇❡❡♥
♠❛r❣✐♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s s❡❡♠s ✐♥❛❞❡q✉❛t❡❀ ✐t ❛s ✐❢ ②♦✉ ❝♦✉❧❞ ❝❤❛♥❣❡ t❤❡ ❝♦♥t❡♥t
♦❢ t❤❡ ❢r♦❣✬s ♠❡♥t❛❧ st❛t❡ ❜② s✐♠♣❧② ❦✐❧❧✐♥❣ s♦♠❡ ❜❡❡s✳
❆s ❛ r❡s♣♦♥s❡✱ ♦♥❡ ♠✐❣❤t ❜✐t❡ t❤❡ ❜✉❧❧❡t ❛♥❞ ❞❡❢❡♥❞ t❤❛t ♣r❡❞❛t♦rs ❞♦ r❡♣✲
r❡s❡♥t t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❞r♦♥❡✢✐❡s✳ ❘✉♣❡rt ✭✶✾✾✾✱ ♣✳ ✸✸✻✮✱ ❢♦r ✐♥st❛♥❝❡✱ ❛t s♦♠❡
♣♦✐♥t s❡❡♠s t♦ s✉❣❣❡st t❤✐s str❛t❡❣②✳ ❍❡ ❞✐s❝✉ss❡s ❛ s✐♠✐❧❛r ❡①❛♠♣❧❡ ✐♥ ✇❤✐❝❤
❢❡♠❛❧❡s ♦❢ ❝❡rt❛✐♥ s♣❡❝✐❡s t♦❦❡♥ ❛ ♠❡♥t❛❧ st❛t❡ t♦ ❞❡❝✐❞❡ ✇✐t❤ ✇❤♦♠ t♦ ♠❛t❡
❛♥❞ ❝❧❛✐♠s t❤❛t ✐❢ t❤✐s ♠❡♥t❛❧ st❛t❡ ❤❛s ❛ ❤✐❣❤❡r ❝♦rr❡❧❛t✐♦♥ ✇✐t❤ ❛ ♠❛❧❡ ♦❢ ❛
❞✐✛❡r❡♥t s♣❡❝✐❡s✱ t❤❛t ♠✐❣❤t ❜❡ ✇❤❛t t❤❡② ✐♥ ❢❛❝t r❡♣r❡s❡♥t✳ ◆♦♥❡t❤❡❧❡ss✱ t❤❡r❡
❛r❡ ♣♦✇❡r❢✉❧ r❡❛s♦♥s ❢♦r r❡s✐st✐♥❣ t❤✐s ♠♦✈❡✳ ■♥ ❘✉♣❡rt✬s ❡①❛♠♣❧❡✱ t❤❡ ❢❡♠❛❧❡✬s
❜❡❤❛✈✐♦r ✭❛♥❞✱ ✐♥❞❡❡❞ t❤❡ ✈❡r② ❡①✐st❡♥❝❡ ♦❢ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♠❡❝❤❛♥✐s♠✮
✇♦✉❧❞ ❜❡ ❤❛r❞ t♦ ❡①♣❧❛✐♥ ✉♥❧❡ss ✐t ✐s ❛ss✉♠❡❞ t❤❛t ✐t r❡♣r❡s❡♥ts ❝♦♥s♣❡❝✐✜❝
♠❛❧❡s✳ ❙✐♠✐❧❛r❧②✱ ✐♥ ♠✐♠✐❝r②✱ ✉♥❧❡ss ❘ ♠❡❛♥s ❜❡❡ ✭♦r ❞❛♥❣❡r♦✉s ❛♥✐♠❛❧ ♦r t❤❡
❧✐❦❡✮✱ t❤❡ ❡①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ♦r❣❛♥✐s♠✬s ♠❡♥t❛❧ st❛t❡s ❛♥❞ ❜❡❤❛✈✐♦r ✇♦✉❧❞ r❡✲
♠❛✐♥ ♠②st❡r✐♦✉s✳ ❲❤② ❞♦ ♣r❡❞❛t♦rs ❢❛✐❧ t♦ ♣r❡② ♦♥ ❛♥ ✐♥s❡❝t ✇❤❡♥ Ri ✐s t♦❦❡♥❡❞
✐❢ ✐t ♠❡❛♥s ❞r♦♥❡✢② ❛♥❞ t❤❡s❡ ✐♥s❡❝ts ❛r❡ ❤❛r♠❧❡ss ❢♦♦❞❄ ■t s❡❡♠s t❤❛t t❤❡ ♦♥❧②
✇❛② t♦ ♠❛❦❡ s❡♥s❡ ♦❢ t❤❡ ✇❤♦❧❡ ♣r♦❝❡ss ✐s ❜② s✉♣♣♦s✐♥❣ t❤❛t t❤❡ ❜r❛✐♥ st❛t❡
♠❡❛♥s s♦♠❡t❤✐♥❣ ❧✐❦❡ ❜❡❡ ✭♦r t❤❡ ❧✐❦❡✮✳✾
❆❧t❡r♥❛t✐✈❡❧②✱ ♦♥❡ ❝♦✉❧❞ tr② t♦ r❡s✐st t❤✐s ♦❜❥❡❝t✐♦♥ ❜② ❝❧❛✐♠✐♥❣ t❤❛t ♣r❡❞❛✲
t♦rs r❡♣r❡s❡♥t ♥❡✐t❤❡r ❜❡❡s ♥♦r ❞r♦♥❡✢✐❡s✱ ❜✉t ✇❤❛t❡✈❡r ❛♣♣❡❛r❛♥❝❡ ♣r♦♣❡rt②
t❤❡② s❤❛r❡ ✭s♦♠❡t❤✐♥❣ ❧✐❦❡ ❜❡❡✲❧♦♦❦✐♥❣ t❤✐♥❣ ✮✳ ❚❤✐s r❡♣❧②✱ ❤♦✇❡✈❡r✱ ♦♥❧② ♠✐❣❤t
s❡❡♠ ♣❧❛✉s✐❜❧❡ ✐❢ ♦♥❡ t❤✐♥❦s ♦❢ ♣r❡❞❛t♦rs ✇✐t❤♦✉t s♦♣❤✐st✐❝❛t❡❞ ❝♦❣♥✐t✐✈❡ ❝❛♣❛❝✲
✐t✐❡s✳ ❜✉t s✉♣♣♦s❡ t❤❛t ♣r❡❞❛t♦rs ❛r❡ ❤✉♠❛♥ ❜❡✐♥❣s✳ ❈❡rt❛✐♥❧② ✇❡ ❝❛♥ t❤✐♥❦
♦❢ ❜❡❡s✱ ♥♦t ♦♥❧② ♦❢ ❜❡❡✲❧♦♦❦✐♥❣ t❤✐♥❣s ✭♦t❤❡r✇✐s❡✱ t❤✐s ✇❤♦❧❡ s❡❝t✐♦♥ ✇♦✉❧❞ ❜❡
✉♥✐♥t❡❧❧✐❣✐❜❧❡ t♦ ②♦✉✮✳ ❍♦✇❡✈❡r✱ ✐t ✐s ❡♥t✐r❡❧② ♣♦ss✐❜❧❡ t❤❛t ♠♦st ♦❢ t❤❡ t✐♠❡ ✇❡
❛♣♣❧② t❤❡ ❝♦♥❝❡♣t ❇❊❊ t♦ ❞r♦♥❡✢✐❡s ❛♥❞ t❤❛t ❛r♦✉♥❞ ✉s ❞r♦♥❡✢✐❡s ❛r❡ ♠♦r❡
❝♦♠♠♦♥ t❤❛♥ ❜❡❡s✳ ■♥ ❛♥② ❝❛s❡✱ t❤❡ s✉❣❣❡st✐♦♥ t❤❛t r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡s ❛❝✲
t✉❛❧❧② tr❛❝❦ ♣r♦①✐♠❛❧ st❛t❡s r❛t❤❡r t❤❛♥ ❞✐st❛❧ ♦♥❡s ♣♦✐♥ts ❛t ❛ s❡r✐♦✉s ♦❜❥❡❝t✐♦♥
t♦ ■♥❢♦ t❤❛t ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳
❈♦♥s❡q✉❡♥t❧②✱ ❙●■❚ ❤❛✈❡ ♣r♦❜❧❡♠s ❛❝❝♦♠♠♦❞❛t✐♥❣ s♦♠❡ ❝❛s❡s ♦❢ ❢r❡q✉❡♥t
♠✐sr❡♣r❡s❡♥t❛t✐♦♥✳ ❆❧t❤♦✉❣❤ ♦❝❝❛s✐♦♥❛❧ ♠✐st❛❦❡s ❛♥❞ s♦♠❡ ❢♦r♠s ♦❢ r❡❣✉❧❛r
♠✐sr❡♣r❡s❡♥t❛t✐♦♥s ❛r❡ ❛❧❧♦✇❡❞✱ ♦t❤❡rs ❛r❡ r❡♥❞❡r❡❞ ✐♠♣♦ss✐❜❧❡✳ ●✐✈❡♥ t❤❛t ♦♥❡
♦❢ t❤❡✐r ♣r✐♠❛r② ♠♦t✐✈❛t✐♦♥s ❢♦r ❙●■❚ ✇❛s t♦ ❢✉❧❧② ❛❝❝♦✉♥t ❢♦r ❡rr♦r✱ t❤✐s ✐s
❛ s✐❣♥✐✜❝❛♥t r❡s✉❧t✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❛r❣✉❡❞ t❤❛t ❙●■❚ s❡❡♠ t♦ ❜❛s❡ ❝♦♥t❡♥t
✾ ❆s ❛ r❡✈✐❡✇❡r s✉❣❣❡st❡❞✱ ❛t ❧❡❛st ♦♥❡ s❤♦✉❧❞ ❣r❛♥t t❤❡ ❝♦♥❝❡♣t✉❛❧ ♣♦ss✐❜✐❧✐t② ♦❢ ♣r❡❞❛t♦rs
r❡♣r❡s❡♥t✐♥❣ ❜❡❡s ❡✈❡♥ ✐❢ t❤❡② ❢r❡q✉❡♥t❧② ♠✐st❛❦❡ ❞r♦♥❡✢✐❡s ❢♦r ❜❡❡s ❛♥❞ t❤❡ ❢♦r♠❡r ♦✉t♥✉♠✲
❜❡r t❤❡ ❧❛tt❡r✳ ❉❡♣❡♥❞✐♥❣ ♦♥ ♦♥❡✬s ♠❡t❛♣❤②s✐❝❛❧ ❛ss✉♠♣t✐♦♥s✱ t❤❡ ♠❡r❡ ❢❛❝t t❤❛t t❤✐s ✐s
❝♦♥❝❡♣t✉❛❧❧② ♣♦ss✐❜❧❡ ♠✐❣❤t ❜❡ ❡♥♦✉❣❤ ❢♦r r❛✐s✐♥❣ ❛ ♣r♦❜❧❡♠ ❢♦r ■♥❢♦✳
✼
❛ttr✐❜✉t✐♦♥ ♦♥ t❤❡ ✇r♦♥❣ ❦✐♥❞ ♦❢ ❝♦♥s✐❞❡r❛t✐♦♥s✳ ❚❤❡ r♦♦t ♦❢ t❤❡ ♣r♦❜❧❡♠ ✐s
♥♦t ❤❛r❞ t♦ ✐❞❡♥t✐❢②✿ ❙●■❚ ❢❛✐❧ t♦ ❧❡❛✈❡ r♦♦♠ ❢♦r t❤✐s ❦✐♥❞ ♦❢ ♠✐st❛❦❡s ❞✉❡ t♦
t❤❡✐r r❡❧✐❛♥❝❡ ♦♥ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝✐❡s✳ ❚❤✉s✱ t❤❡ ♣r♦❜❧❡♠ ❞❡r✐✈❡s ❢r♦♠ t❤❡✐r
❝❡♥tr❛❧ ❛ss✉♠♣t✐♦♥✳✶✵
✸✳✷
❉✐st❛❧✐t②
❚❤❡ s❡❝♦♥❞ ❞✐✣❝✉❧t② ✇❡ ✇✐❧❧ ❞✐s❝✉ss ✐s t❤❡ ✇r♦♥❣ ❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠
❛♥❞ t❤❡ ❜❡st ✇❛② t♦ ♠♦t✐✈❛t❡ ✐t ✐s t♦ ♣r❡s❡♥t ❛ s❧✐❣❤t❧② ❞✐✛❡r❡♥t ♣r♦❜❧❡♠ t❤❛t
❙●■❚ ❞♦ ♥♦t ❤❛✈❡✿ t❤❡ ✐♥❞❡t❡r♠✐♥❛❝② ♣r♦❜❧❡♠✳ ❆ t❤❡♦r② s✉✛❡rs ❢r♦♠ t❤❡ ✐♥❞❡✲
t❡r♠✐♥❛❝② ♣r♦❜❧❡♠ ✇❤❡♥ ✐t ✉♥❞❡r❞❡t❡r♠✐♥❡s ❝♦♥t❡♥t ❛ttr✐❜✉t✐♦♥✱ t❤❛t ✐s✱ ✇❤❡♥
t❤❡r❡ ❛r❡ ❥✉st t♦♦ ♠❛♥② ❡♥t✐t✐❡s t❤❛t ❝♦✉❧❞ ❜❡ r❡♣r❡s❡♥t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ✐t✳ ❙✉♣✲
♣♦s❡✱ ❢♦r ✐♥st❛♥❝❡✱ ❛ ♥❛✐✈❡ ✐♥❢♦r♠❛t✐♦♥❛❧ ❛❝❝♦✉♥t✱ ❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ ❛ ♠❡♥t❛❧
st❛t❡ ❘ r❡♣r❡s❡♥ts ❙ ✐✛ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ st❛t❡ ✐♥❝r❡❛s❡s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❙✳
❙✐♥❝❡ ❛❝t✐✈✐t② ✐♥ ♦✉r ♣❤♦t♦r❡❝❡♣t♦r ❝❡❧❧s r❛✐s❡s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❝❡rt❛✐♥ ♣❤♦✲
t♦♥s str✐❦✐♥❣ t❤❡ r❡t✐♥❛✱ ❜✉t ❛❧s♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ✐t ❜❡✐♥❣ s✉♥♥②✱ ♦❢ t❤❡ s✉❜❥❡❝t
❜❡✐♥❣ ❛✇❛❦❡✱ ♦❢ s❤♦♣s ❜❡✐♥❣ ♦♣❡♥ ❛♥❞ ♠❛♥② ♦t❤❡rs✱ t❤❡ ❝♦♥t❡♥t ♦❢ ❘ ✇♦✉❧❞ ❜❡
❤✐❣❤❧② ✐♥❞❡t❡r♠✐♥❛t❡✳ ❙♦ t❤❡ ✐♥❞❡t❡r♠✐♥❛❝② ♣r♦❜❧❡♠ s❡❡♠s t♦ ❥❡♦♣❛r❞✐③❡ t❤✐s
♥❛✐✈❡ ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r②✳ ■♥ t❤❛t r❡s♣❡❝t✱ ■♥❢♦ ❢❛r❡s ♠✉❝❤ ❜❡tt❡r t❤❛♥ t❤❡
♥❛✐✈❡ ❛♣♣r♦❛❝❤✱ ❜❡❝❛✉s❡ ✐t ♣✐❝❦s ✉♣ t❤❡ s✐♥❣❧❡ st❛t❡ t❤❛t ❤❛s ❛ ❣r❡❛t❡st st❛t✐st✐❝❛❧
❞❡♣❡♥❞❡♥❝❡ ✇✐t❤ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤✐s s♦❧✉t✐♦♥ t♦ t❤❡
♣r♦❜❧❡♠ ♦❢ ✐♥❞❡t❡r♠✐♥❛❝② ❤❛s t✇♦ str✐❦✐♥❣ ✉♥✇❡❧❝♦♠❡ ❝♦♥s❡q✉❡♥❝❡s✿ t❤❡ ✇r♦♥❣
❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❞❡✈❡❧♦♣❡❞ ❤❡r❡✱ ❛♥❞ t❤❡ ♣r♦❜❧❡♠ ♦❢
❛♠❜✐❣✉♦✉s r❡♣r❡s❡♥t❛t✐♦♥s✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳
❈♦♥s✐❞❡r t❤❡ ❋✉s✐❢♦r♠ ❋❛❝❡ ❆r❡❛ ✭❋❋❆✮✱ ✇❤✐❝❤ ✐s ✉s✉❛❧❧② t❤♦✉❣❤t t♦ r❡♣r❡✲
s❡♥t ❢❛❝❡s ✭❑❛♥✇✐s❤❡r ❡t ❛❧✳ ✶✾✾✼❀ ❉❡s✐♠♦♥❡✱ ✶✾✾✶✮✳ ❙✉♣♣♦s❡ ✇❡ ❞✐s❝♦✈❡r t❤❛t
❛ ❝❡rt❛✐♥ ♥❡✉r❛❧ ♥❡t✇♦r❦ ❘ ✐♥ t❤❡ ❋❋❆ s❡❧❡❝t✐✈❡❧② ✜r❡s ✇✐t❤ s✐❣♥✐✜❝❛♥t ✐♥t❡♥s✐t②
✇❤❡♥ t❤❡r❡ ✐s ❛ ❢❛❝❡ ❛♥❞ ❛❧s♦ t❤❛t✱ ❣✐✈❡♥ t❤❛t ❘ ✐s ❛❝t✐✈❡✱ t❤❡ ❡♥t✐t② t❤❛t ✐s
♠♦r❡ ❧✐❦❡❧② t♦ ❜❡ ♣r❡s❡♥t ✐s ❛ ❢❛❝❡✳ ❖♥❡ ♠✐❣❤t t❤✐♥❦ t❤❡s❡ ♦❜s❡r✈❛t✐♦♥s s✉✣❝❡
❢♦r ❡st❛❜❧✐s❤✐♥❣ t❤❡ ❢❛❝t t❤❛t t❤❡ ❜r❛✐♥ st❛t❡ r❡♣r❡s❡♥ts ❢❛❝❡ ❛❝❝♦r❞✐♥❣ t♦ ❙●■❚✳
❯♥❢♦rt✉♥❛t❡❧②✱ ✐t ✐s ✉♥❝❧❡❛r t❤❛t ❙●■❚ ❝❛♥ ❞❡❧✐✈❡r t❤✐s r❡s✉❧t✳ ❈♦♥s✐❞❡r✱ ❢♦r ✐♥✲
st❛♥❝❡✱ t❤❡ s❡t ♦❢ ♥❡✉r♦♥❛❧ str✉❝t✉r❡s ✐♥ t❤❡ t❤❛❧❛♠✉s t❤❛t ❛r❡ ❛❝t✐✈❡ ✇❤❡♥❡✈❡r
t❤❡r❡ ✐s ❛ ❢❛❝❡ ✐♥ ❢r♦♥t ♦❢ t❤❡ s✉❜❥❡❝t✳ ■❢ ❘ ❤❛s t❤❡ ❤✐❣❤❡st st❛t✐st✐❝❛❧ ❞❡♣❡♥✲
❞❡♥❝❡ ✇✐t❤ ❢❛❝❡s✱ ✐t ✇✐❧❧ ❛❧s♦ ♥♦r♠❛❧❧② ❤❛✈❡ ❛ ❤✐❣❤❡r st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ✇✐t❤
t❤❡s❡ ♥❡✉r♦♥❛❧ st❛t❡s ✐♥ ❡❛r❧② ✈✐s✐♦♥✳ ❚❤✉s✱ ❙●■❚ ✇♦✉❧❞ ❡♥t❛✐❧ t❤❛t t❤✐s ❛❝t✐✈✲
✐t② ✐♥ ❋❋❆ r❡♣r❡s❡♥ts ♥❡✉r♦♥❛❧ ❛❝t✐✈❛t✐♦♥ ✐♥ ❛♥♦t❤❡r ♣❛rt ♦❢ t❤❡ ❜r❛✐♥✳ ❚❤✐s ✐s
♦❢ ❝♦✉rs❡ ❛♥ ❡①tr❡♠❡❧② ❝♦✉♥t❡r✐♥t✉✐t✐✈❡ r❡s✉❧t✳ ■♥❞❡❡❞✱ ❡✈❡♥ ✐❢ t❤❡r❡ ✇❛s s♦♠❡
♣r✐♥❝✐♣❧❡❞ ✇❛② ♦❢ ❡①❝❧✉❞✐♥❣ ♦t❤❡r ❜r❛✐♥ st❛t❡s ❢r♦♠ ❜❡✐♥❣ r❡♣r❡s❡♥t❡❞✱ ♦t❤❡r
✐♥❛❞❡q✉❛t❡ ❝♦♥t❡♥ts s✉❝❤ ❛s ❢❛❝❡✲❧♦♦❦✐♥❣ t❤✐♥❣ ❝♦✉❧❞ ♣r♦❜❛❜❧② ♥♦t ❜❡ ❛✈♦✐❞❡❞✳
▼♦r❡ ❣❡♥❡r❛❧❧②✱ ❛ t❤❡♦r② s✉✛❡rs ❢r♦♠ t❤❡ ✇r♦♥❣ ❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠
✇❤❡♥ ✐t s②st❡♠❛t✐❝❛❧❧② ❞❡❧✐✈❡rs ❝♦♥t❡♥t ❛t t❤❡ ✇r♦♥❣ ❧❡✈❡❧ ♦❢ ❞✐st❛❧✐t②✳✶✶ ❙✐♥❝❡
✶✵ ■t ❤❛s ❜❡❡♥ ❛r❣✉❡❞ t❤❛t t❡❧❡♦❧♦❣✐❝❛❧ t❤❡♦r✐❡s ❛r❡ ❛❧s♦ ✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ s♦♠❡ ❢♦r♠s ♦❢
s②st❡♠❛t✐❝
♠✐sr❡♣r❡s❡♥t❛t✐♦♥ ✭▼❡♥❞❡❧♦✈✐❝✐✱ ✷✵✶✸❀ ✷✵✶✻✮✳ ❋♦r ❛ r❡s♣♦♥s❡✱ s❡❡ ❬❛✉t❤♦r✶❪✳
✶✶ ❙♦♠❡ ❡♠♣❧♦② t❤❡ ❧❛❜❡❧ ✬❞✐st❛❧✐t② ♣r♦❜❧❡♠✬ t♦ r❡❢❡r t♦ t❤✐s ❞✐✣❝✉❧t②✱ ❜✉t t❤✐s ❡①♣r❡ss✐♦♥
✐s ❛❧s♦ ❢r❡q✉❡♥t❧② ✉s❡❞ ❢♦r ✈❡rs✐♦♥ ♦❢ t❤❡ ✐♥❞❡t❡r♠✐♥❛❝② ♣r♦❜❧❡♠ ✭❡✳❣✳ ◆❡❛♥❞❡r✱ ✷✵✶✼✱ ❝❤✳✾❀
❙❝❤✉❧t❡ ✷✵✶✽✮✳ ❚♦ ❛✈♦✐❞ ❛♥② s♦rt ♦❢ ♠✐s✉♥❞❡rst❛♥❞✐♥❣✱ ✇❡ ❝❛❧❧ t❤✐s ♦❜❥❡❝t✐♦♥ t❤❡ ✬✇r♦♥❣
✽
♠❡♥t❛❧ st❛t❡s ✇✐❧❧ ♥♦r♠❛❧❧② ❤❛✈❡ ❛ ❤✐❣❤❡r st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ✇✐t❤ ♠♦st ♣r♦①✲
✐♠❛❧ st✐♠✉❧✐✱ ❙●■❚ ✇♦✉❧❞ t❡♥❞ t♦ ❞❡❧✐✈❡r ❝♦♥t❡♥ts t❤❛t ❛r❡ t♦♦ ♣r♦①✐♠❛❧✳ ❚❤✐s
r❡s✉❧t ✐s ❛t ♦❞❞s ✇✐t❤ ♦✉r ✐♥t✉✐t✐♦♥s ❛♥❞ s❡❡♠s t♦ ❜❡ ✐♥ t❡♥s✐♦♥ ✇✐t❤ t❤❡ ❝❧❛✐♠s
♠❛❞❡ ❜② ❝♦❣♥✐t✐✈❡ s❝✐❡♥t✐sts ✭s❡❡ ❬❛✉t❤♦r✶❛♥❞✷✱ ②❡❛r❪✮✳
❖❢ ❝♦✉rs❡✱ s✉♣♣♦rt❡rs ♦❢ ❙●■❚ ❝❛♥ ✐♥s✐st t❤❛t✱ ✐❢ ❛ r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡
❛❝t✉❛❧❧② ♠❡❛♥s
❢❛❝❡✱ ✐t ✇✐❧❧ s✉r❡❧② ❤❛✈❡ ❤✐❣❤❡r st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ✇✐t❤ ❢❛❝❡s
t❤❛♥ ✇✐t❤ ❢❛❝❡✲❧♦♦❦✐♥❣ t❤✐♥❣s✳ ❍♦✇❡✈❡r✱ t❤✐s ✐s ❛ ❞♦✉❜t❢✉❧ ❛ss✉♠♣t✐♦♥✳ ❋✐rst✱
❢♦r t❤❡ r❡❛s♦♥s ♣r♦✈✐❞❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ✐t ✐s ♥♦t ♦❜✈✐♦✉s t❤❛t ❛ r❡♣✲
r❡s❡♥t❛t✐♦♥ ♦❢ ❢❛❝❡s ♥❡❡❞s t♦ ❝♦rr❡❧❛t❡ ❜❡tt❡r ✇✐t❤ ❢❛❝❡s t❤❛♥ ✇✐t❤ ♥♦♥✲❢❛❝❡s✳
❙❡❝♦♥❞❧②✱ s✐♥❝❡ ✇❡ ✉s✉❛❧❧② ✐❞❡♥t✐❢② ❢❛❝❡s ❜② ❞❡t❡❝t✐♥❣ ❢❛❝❡✲❧♦♦❦✐♥❣ t❤✐♥❣s✱ ✐t ✐s
♥♦t ✉♥r❡❛s♦♥❛❜❧❡ t♦ s✉♣♣♦s❡ t❤❛t t❤❡ ❝♦rr❡❧❛t✐♦♥ ✇✐t❤ ♣r♦①✐♠❛❧ st✐♠✉❧✐ ✐s ❛t
❧❡❛st ❛s ❣♦♦❞ ❛s t❤❡ ❝♦rr❡❧❛t✐♦♥ ✇✐t❤ ❞✐st❛❧ st✐♠✉❧✐✳
❙✐♠✐❧❛r❧②✱ ❣✐✈❡♥ t❤❛t ✇❡
✐❞❡♥t✐❢② ❢❛❝❡✲❧♦♦❦✐♥❣ t❤✐♥❣s ❜② ❞❡t❡❝t✐♥❣ ❝❡rt❛✐♥ ❢❡❛t✉r❡s✱ t❤❡ ❝♦rr❡❧❛t✐♦♥ ✇✐t❤
t❤❡ ❧❛tt❡r ✇✐❧❧ ❜❡ str♦♥❣❡r✳ ❆♥❞ s♦ ♦♥✳ ❚❤✐s r❡❛s♦♥✐♥❣ ❝❛♥ ❜❡ ✐t❡r❛t❡❞ ✉♥t✐❧ t❤❡
♠♦st ♣r♦①✐♠❛❧ st❛t❡ t❤❛t ❛❝t✉❛❧❧② tr✐❣❣❡rs t❤❡ ❜r❛✐♥ st❛t❡ ✭✇❤✐❝❤ ✐♥ ♠❛♥② ❝❛s❡s
✇✐❧❧ ❜❡ ❛♥♦t❤❡r ♠❡♥t❛❧ st❛t❡✮ ❤♦❧❞s✳ ❈♦♥s❡q✉❡♥t❧②✱ ❙●■❚ ❝❧❡❛r❧② s✉✛❡r ❢r♦♠ t❤❡
✇r♦♥❣ ❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠✳
❋✐♥❛❧❧②✱ ♦♥❡ ♠✐❣❤t t❤✐♥❦ t❤❛t t❤❡ ✇r♦♥❣ ❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠ ✐s ❧❡ss
tr♦✉❜❧✐♥❣ ❢♦r ❘✉♣❡rt✬s ❛❝❝♦✉♥t✱ s✐♥❝❡ ❤❡ r❡str✐❝ts ❤✐s ❛♣♣r♦❛❝❤ t♦ r❡♣r❡s❡♥t❛✲
t✐♦♥s ♦❢ ♥❛t✉r❛❧ ❦✐♥❞s✳
❚❤✐s ✐s ❞♦✉❜t❢✉❧✳
❍✐s ♣r♦♣♦s❛❧ st✐❧❧ ❢❛❝❡s ❛ ❞❛✉♥t✐♥❣
❞✐❧❡♠♠❛✱ ✇❤♦s❡ ❤♦r♥s ❞❡♣❡♥❞ ♦♥ ✇❤❡t❤❡r t❤❡ s❡t ♦❢ ❝❛♥❞✐❞❛t❡ ♣r♦♣❡rt✐❡s t❤❛t
❣✐✈❡ r✐s❡ t♦ t❤❡ ✇r♦♥❣ ❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠ q✉❛❧✐❢② ❛s ♥❛t✉r❛❧ ❦✐♥❞s ♦r
♥♦t✳ ■❢✱ ❢♦r ✐♥st❛♥❝❡✱ ❧✐❣❤t ♣❤♦t♦♥s ♦r ❝❡rt❛✐♥ ❦✐♥❞ ♦❢ ♥❡✉r♦♥❛❧ st❛t❡s ✐♥ ❡❛r❧②
✈✐s✐♦♥ ❝♦✉♥t ❛s ♥❛t✉r❛❧ ❦✐♥❞s✱ t❤❡ ♣r♦❜❧❡♠ r❡♠❛✐♥s ✐♥ ❛❧❧ ✐ts ❢♦r❝❡✳
✶✷ ■❢ t❤❡②
❞♦ ♥♦t q✉❛❧✐❢② ❛s ♥❛t✉r❛❧ ❦✐♥❞s✱ t❤❡♥ t❤❡ ❛♣♣r♦❛❝❤ ❝❧❡❛r❧② ❡①❝❧✉❞❡s t♦♦ ♠✉❝❤✿
t❤❡s❡ ❡♥t✐t✐❡s ❛r❡ s♦♠❡t✐♠❡s r❡♣r❡s❡♥t❡❞ ❜② ❝♦❣♥✐t✐✈❡ st❛t❡s✱ s♦ r✉❧✐♥❣ t❤❡♠ ♦✉t
❜② ❞❡✜♥✐t✐♦♥
✐s ❝❧❡❛r❧② ✐♥❛❞❡q✉❛t❡✳ ❆ ❞✐✛❡r❡♥t ❛♥s✇❡r t♦ t❤✐s ♣r♦❜❧❡♠ ✐s ❜r✐❡✢②
s❦❡t❝❤❡❞ ❜② ❊❧✐❛s♠✐t❤✳ ❍❡ s✉❣❣❡sts ❛♥ ❛❞❞✐t✐♦♥❛❧ r❡q✉✐r❡♠❡♥t✿ t❤❡ r❡❢❡r❡♥t ❝❛♥✲
♥♦t ✏❢❛❧❧ ✉♥❞❡r t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❡s❝r✐♣t✐♦♥✑✱ t❤❛t ✐s✱ t❤❡r❡ ♠✉st ♥♦t ❜❡ ❛♥②
✐♥t❡r♥❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❡s❝r✐♣t✐♦♥ r❡❧❛t✐♥❣ t❤❡ r❡❢❡r❡♥t ✇✐t❤ t❤❡ ♠❡♥t❛❧ st❛t❡
s✉❝❤ t❤❛t ✐t ❝♦✉❧❞ ❛❝❝♦✉♥t ❢♦r t❤❡ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ✭❊❧✐❛s♠✐t❤✱ ✷✵✵✺❛✱ ♣✳
✶✵✹✼❀ ❊❧✐❛s♠✐t❤✱ ✷✵✵✵ ♣✳ ✺✾✲✻✵✮✳ ❍♦✇❡✈❡r✱ t❤✐s ♣r♦♣♦s❛❧ ✐s st✐❧❧ ✉♥s❛t✐s❢❛❝t♦r②✳
❋✐rst ♦❢ ❛❧❧✱ ♠❛♥② t❤✐♥❦ t❤❛t ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ♦✈❡r r❡♣r❡s❡♥t❛t✐♦♥s✳
❋♦r t❤✐s r❡❛s♦♥✱ t♦ ❦♥♦✇ ✇❤❡t❤❡r t✇♦ ❝❛✉s❛❧❧② r❡❧❛t❡❞ ❜r❛✐♥ st❛t❡s ❛r❡ ❝♦♠♣✉✲
t❛t✐♦♥❛❧❧② r❡❧❛t❡❞✱ ♦♥❡ s❤♦✉❧❞ ❦♥♦✇ ✇❤❡t❤❡r t❤❡② ❛r❡ r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ❤♦✇
t❤❡✐r ❝♦♥t❡♥t ✐s r❡❧❛t❡❞✳
❨❡t t❤✐s ✐s ♣r❡❝✐s❡❧② ✇❤❛t t❤✐s ❝♦♥❞✐t✐♦♥ ✐s s✉♣♣♦s❡❞
t♦ ❡st❛❜❧✐s❤✳ ❙❡❝♦♥❞❧②✱ t❤✐s ❛♣♣r♦❛❝❤ ✐s ❛❧s♦ t♦♦ ❡①❝❧✉s✐✈❡ ❛♥❞ t♦♦ ✐♥❝❧✉s✐✈❡ ❛t
❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠✬✳
❆s ❛ r❡✈✐❡✇❡r ♣♦✐♥t❡❞ ♦✉t✱ ♦♥❡ ♠✐❣❤t q✉❡st✐♦♥ ✇❤❡t❤❡r ✐♥ ❛❧❧ ❝❛s❡s t❤❡ ♣r♦❜❧❡♠ ✇❡ ♣r❡s❡♥t
❝♦♥❝❡r♥s ♠♦r❡ ❞✐st❛❧ ✈s✳ ❧❡ss ❞✐st❛❧ ❢❡❛t✉r❡s ✭❝♦♥s✐❞❡r✱ ❢♦r ✐♥st❛♥❝❡✱ t❤❡ ❝♦♥tr❛st ❜❡t✇❡❡♥ ❢❛❝❡s
❛♥❞ ❢❛❝❡✲❧♦♦❦✐♥❣ t❤✐♥❣s✮✳ ❲❡ ❛❞♦♣t❡❞ t❤✐s t❡r♠✐♥♦❧♦❣② ❤❡r❡ ❜❡❝❛✉s❡ ✐t ✐s ✈❡r② ✉s✉❛❧ ✐♥ t❤❡
❧✐t❡r❛t✉r❡ ❛♥❞ ✐♥ ♠❛♥② ❝❛s❡s t❤❡r❡ ✐s ❝❧❡❛r❧② ❛ ❝♦♥tr❛st ✐♥ ❞✐st❛❧✐t② ✭❡✳❣✳ ❢❛❝❡s ✈s✳ ♥❡✉r♦♥❛❧
♣❛tt❡r♥s✮✳ ◆♦♥❡t❤❡❧❡ss✱ t❤❡ ♥❛♠❡ ✐s ♥♦t ✐♠♣♦rt❛♥t❀ t❤❡ ❦❡② ♣♦✐♥t ✐s t❤❛t t❤❡ t❤❡♦r② ❞❡❧✐✈❡rs
t❤❡ ✇r♦♥❣ ❝♦♥t❡♥t✱ ❜✉t ♥♦t ✐♥ ✈✐rt✉❡ ♦❢ ✐t ❜❡✐♥❣ ✐♥❞❡t❡r♠✐♥❛t❡✳
✶✷ ❆❝t✉❛❧❧②
❘✉♣❡rt ✭✶✾✾✾✱ ♣✳
✸✹✵✮ ❛❝❝❡♣ts ❛ ❡①tr❡♠❡❧② ❧✐❜❡r❛❧ ❛♣♣r♦❛❝❤ t♦ ♥❛t✉r❛❧ ❦✐♥❞s✱
❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ ✏♥❛t✉r❛❧ ❦✐♥❞s ❛r❡ ❛♥② ❦✐♥❞s t❤❛t s✉❝❝❡ss❢✉❧ ♥♦♥✲✐♥t❡♥t✐♦♥❛❧ s❝✐❡♥❝❡ ✜♥❞s
t❤❡♦r❡t✐❝❛❧❧② ✐♥t❡r❡st✐♥❣ ❛♥❞ ✉s❡❢✉❧✑✳ ❚❤✉s✱ ❤❡ ♣r♦❜❛❜❧② ❢❛❝❡s t❤❡ ✜rst ❤♦r♥ ♦❢ t❤❡ ❞✐❧❡♠♠❛✳
✾
t❤❡ s❛♠❡ t✐♠❡✳ ❖♥❡ t❤❡ ❤❛♥❞✱ ✐t st✐❧❧ ✐♥❝❧✉❞❡s t♦♦ ♠✉❝❤ ❜❡❝❛✉s❡ t❤❡ ♣r♦❜❧❡♠ ✐s
♥♦t ♦♥❧② ❝❛✉s❡❞ ❜② ♥❡✉r♦♥❛❧ st❛t❡s✱ ❜✉t ❛❧s♦ ❜② ❡①t❡r♥❛❧ ❡♥t✐t✐❡s ✭❡✳❣✳ ♣❤♦t♦♥s✱
♦❜❥❡❝t✲❧♦♦❦✐♥❣ t❤✐♥❣s✱ ❡t❝✳✳✳✮✳ ❖♥ t❤❡ ♦t❤❡r✱ ✐t ❡①❝❧✉❞❡s t♦♦ ♠✉❝❤ ❜❡❝❛✉s❡ s♦♠❡
♥❡✉r♦♥❛❧ st❛t❡s ✐♥❞❡❡❞ r❡♣r❡s❡♥t ♦t❤❡r st❛t❡s ♦❢ t❤❡ ❜r❛✐♥ ♦r t❤❡ ❜♦❞② ✭❉❛♠❛s✐♦
✭✷✵✶✵✮❀ ❘♦s❡♥t❤❛❧ ✭✷✵✵✺✮❀ Pr✐♥③ ✭✷✵✵✹✮✮✳ ❋♦r ❛ ♠♦r❡ ❞❡t❛✐❧❡❞ ❞✐s❝✉ss✐♦♥✱ s❡❡ ✭s❡❡
❬❛✉t❤♦r✶❛♥❞✷✱ ②❡❛r❪✮✳
❍♦✇ ❝♦✉❧❞ ♦♥❡ ❛tt❡♠♣t t♦ s♦❧✈❡ t❤✐s ♣r♦❜❧❡♠❄ ❲❡ t❤✐♥❦ ♣r♦❣r❡ss ❝♦✉❧❞ ❜❡
♠❛❞❡ ❜② ❛♣♣❡❛❧✐♥❣ t♦ s♦♠❡ ♦t❤❡r ♥♦t✐♦♥s s✉❝❤ ❛s ❢✉♥❝t✐♦♥✳ ■♥ ❛ ♥✉ts❤❡❧❧✱ t❤❡
✐❞❡❛ ✐s t❤❛t ❛❧t❤♦✉❣❤ ❛ ❣✐✈❡♥ ♠❡♥t❛❧ st❛t❡ ▼ ❤❛s ❛ ❤✐❣❤❡r st❛t✐st✐❝❛❧ ❝♦rr❡❧❛t✐♦♥
✇✐t❤ ❛ ♣r♦①✐♠❛❧ ❢❡❛t✉r❡ ❋✱ ✐t ♠✐❣❤t st✐❧❧ ❜❡ ✐ts ❢✉♥❝t✐♦♥ t♦ r❡♣r❡s❡♥t s♦♠❡t❤✐♥❣
♠♦r❡ ❞✐st❛❧✳ ❚❛❦✐♥❣ t❤✐s ♣❛t❤✱ ❤♦✇❡✈❡r✱ ✇♦✉❧❞ ❤❛✈❡ s✐❣♥✐✜❝❛♥t ❝♦♥s❡q✉❡♥❝❡s✿
✐❢ ♠❡♥t❛❧ st❛t❡s ❤❛✈❡ ❛ ❤✐❣❤❡r st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ✇✐t❤ t❤❡✐r ♠♦st ♣r♦①✐♠❛❧
❝❛✉s❡ ✇❤✐❧❡ t❤❡✐r ❢✉♥❝t✐♦♥ ❛♥❞ ❝♦♥t❡♥t ❝♦♥❝❡r♥s ❛ ❞✐st❛❧ ❢❡❛t✉r❡✱ ✐t ✐s ♥♦t ♦❜✈✐♦✉s
t❤❛t t❤❡ ♥♦t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✐s ❛❝t✉❛❧❧② ♣❧❛②✐♥❣ ❛♥② ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ❛ t❤❡♦r②
♦❢ r❡♣r❡s❡♥t❛t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ t❤❡ ♠❡♥t❛❧ st❛t❡s t❡♥❞ t♦ ❤❛✈❡ ❛ str♦♥❣❡r
❝♦rr❡❧❛t✐♦♥ ✇✐t❤ t❤❡✐r ♣r♦①✐♠❛❧ ❝❛✉s❡s✱ ❜✉t t❡♥❞ t♦ ❤❛✈❡ ❞✐st❛❧ ❝♦♥t❡♥ts✱ t❤❡♥ ✐t
✐s ✉♥❝❧❡❛r ✇❤❡t❤❡r t❤❡ ❢♦r♠❡r ❝❛♥ ❜❡ ✉s❡❞ t♦ ♣r♦✈✐❞❡ ❛♥ ❛❝❝♦✉♥t ♦❢ t❤❡ ❧❛tt❡r✳
❖❢ ❝♦✉rs❡✱ ♠✉❝❤ ♠♦r❡ s❤♦✉❧❞ ❜❡ s❛✐❞ t♦ ♠❛❦❡ t❤✐s t❤♦✉❣❤t ❢✉❧❧② ❝♦♠♣❡❧❧✐♥❣❀ ♦♥❡
✇♦✉❧❞ ♥❡❡❞ t♦ s♣❡❧❧ ♦✉t ✐♥ ❞❡t❛✐❧ t❤✐s ♥♦t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥ ❛♥❞ s❤♦✇ ❤♦✇ ✐t ❝❛♥
❞❡❧✐✈❡r t❤❡ r✐❣❤t ❧❡✈❡❧ ♦❢ ❞✐st❛❧✐t②✳ ◆♦♥❡t❤❡❧❡ss✱ ✇❡ t❤✐♥❦ t❤❛t s✉♣♣♦rt❡rs ♦❢ ❙●■❚
s❤♦✉❧❞ s❡r✐♦✉s❧② ❝♦♥s✐❞❡r t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t t❤❡ ✇r♦♥❣ ❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥
♣r♦❜❧❡♠ ♠✐❣❤t s❤♦✇ t❤❛t t❤❡ ♥♦t✐♦♥ ♦❢ ❤✐❣❤❡r st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ✐s ❛❝t✉❛❧❧②
❛♥ ✐♥❛♣♣r♦♣r✐❛t❡ t♦♦❧ ❢♦r ❛ t❤❡♦r② ♦❢ ❝♦♥t❡♥t✳
✸✳✸
▼✉❧t✐♣❧❡ ❘❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ❈♦♥t❡♥ts
❲❡ s❛✇ t❤❛t t❤❡ ❢❛❝t t❤❛t ❙●■❚ r❡str✐❝t r❡♣r❡s❡♥t❛t✐♦♥❛❧ ❝♦♥t❡♥t t♦ t❤❡
❡♥t✐t② t❤❛t ❤❛s ❛ ❤✐❣❤❡r st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ✇✐t❤ ❛ ♠❡♥t❛❧ st❛t❡ ❤❡❧♣s
t❤❡♠ r❡❞✉❝❡ t❤❡ ✐♥❞❡t❡r♠✐♥❛❝② ❢❛❝❡❞ ❜② t❤❡ ♥❛✐✈❡ ✐♥❢♦r♠❛t✐♦♥❛❧ ❛♣♣r♦❛❝❤✱ ❜✉t
✐t ❣✐✈❡s r✐s❡ t♦ t❤❡ ✇r♦♥❣ ❞✐st❛❧✐t② ❛ttr✐❜✉t✐♦♥ ♣r♦❜❧❡♠✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇♦✉❧❞
❧✐❦❡ t♦ ❤✐❣❤❧✐❣❤t ❛ s❡❝♦♥❞ ♥❡❣❛t✐✈❡ ❝♦♥s❡q✉❡♥❝❡✳ ■t ✐s ❛ ♣❧❛t✐t✉❞❡ t❤❛t✱ ✐♥ ♠❛♥②
❝❛s❡s✱ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ t❤❡✐r ❝♦♥t❡♥ts ✐s ♥♦t ♦♥❡✲t♦✲♦♥❡✱
❜✉t ♠❛♥②✲t♦✲♦♥❡ ♦r ♦♥❡✲t♦✲♠❛♥②✳ ❆s ❛ ♠❛tt❡r ♦❢ ❢❛❝t s♦♠❡ r❡♣r❡s❡♥t❛t✐♦♥s ❤❛✈❡
♠✉❧t✐♣❧❡ r❡❢❡r❡♥ts ❛♥❞ t❤❡ s❛♠❡ r❡❢❡r❡♥t ✐s s♦♠❡t✐♠❡s r❡♣r❡s❡♥t❡❞ ❜② ❞✐✛❡r❡♥t
st❛t❡s✳ ❍♦✇❡✈❡r✱ ❙●■❚ r❡♥❞❡r t❤❡s❡ ❢❛❝ts ✐♠♣♦ss✐❜❧❡ ❜② ❞❡✜♥✐t✐♦♥✳ ❇② r❡q✉✐r✐♥❣
❝♦♥t❡♥t t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s✐♥❣❧❡ ❡♥t✐t② t❤❛t ❤❛s ❛ ❤✐❣❤❡r st❛t✐st✐❝❛❧ ❞❡✲
♣❡♥❞❡♥❝❡✱ ❙●■❚ ❤❛✈❡ ❞✐✣❝✉❧t✐❡s ✐♥ ❛❝❝♦✉♥t✐♥❣ ❢♦r r❡♣r❡s❡♥t❛t✐♦♥s ✇✐t❤ ♠✉❧t✐♣❧❡
❝♦♥t❡♥ts ❛♥❞ ❝♦♥t❡♥ts s❤❛r❡❞ ❜② ♠✉❧t✐♣❧❡ r❡♣r❡s❡♥t❛t✐♦♥s✳ ❚❤✐s ♣r♦❜❧❡♠ ❞❡r✐✈❡s
❢r♦♠ ❡❛❝❤ ♦❢ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✐♥❝❧✉❞❡❞ ✐♥
❆♥ ❡①❛♠♣❧❡ ❢r♦♠ ♦✉r ✈✐s✉❛❧ s②st❡♠ ♠✐❣❤t ❤❡❧♣ ✐❧❧✉str❛t❡ t❤❡ ♣r♦❜❧❡♠✳ ❈♦♥✲
s✐❞❡r t❤❡ ❝❛s❡ ♦❢
✳ ❚❤❡ ❤✉♠❛♥ ❡②❡ ❝♦♥t❛✐♥s ♦♥❧② t❤r❡❡ t②♣❡s ♦❢ ❝♦♥❡
❝❡❧❧s✱ ✇❤✐❝❤ ❛r❡ r❡s♣♦♥s✐❜❧❡ ❢♦r ❝♦❧♦r ✈✐s✐♦♥✳ ❊❛❝❤ ♦❢ t❤❡s❡ t❤❡s❡ t②♣❡s ♦❢ ❝❡❧❧
r❡s♣♦♥❞ t♦ t❤❡ ❝✉♠✉❧❛t✐✈❡ ❡♥❡r❣② ❢r♦♠ ❛ ❜r♦❛❞ r❛♥❣❡ ♦❢ ✇❛✈❡❧❡♥❣t❤s✳ ◆♦✇✱ ❛s
✐t ❤❛♣♣❡♥s✱ ❞✐✛❡r❡♥t ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❧✐❣❤t ❛❝r♦ss ❛❧❧ ✇❛✈❡❧❡♥❣t❤s ❝❛♥ ♣r♦❞✉❝❡
❛♥ ❡q✉✐✈❛❧❡♥t r❡❝❡♣t♦r r❡s♣♦♥s❡✳ ❈♦♥s❡q✉❡♥t❧②✱✖♦♥ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❝♦❧✲
♦rs ❛r❡ s✉r❢❛❝❡ r❡✢❡❝t❛♥❝❡s✖t❤❡r❡ ❛r❡ ❝♦❧♦rs ✭❝❛❧❧❡❞ ❵♠❡t❛♠❡rs✬✮ t❤❛t ❞❡s♣✐t❡
s✐♥✲
❣❧❡
✐♥❢♦✳
♠❡t❛♠❡r✐s♠
✶✵
❤❛✈✐♥❣ ❞✐✛❡r❡♥t s♣❡❝tr❛❧ ♣♦✇❡r ❞✐str✐❜✉t✐♦♥s ♣r♦❞✉❝❡ t❤❡ ✈❡r② s❛♠❡ ♥❡✉r❛❧ ♥❡t✲
✇♦r❦ ❛❝t✐✈❛t✐♦♥ ✐♥ t❤❡ ✈✐s✉❛❧ ❝♦rt❡①✳ ❈❛❧❧ ❵C1 ✬ ❛♥❞ ❵C2 ✬ t✇♦ s✉❝❤ ♠❡t❛♠❡rs✳
❈♦♥s✐❞❡r t❤❡ ♥❡✉r❛❧ ♥❡t✇♦r❦✱ Ri ✱ ✐♥ t❤❡ ✈✐s✉❛❧ ❝♦rt❡① t❤❛t s❛t✐s✜❡s ❝♦♥❞✐t✐♦♥ ✷
♦❢ ✐♥❢♦✿ P (C1 | Ri ) > P (C1 | Rj ) ❢♦r ❡✈❡r② Rj 6= Ri ✳ ❚❤✉s✱ Ri ✐s t❤❡ ♥❡t✲
✇♦r❦ t❤❛t ♠♦st str♦♥❣❧② ✐♥❝r❡♠❡♥ts t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ C1 ✳ ■♠❛❣✐♥❡ ❛❧s♦ t❤❛t
P (Ri | C1 ) > P (Ri | C2 )✳ ❈♦♥❞✐t✐♦♥ ✶ ✐s ❛❧s♦ s❛t✐s✜❡❞✱ ❛♥❞ s♦✱ ❛❝❝♦r❞✐♥❣ t♦
t❤❡ t❤❡♦r②✱ Ri r❡♣r❡s❡♥ts C1 ✳ ❍♦✇❡✈❡r✱ ✐♥t✉✐t✐✈❡❧②✱ t❤❡ ❝♦♥t❡♥t ♦❢ Ri ✐s r❛t❤❡r
C1 ∨ C2 ✳ ❚❤❡ t❤❡♦r② ❝❛♥♥♦t ♣r♦✈✐❞❡ t❤✐s r❡s✉❧t✱ ❛t ❧❡❛st ❛s t❤❡r❡ ✐s ❛ s❧✐❣❤t ❞✐❢✲
❢❡r❡♥❝❡ ✐♥ ♣r♦❜❛❜✐❧✐t✐❡s ❜❡t✇❡❡♥ C1 ❛♥❞ C2 ✳ ❯♥❞❡r t❤❡s❡ ❝♦♥❞✐t✐♦♥s✱ ♦r❣❛♥✐s♠s
❝❛♥♥♦t r❡♣r❡s❡♥t C2 ✱ ❡✈❡♥ ✐❢ C2 ✇❡r❡ ✐♥ ❢❛❝t t❤❡ ♠♦st ❝♦♠♠♦♥ ❝♦❧♦r ❛❢t❡r C1
✐♥ t❤❡ ❡♥✈✐r♦♥♠❡♥t✳
■❢ ❙●■❚ ❢❛❝❡ ❞✐✣❝✉❧t✐❡s ✇✐t❤ r❡♣r❡s❡♥t❛t✐♦♥s ✇✐t❤ ❛ ❞✐s❥✉♥❝t✐✈❡ ❝♦♥t❡♥t✱ ❛♥
❛♥❛❧♦❣♦✉s ♣r♦❜❧❡♠ ❛r✐s❡s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♠✉❧t✐♣❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ✇✐t❤ t❤❡
s❛♠❡ ❝♦♥t❡♥t✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛❝❝♦r❞✐♥❣ t♦ ❝♦♥❞✐t✐♦♥ ✷ ♦❢ ■♥❢♦✱ R r❡♣r❡s❡♥ts ❛
st✐♠✉❧✉s ❙ ♦♥❧② ✐❢ t❤❡r❡ ✐s ♥♦ ♦t❤❡r st❛t❡ Rj ✱ s✉❝❤ t❤❛t P (Si | Rj ) > P (Si | Ri )✳
❚❤❛t ♠❡❛♥s t❤❛t ♦♥❧② ♦♥❡ r❡♣r❡s❡♥t❛t✐♦♥ ❝❛♥ ❤❛✈❡ ❙ ❛s ✐ts ❝♦♥t❡♥t✳ ❚❤✉s✱
❛❝❝♦r❞✐♥❣ t♦ t❤❡s❡ t❤❡♦r✐❡s ❜② ❞❡✜♥✐t✐♦♥ t✇♦ r❡♣r❡s❡♥t❛t✐♦♥s ❝❛♥♥♦t ❤❛✈❡ t❤❡
s❛♠❡ r❡❢❡r❡♥t✱ ②❡t ✇❡ ❞♦ ❤❛✈❡ ❞✐✛❡r❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ✇✐t❤ ❛ ❝♦♠♠♦♥ r❡❢❡r❡♥❝❡✳
❆ str✐❦✐♥❣ ❡①❛♠♣❧❡ ✐s ♣r♦✈✐❞❡❞ ❜② r❡❞✉♥❞❛♥t ♣r♦❝❡ss✐♥❣✳ ❘❡❞✉♥❞❛♥❝② ✐s t❤❡
❞✉♣❧✐❝❛t✐♦♥ ♦❢ ❝r✐t✐❝❛❧ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ ♣✉r♣♦s❡ ♦❢ ✐♥❝r❡❛s✐♥❣ t❤❡ r❡❧✐❛❜✐❧✐t②
♦❢ t❤❡ s②st❡♠✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ ✈✐s✐♦♥ ✐s ❧♦st ✐♥ ♦♥❡ ❡②❡ ✇❡ ❞♦ ♥♦t ❜❡❝♦♠❡
❜❧✐♥❞ ✭❛❧t❤♦✉❣❤ ❞❡♣t❤ ♣❡r❝❡♣t✐♦♥ ✐s ✐♠♣❛✐r❡❞✮ ❛♥❞ ♦❢t❡♥ t❤❡ s❛♠❡ ✐♥❢♦r♠❛t✐♦♥
✐s ♣r❡s❡♥t❡❞ t♦ ❜♦t❤ ❡②❡s✳ ❘❡❞✉♥❞❛♥❝② ✐s ❛♥ ✐♠♣♦rt❛♥t ✜❡❧❞ ♦❢ st✉❞② ✐♥ ❝♦❣♥✐t✐✈❡
s❝✐❡♥❝❡✳ ❚❤❡r❡ ✐s str♦♥❣ ❡✈✐❞❡♥❝❡✱ ❢♦r ✐♥st❛♥❝❡✱ t❤❛t r❡❞✉♥❞❛♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢
✐♥❢♦r♠❛t✐♦♥ ❛❝r♦ss ♠♦❞❛❧✐t✐❡s r❡❝r✉✐ts ❛tt❡♥t✐♦♥ ❛♥❞ ❡♥❤❛♥❝❡s ❧❡❛r♥✐♥❣ ✭✇❤❛t
✐s ❝❛❧❧❡❞ t❤❡ ❵✐♥t❡rs❡♥s♦r② r❡❞✉♥❞❛♥❝② ❤②♣♦t❤❡s✐s✬ ❬■❘❍❪✿ ❇❛❤r✐❝❦ ❡t ❛❧✳ ✷✵✵✷❀
❇❛❤r✐❝❦ ❛♥❞ ▲✐❝❦❧✐t❡r ✷✵✵✵❀ ❇❛❤r✐❝❦ ❡t ❛❧✳ ✷✵✵✹✮✳ ❆s ❇r❡♠❡r ❛♥❞ ❝♦❧❧❡❛❣✉❡s
✭✷✵✶✷✱ ♣✳ ✾✺✺✮ s✉❣❣❡st✿
❆❝❝♦r❞✐♥❣ t♦ t❤❡ ✐♥t❡rs❡♥s♦r② r❡❞✉♥❞❛♥❝② ❤②♣♦t❤❡s✐s✱ ♦♣t✐♠✉♠
❝♦♥❞✐t✐♦♥s ❢♦r ❞❡r✐✈✐♥❣ ❜❡♥❡✜t ❢r♦♠ ♣r♦✈✐s✐♦♥ ♦❢ ♠✉❧t✐s❡♥s♦r② ✐♥❢♦r✲
♠❛t✐♦♥ ✇♦✉❧❞ ❜❡ t❤♦s❡ ✐♥ ✇❤✐❝❤ ❜♦t❤ ✈✐s✉❛❧ ❛♥❞ ❛✉❞✐t♦r② ✐♥❢♦r♠❛t✐♦♥
♣r♦✈✐❞❡ ❝♦♥❣r✉❡♥t ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❛♥ ♦❜❥❡❝t✬s tr❛❥❡❝t♦r②✳ ❯♥❞❡r
s✉❝❤ ❝♦♥❞✐t✐♦♥s✱ ✈✐s✉❛❧ ❛♥❞ ❛✉❞✐t♦r② ✐♥❢♦r♠❛t✐♦♥ ✇♦✉❧❞ s♣❡❝✐❢② t❤❡
♦❜❥❡❝t✬s tr❛❥❡❝t♦r② r❡❞✉♥❞❛♥t❧②✱ ❛♥❞ s♦ ❝♦✉❧❞ ❜❡ ❡①♣❡❝t❡❞ t♦ ❡♥❤❛♥❝❡
♣❡r❝❡♣t✐♦♥ ♦❢ tr❛❥❡❝t♦r② ❝♦♥t✐♥✉✐t② ❛s t❤❡ ♦❜❥❡❝t ♣❛ss❡❞ ❜❡❤✐♥❞ ❛♥
♦❝❝❧✉❞❡r✳
❲❤❛t❡✈❡r t❤❡ ♠❡r✐ts ❛r❡ ♦❢ t❤❡ ■♥t❡rs❡♥s♦r② ❘❡❞✉♥❞❛♥❝② ❍②♣♦t❤❡s✐s✱ ✐t ✐s ❛
❝♦❤❡r❡♥t t❤❡♦r② ✇❤♦s❡ tr✉t❤ ♦r ❢❛❧s✐t② s❤♦✉❧❞ ❜❡ ❛ss❡ss❡❞ ❡♠♣✐r✐❝❛❧❧②✳ ❍♦✇❡✈❡r✱
❙●■❚ s❡❡♠ t♦ ❜❡ ✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ✐ts tr✉t❤✳ ❚♦ t❛❦❡ ❛ ♣❛rt✐❝✉❧❛r ❡①❛♠♣❧❡✱
❝♦♥s✐❞❡r t✇♦ ❞✐✛❡r❡♥t ♥❡t✇♦r❦s r❡♣r❡s❡♥t✐♥❣ t❤❡ ♦❜❥❡❝t✬s tr❛❥❡❝t♦r②✱ ♦♥❡ ✐♥ t❤❡
✈✐s✉❛❧ ❝♦rt❡①✱ Vt ✱ ❛♥❞ ♦♥❡ ✐♥ t❤❡ ❛✉❞✐t♦r② ❝♦rt❡①✱ At ✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ■♥t❡r✲
s❡♥s♦r② ❘❡❞✉♥❞❛♥❝② ❍②♣♦t❤❡s✐s✱ t❤✐s r❡❞✉♥❞❛♥❝② ❡♥❤❛♥❝❡s ♣❡r❝❡♣t✐♦♥ ♦❢ t❤❡
tr❛❥❡❝t♦r② ✇❤❡♥ t❤❡ ♦❜❥❡❝t ♣❛ss❡s ❜❡❤✐♥❞ ❛♥ ♦❝❝❧✉❞❡r✳ ■♥ ❝❡rt❛✐♥ ❝❛s❡s ♦❢ ♠❛①✲
✐♠❛❧ ❝♦♥❣r✉❡♥❝❡✱ Vt ❛♥❞ At ♣r♦✈✐❞❡ t❤❡ s❛♠❡ ✐♥❢♦r♠❛t✐♦♥ ❛♥❞ ❜♦t❤ r❡♣r❡s❡♥t
t❤❡ s❛♠❡ tr❛❥❡❝t♦r②✱ T ✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ♥♦✇ str❛✐❣❤t❢♦r✇❛r❞✿ ✐❢✱ ♦♥ t❤❡ ♦♥❡
✶✶
❤❛♥❞✱ P (T | Vt ) > P (T | At )✱ t❤❡♥ At ❞♦❡s ♥♦t r❡♣r❡s❡♥t T ✱ ❛♥❞ ✐❢✱ ♦♥ t❤❡ ♦t❤❡r
❤❛♥❞ P (T | Vt ) < P (T | At )✱ t❤❡♥ Vt ❞♦❡s ♥♦t r❡♣r❡s❡♥t T ✳ ❲❤❛t ❙●■❚ ❝❛♥♥♦t
❛❝❝❡♣t ✐s t❤❛t ❜♦t❤ ♥❡✉r❛❧ ♥❡t✇♦r❦s r❡♣r❡s❡♥t t❤❡ tr❛❥❡❝t♦r② ♦❢ t❤❡ ♦❜❥❡❝t✳
❆♥t✐❝✐♣❛t✐♥❣ ❛ s✐♠✐❧❛r ♣r♦❜❧❡♠✱ ❘✉♣❡rt ✭✶✾✾✾✱ ♣✳ ✸✹✾✲✸✺✵✮ s✉❣❣❡st❡❞ t❤❡
❢♦❧❧♦✇✐♥❣ ❛♠❡♥❞♠❡♥t ✐♥ t❤❡ t❤❡♦r②✿
❘ ❤❛s ❛ tr✉❧② ❞✐s❥✉♥❝t✐✈❡ ❡①t❡♥s✐♦♥ ✇❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✱
✭❛✮✱ ✭❜✮✱ ❛♥❞ ✭❝✮✱ ❛r❡ ♠❡t✿ ✭❛✮ t✇♦ ♦r ♠♦r❡ ♥❛t✉r❛❧ ❦✐♥❞s ❤❛✈❡ ❡q✉❛❧
♦r r♦✉❣❤❧② ❡q✉❛❧ s✉❝❝❡ss r❛t❡s r❡❧❛t✐✈❡ t♦ t❀ ✭❜✮ ♥♦ ♦t❤❡r ♥❛t✉r❛❧
❦✐♥❞ ❤❛s ❛ s✉❝❝❡ss r❛t❡ s✉❜st❛♥t✐❛❧❧② ❤✐❣❤❡r t❤❛♥ t❤♦s❡ ❦✐♥❞s ✇❤♦s❡
s✉❝❝❡ss r❛t❡s ❛r❡ ❡q✉❛❧ ♦r r♦✉❣❤❧② ❡q✉❛❧ r❡❧❛t✐✈❡ t♦ t❀ ❛♥❞ ✭❝✮ t❤❡
❣❛♣ ❜❡t✇❡❡♥ t❤❡ ❣r♦✉♣ ♦❢ s✉❝❝❡ss r❛t❡s ❛t t❤❡ t♦♣ ❛♥❞ t❤♦s❡ ❢❛rt❤❡r
❞♦✇♥ ✐s s✉❜st❛♥t✐❛❧✳
●❡♥❡r❛❧✐③✐♥❣ t❤✐s ✐❞❡❛✱ t❤❡ s✉❣❣❡st✐♦♥ ✐s t♦ q✉❛❧✐❢② ❝♦♥❞✐t✐♦♥ ✶ ♦❢ ✐♥❢♦ ❛♥❞
❛❝❝❡♣t t❤❛t t❤❡ r❡q✉✐r❡❞ ♣r♦❜❛❜✐❧✐t② ♥❡❡❞ ♥♦t ❜❡ str✐❝t❧② ❤✐❣❤❡r❀ ✐t s✉✣❝❡s ✐❢
P (Ri | Si ) ≥ P (Ri | Sj )✳ ■♥ t❤✐s ✇❛②✱ ❙●■❚ ❝♦✉❧❞ ❛❝❝♦✉♥t ❢♦r s♦♠❡ ❛♠❜✐❣✉♦✉s
r❡♣r❡s❡♥t❛t✐♦♥s✳ ❆ s✐♠✐❧❛r ♠♦❞✐✜❝❛t✐♦♥ ♦❢ ❝♦♥❞✐t✐♦♥ ✷ ✭✐✳❡✳ P (Si | R) ≥ P (Si |
Rj ) ✮ ❝♦✉❧❞ ❧❡❛✈❡ r♦♦♠ ❢♦r ♠✉❧t✐♣❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ✇✐t❤ t❤❡ s❛♠❡ ❝♦♥t❡♥t✳
❉♦❡s t❤✐s s✉❣❣❡st✐♦♥ ♣r♦✈✐❞❡ ❛ ❝♦♥✈✐♥❝✐♥❣ ✇❛② ♦✉t t♦ t❤❡ ♣r♦❜❧❡♠❄ ■t ✐s
✉♥❝❧❡❛r t❤❛t t❤✐s ❛❞❞✐t✐♦♥❛❧ ❝❧❛✉s❡ ❝❛♥ ❣✐✈❡ t❤❡ r✐❣❤t r❡s✉❧ts✳ ❋♦r ♦♥❡ t❤✐♥❣✱
t❤❡r❡ ❛r❡ s✉r❡❧② ♠❛♥② ❝❛s❡s ♦❢ ❛♠❜✐❣✉♦✉s ♦r r❡❞✉♥❞❛♥t r❡♣r❡s❡♥t❛t✐♦♥s ✐♥ ✇❤✐❝❤
t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❞✐✛❡r❡♥t st✐♠✉❧✐ ♦r r❡♣r❡s❡♥t❛t✐♦♥s ❛r❡ ✉♥❡q✉❛❧✳ ❋♦r ❛♥♦t❤❡r✱
❡q✉❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ❞♦❡s ♥♦t ❡♥s✉r❡ ❝♦r❡❢❡r❡♥❝❡✱ ♣r❡❝✐s❡❧② ❜❡❝❛✉s❡ ♦❢ t❤❡ ❡①✐st❡♥❝❡
♦❢ r❡❣✉❧❛r ♠✐sr❡♣r❡s❡♥t❛t✐♦♥s✳ ■♥ ❘✉♣❡rt✬s ❢♦r♠✉❧❛t✐♦♥✱ t❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡
❡♥t✐t✐❡s t❤❛t r❡❧❛t❡ t♦ ❞✐✛❡r❡♥t ♠❡❛♥✐♥❣s ✐♥ ❛♥ ❛♠❜✐❣✉♦✉s r❡♣r❡s❡♥t❛t✐♦♥ ❤❛s
t♦ ❜❡ ❡q✉❛❧ ♦r r♦✉❣❤❧② ❡q✉❛❧✱ ✐✳❡✳ ❘ r❡❢❡rs t♦ t✇♦ ❞✐✛❡r❡♥t ❡♥t✐t✐❡s ✐✛ ❛ s✉❜❥❡❝t
r❡❣✉❧❛r❧② ❛♥❞ ❝♦♥s✐st❡♥t❧② ❛♣♣❧✐❡s ❘ t♦ Si ❛♥❞ Sj ❛♥❞ t❤✐s ❢❛❝t ❝♦♥tr❛sts ✇✐t❤ ❤♦✇
♦❢t❡♥ Si ❛♥❞ Sj ❝❛✉s❡ ♦t❤❡r r❡♣r❡s❡♥t❛t✐♦♥s✳ ❨❡t t❤✐s s✉❣❣❡st✐♦♥ ❞♦❡s ♥♦t t❛❦❡
t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ r❡❝✉rr❡♥t ♠✐sr❡♣r❡s❡♥t❛t✐♦♥ s❡r✐♦✉s❧② ❡♥♦✉❣❤✳ ■♥ ♣❛rt✐❝✉❧❛r✱
t❤❡r❡ ✐s ♥♦t❤✐♥❣ ✐♥ t❤✐s ❛♣♣r♦❛❝❤ t❤❛t ❝♦✉❧❞ ❞✐st✐♥❣✉✐s❤ ❛♠❜✐❣✉♦✉s ❝♦♥❝❡♣ts
❢r♦♠ s②st❡♠❛t✐❝ ♠✐st❛❦❡s✳ ❚❛❦✐♥❣ t❤❡ ❡①❛♠♣❧❡ ❞✐s❝✉ss❡❞ ❡❛r❧✐❡r✱ ❝♦♥s✐❞❡r t❤❡
♣r❡❞❛t♦r✬s r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❜❡❡s✿ t❤❡ ♣r❡❞❛t♦r ♠✐❣❤t s②st❡♠❛t✐❝❛❧❧② ❝♦♥❢✉s❡
❞r♦♥❡✢✐❡s ❢♦r ❜❡❡s ❛♥❞ t❤❛t ❞♦❡s ♥♦t ♠❡❛♥ t❤❛t ✐ts r❡♣r❡s❡♥t❛t✐♦♥ ✐s ❛♠❜✐❣✉♦✉s❀
✐t ❥✉st ♠❡❛♥s t❤❛t ✐t ✐s ✇r♦♥❣ r♦✉❣❤❧② ❤❛❧❢ ♦❢ t❤❡ t✐♠❡✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ❛♥
❡①❝❧✉s✐✈❡ r❡❧✐❛♥❝❡ ♦♥ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝✐❡s r❡♥❞❡rs ✐t ✉♥❛❜❧❡ t♦ ♠❛❦❡ t❤✐s
❞✐st✐♥❝t✐♦♥✳ ❚❤✉s tr②✐♥❣ t♦ ❧❡❛✈❡ r♦♦♠ ❢♦r ♠✉❧t✐♣❧❡ ❝♦♥t❡♥ts ❛♥❞ r❡❞✉♥❞❛♥❝② ❜②
s✐♠♣❧② r❡❧❛①✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ❢♦r r❡♣r❡s❡♥t✐♥❣✱ ❞♦❡s ♥♦t ❡♥s✉r❡ t❤❛t t❤❡ r✐❣❤t
❞✐s❥✉♥❝t✐✈❡ ❝♦♥t❡♥ts ❛r❡ ♣r❡❞✐❝t❡❞✳
❆s ❛ r❡s✉❧t✱ ❙●■❚ ❛r❡ ♣r♦❜❛❜❧② ✉♥❛❜❧❡ t♦ ❛❝❝♦✉♥t ❢♦r ❣❡♥✉✐♥❡ ❝❛s❡s ♦❢ ❛♠❜✐✲
❣✉✐t② ❛♥❞ r❡❞✉♥❞❛♥❝②✳ ❆❣❛✐♥✱ ♦t❤❡r ♥♦t✐♦♥s ❧✐❦❡ ❢✉♥❝t✐♦♥❛❧ r♦❧❡ ♦r t❡❧❡♦❢✉♥❝t✐♦♥
✭▼✐❧❧✐❦❛♥✱ ✷✵✵✵✮ ♠✐❣❤t ❝♦♥tr✐❜✉t❡ t♦ s♦❧✈✐♥❣ t❤✐s ♣r♦❜❧❡♠✳ ❇✉t st❛t✐st✐❝❛❧ ❞❡✲
♣❡♥❞❡♥❝❡ ❞♦❡s ♥♦t s❡❡♠ t♦ ❜❡ t❤❡ r✐❣❤t t♦♦❧ ❢♦r t❤❡ t❛s❦ ❛t ❤❛♥❞✳
✶✷
✸✳✹
▼❡t❛s❡♠❛♥t✐❝s
❚❤❡ ❧❛st ♣r♦❜❧❡♠ ♦❢ ❙●■❚ ✐s t❤❛t t❤❡② ❞♦ ♥♦t ❡✈❡♥ ❛❞❞r❡ss t❤❡ ♠♦st ♣r❡ss✲
✐♥❣ q✉❡st✐♦♥ ❢♦r ❛ t❤❡♦r② ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✱ ♥❛♠❡❧②✱ ✇❤❛t ♠❛❦❡s ❝❡rt❛✐♥ st❛t❡s
r❡♣r❡s❡♥t❛t✐♦♥❛❧✳ ▲❡t ✉s ❡❧❛❜♦r❛t❡✳
●❡♥❡r❛❧❧②✱ t❤❡ ❞✐s❝✉ss✐♦♥ ♦♥ ♥❛t✉r❛❧✐st✐❝ t❤❡♦r✐❡s ♦❢ ❝♦♥t❡♥t ❤❛s ❢❛✐❧❡❞ t♦
♠❛❦❡ ❡①♣❧✐❝✐t ❛♥ ✐♠♣♦rt❛♥t ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❞✐✛❡r❡♥t ❣♦❛❧s✳ ❋✐rst ♦❢ ❛❧❧✱
❛ t❤❡♦r② ♦❢ r❡♣r❡s❡♥t❛t✐♦♥ ♠✉st ❡①♣❧❛✐♥ ✇❤② ❛ st❛t❡ R r❡♣r❡s❡♥ts S1 r❛t❤❡r t❤❛♥
❛ ❞✐✛❡r❡♥t st❛t❡ S2 ✳ ❋♦r ✐♥st❛♥❝❡✱ ♦♥❡ ♠✐❣❤t ❛tt❡♠♣t t♦ ❡①♣❧❛✐♥ ✇❤② ❝❡rt❛✐♥
♥❡✉r♦♥❛❧ ❛❝t✐✈✐t② ✐♥ t❤❡ str✐❛t❡ ❝♦rt❡① ✐♥❞✐❝❛t❡s t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡
r❛t❤❡r t❤❛♥ ❛ ❝✐r❝❧❡✳ ❙❡❝♦♥❞❧②✱ ❛ s❛t✐s❢❛❝t♦r② t❤❡♦r② ❛❧s♦ ❤❛s t♦ ♣r♦✈✐❞❡ ❛♥
❛❝❝♦✉♥t ♦❢ ✇❤② R r❡♣r❡s❡♥ts s♦♠❡t❤✐♥❣ ❛t ❛❧❧✳ ❆❝❝♦r❞✐♥❣ t♦ t❤✐s s❡❝♦♥❞ ✇❛②
♦❢ ❛❞❞r❡ss✐♥❣ t❤❡ ♣r♦❜❧❡♠✱ t❤❡ ❣♦❛❧ ✐s ♥♦t t♦ ❡①♣❧❛✐♥ ✇❤② ♥❡✉r♦♥❛❧ ❛❝t✐✈✐t② ✐s
s✉♣♣♦s❡❞ t♦ ❞❡t❡❝t ✈❡rt✐❝❛❧ ❧✐♥❡s r❛t❤❡r t❤❛♥ s♦♠❡t❤✐♥❣ ❡❧s❡✱ ❜✉t ✇❤② t❤✐s st❛t❡
✐s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ❛t ❛❧❧✳ ▲❡t ✉s ❝❛❧❧ t❤❡ ✜rst ❦✐♥❞ ♦❢ t❤❡♦r② ❵s❡♠❛♥t✐❝✬ ❛♥❞ t❤❡
s❡❝♦♥❞ ♦♥❡ ❵♠❡t❛s❡♠❛♥t✐❝✬ ❬❛✉t❤♦r✶❪✳
◆♦✇✱ t❤❛t ❞✐st✐♥❝t✐♦♥ ✐s ✐♠♣♦rt❛♥t ❜❡❝❛✉s❡ s♦❧✈✐♥❣ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✐♥t❡♥t✐♦♥✲
❛❧✐t② r❡q✉✐r❡s ♣r♦✈✐❞✐♥❣ ❛ s❡♠❛♥t✐❝ ❛♥❞ ❛ ♠❡t❛s❡♠❛♥t✐❝ t❤❡♦r②✳ ■❢ ✇❡ ✇❛♥t ❛
t❤❡♦r② t♦ ❢✉❧❧② ❡①♣❧❛✐♥ ✇❤② ❝❡rt❛✐♥ st❛t❡s ❛r❡ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ❛♥❞ ❤♦✇ t❤❡✐r
❝♦♥t❡♥t ✐s ❞❡t❡r♠✐♥❡❞✱ ✇❡ ♥❡❡❞ t♦ ♣r♦✈✐❞❡ ❜♦t❤ ❦✐♥❞s ♦❢ t❤❡♦r✐❡s✳ ▼❡r❡❧② ❡①✲
♣❧❛✐♥✐♥❣ ✇❤② ❛ ❝❡rt❛✐♥ st❛t❡ r❡♣r❡s❡♥ts ❧✐♥❡s r❛t❤❡r ❝✐r❝❧❡s ❞♦❡s ♥♦t ❝♦♠♣❧❡t❡❧②
❞✐s♣❡❧ t❤❡ ♠②st❡r② ♣♦s❡❞ ❜② r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♣❤❡♥♦♠❡♥❛✱ s✐♥❝❡ t❤❡r❡ ✐s st✐❧❧ ❛
❢✉♥❞❛♠❡♥t❛❧ q✉❡st✐♦♥ t❤❛t r❡♠❛✐♥s ✉♥❛♥s✇❡r❡❞✿ ✇❤② ❞♦❡s t❤✐s st❛t❡ r❡♣r❡s❡♥t
s♦♠❡t❤✐♥❣ ❛t ❛❧❧❄ ❯♥❧❡ss t❤✐s ✐ss✉❡ ✐s ❛❞❞r❡ss❡❞✱ ✇❡ ✇✐❧❧ ❧❛❝❦ ❛ s♦❧✉t✐♦♥ t❤❡
♣r♦❜❧❡♠ ♦❢ ✐♥t❡♥t✐♦♥❛❧✐t②✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❛r✐s♦♥ ♠✐❣❤t ❜❡ ✐❧❧✉♠✐♥❛t✐♥❣✿ t❤❡
♣r♦❜❧❡♠ ♦❢ ❝♦♥s❝✐♦✉s♥❡ss ❝❛♥♥♦t ❜❡ ❝♦♠♣❧❡t❡❧② s♦❧✈❡❞ ❜② ♠❡r❡❧② ♣r♦✈✐❞✐♥❣ ❛
t❤❡♦r② ♦❢ ✇❤② ❛ s✉❜❥❡❝t ❡①♣❡r✐❡♥❝❡s ❜❧✉❡ r❛t❤❡r t❤❛♥ r❡❞ ✭❡✳❣✳ ❜② ♠❡♥t✐♦♥✐♥❣
t❤❡ ❢❛❝t t❤❛t ❞✐✛❡r❡♥t ♣❛rts ♦❢ t❤❡ ❜r❛✐♥ ❛r❡ ❛❝t✐✈❡ ♦r t❤❛t t❤❡② r❡♣r❡s❡♥t ❛
❞✐✛❡r❡♥t ❝♦♥t❡♥t✮✳ ❖♥ t♦♣ ♦❢ t❤❛t ❛♥❞ ♠♦r❡ ✐♠♣♦rt❛♥t❧② ✭❑r✐❡❣❡❧✱ ✷✵✵✾✮✱ ✇❡
♥❡❡❞ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ♦❢ ✇❤❛t ♠❛❦❡s ✐t ❛ ❝♦♥s❝✐♦✉s st❛t❡✳ ❲❡ ❛❜str❛❝t
❢r♦♠ t❤❡ ♣❛rt✐❝✉❧❛r ✇❛②s ❤❛✈✐♥❣ ❞✐✛❡r❡♥t ❡①♣❡r✐❡♥❝❡s ❢❡❡❧ ❛♥❞ ❝♦♥❝❡♥tr❛t❡ ♦♥
t❤❡ ♣r♦❜❧❡♠ ♦❢ ✇❤❛t ♠❛❦❡s ✐t t❤❡ ❝❛s❡ t❤❛t ❤❛✈✐♥❣ ❛ ❝♦♥s❝✐♦✉s ❡①♣❡r✐❡♥❝❡ ❢❡❡❧s
❛♥② ✇❛② ❛t ❛❧❧✳ ▲✐❦❡✇✐s❡✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✐♥t❡♥t✐♦♥❛❧✐t② ❝♦♥❝❡r♥s t❤❡ ✈❡r② ♥❛t✉r❡
♦❢ r❡♣r❡s❡♥t❛t✐♦♥❛❧ st❛t❡s✳ Pr♦✈✐❞✐♥❣ ❛ s❡♠❛♥t✐❝ t❤❡♦r② ✐s ♦❢ ❝♦✉rs❡ ❛ st❡♣ ✐♥ t❤❡
r✐❣❤t ❞✐r❡❝t✐♦♥✱ ❜✉t ✐t ❧❡❛✈❡s ✉♥r❡s♦❧✈❡❞ ❛ ♠❛❥♦r q✉❡st✐♦♥✳
❲✐t❤ t❤✐s ❞✐st✐♥❝t✐♦♥ ✐♥ ♠✐♥❞✱ ✐t s❤♦✉❧❞ ❜❡ ♦❜✈✐♦✉s ✇❤② ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡✲
♦r✐❡s ❞♦ ♥♦t ♣r♦✈✐❞❡ ❛ ♠❡t❛s❡♠❛♥t✐❝ t❤❡♦r② ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✳ ❊✈❡♥ ✐❢ t❤❡②
❝♦rr❡❝t❧② ✐❞❡♥t✐✜❡❞ t❤❡ ❝♦♥❞✐t✐♦♥s t❤❛t s❤♦✉❧❞ ❜❡ s❛t✐s✜❡❞ ❢♦r ❛ r❡♣r❡s❡♥t❛t✐♦♥❛❧
st❛t❡ R t♦ r❡❢❡r t♦ S1 r❛t❤❡r t❤❛♥ t♦ S2 ✱ t❤❡② ❞♦ ♥♦t ♣✉t ❢♦r✇❛r❞ ❛♥② ❝r✐t❡r✐♦♥
❢♦r ❞❡t❡r♠✐♥✐♥❣ ✇❤❡♥ ❛ ❝❡rt❛✐♥ st❛t❡ ✐s ✐♥❞❡❡❞ ❛ r❡♣r❡s❡♥t❛t✐♦♥✳ ■♥❢♦r♠❛t✐♦♥❛❧
t❤❡♦r✐❡s ♦✛❡r s❡♠❛♥t✐❝ t❤❡♦r✐❡s✱ ❜✉t ❢❛✐❧ t♦ ❞❡❧✐✈❡r ♠❡t❛s❡♠❛♥t✐❝ ♦♥❡s✳
❖❢ ❝♦✉rs❡✱ ✐t ❝♦✉❧❞ ❜❡ r❡♣❧✐❡❞ t❤❛t t❤❡ ❢❛❝t t❤❛t t❤❡r❡ ✐s ❛♥ ❛s♣❡❝t ♦❢ t❤❡
♣r♦❥❡❝t t❤❛t t❤❡s❡ t❤❡♦r✐❡s ❤❛✈❡ ♥♦t ②❡t ❛❞❞r❡ss❡❞ ❞♦❡s ♥♦t ♠❡❛♥ t❤❛t t❤❡②
❝♦✉❧❞ ♥♦t ❞♦ ✐t✳ ❋✉t✉r❡ ✇♦r❦ ♠✐❣❤t ✜❧❧ t❤✐s ❣❛♣ ✇✐t❤♦✉t ❛❜❛♥❞♦♥✐♥❣ ❛ ♣✉r❡❧②
✐♥❢♦r♠❛t✐♦♥❛❧ ❢r❛♠❡✇♦r❦✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ♣r♦❜❧❡♠ s❡❡♠s t♦ ❣♦ ♠✉❝❤ ❞❡❡♣❡r
t❤❛♥ t❤❛t✿ ♥♦t ♦♥❧② ❤❛✈❡ ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ❛❝t✉❛❧❧② ❢❛✐❧❡❞ t♦ ♣r♦✈✐❞❡ ❛
✶✸
♠❡t❛s❡♠❛♥t✐❝ ❛❝❝♦✉♥t✱ ❜✉t t❤❡ ✐♥❢♦r♠❛t✐♦♥❛❧ t♦♦❧s t❤❡② ❡♠♣❧♦② s❡❡♠ t♦ ❜❡
✐♥❛❞❡q✉❛t❡ ❢♦r ❝❛rr②✐♥❣ ✐t ♦✉t✳ ❚❤❡ r❡❛s♦♥ ✐s ✇❡❧❧✲❦♥♦✇♥✿ ✐♥❢♦r♠❛t✐♦♥❛❧ r❡❧❛t✐♦♥s
❛r❡ ✉❜✐q✉✐t♦✉s✳ ❆♥② st❛t❡ t❤❛t ✐s s✉❜❥❡❝t t♦ ❜❡ ✐♥ ❛ ❝❛✉s❛❧ ❝❤❛✐♥✖t❤❛t ✐s✱ ❛♥②
st❛t❡ t❤❛t ✐s ♥♦t ♦✉ts✐❞❡ t❤❡ ❝❛✉s❛❧ ♦r❞❡r✖❝❛rr✐❡s ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✐ts ♣♦ss✐❜❧❡
❝❛✉s❡s ❛♥❞ ❡✛❡❝ts ✭❛♠♦♥❣ ♦t❤❡r t❤✐♥❣s✮✳ ❍♦✇❡✈❡r✱ ✇❡ ❞♦ ♥♦t ✇❛♥t t♦ ♠❛✐♥t❛✐♥
t❤❛t r❡♣r❡s❡♥t❛t✐♦♥s ❛r❡ ✉❜✐q✉✐t♦✉s✳ ❚❤✉s✱ t❤❡ ❝♦♥❝❡♣t ♦❢ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡
✐s ♥♦t ✜♥❡✲❣r❛✐♥❡❞ ❡♥♦✉❣❤ ❢♦r ❞✐st✐♥❣✉✐s❤✐♥❣ ✇❤❛t ✐s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ❢r♦♠ ✇❤❛t
✐s ♥♦t✳ ❈❛rr②✐♥❣ ✐♥❢♦r♠❛t✐♦♥✱ ❧✐❦❡ t❤❡ ♣r♦♣❡rt✐❡s ❤❛✈✐♥❣ ❛ ❝❡rt❛✐♥ ✇❡✐❣❤t ♦r
❜❡✐♥❣ ❛♥ ♦❜❥❡❝t ❛r❡ ❥✉st t♦♦ ❝♦❛rs❡✲❣r❛✐♥❡❞ ❢♦r t❤❡ t❛s❦ ❛t ❤❛♥❞✳ ❲❤❛t ❛t ✜rst
❣❧❛♥❝❡ ♠✐❣❤t ❧♦♦❦ ❧✐❦❡ ❛♥ ♦✈❡rs✐❣❤t ❛❝t✉❛❧❧② ❝♦♥st✐t✉t❡s ❛♥ ✐♠♣♦rt❛♥t ❣❛♣ ✐♥ t❤❡
t❤❡♦r②✳
❈❡rt❛✐♥❧②✱ ✇❡ ❞♦ ♥♦t ✇❛♥t t♦ ❞❡♥② t❤❛t t❤❡r❡ ❛r❡ ♠❡t❛s❡♠❛♥t✐❝ t❤❡♦r✐❡s
❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ✐♥❢♦r♠❛t✐♦♥❛❧ ❛❝❝♦✉♥ts✳ ❆❣❛✐♥✱ ♦♥❡ ❝♦✉❧❞ ❛❞❞ ❢✉♥❝t✐♦♥❛❧ r♦❧❡s✱
t❡❧❡♦❢✉♥❝t✐♦♥s ♦r ♦t❤❡r t♦♦❧s ✐♥ ♦r❞❡r t♦ ❞❡✜♥❡ ✇❤❛t r❡♣r❡s❡♥t❛t✐♦♥s ❛❝t✉❛❧❧②
❛r❡✳ ❇✉t✱ ✐♥ ❛♥② ❝❛s❡✱ t❤❡s❡ ❤②❜r✐❞ t❤❡♦r✐❡s ✇♦✉❧❞ ♣r♦✈✐❞❡ ❛ s♦❧✉t✐♦♥ t♦ t❤❡
♠❡t❛s❡♠❛♥t✐❝ ♣r♦❜❧❡♠ ♦♥❧② ✐❢ t❤❡② ❣♦ ❜❡②♦♥❞ t❤❡ ♥♦t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥✳ ❈♦♥✲
s❡q✉❡♥t❧②✱ ♣✉r❡❧② ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ✭s✉❝❤ ❛s ❙●■❚✮ ❛r❡ ✉♥❧✐❦❡❧② t♦ ♣r♦✈✐❞❡
❛ ❢✉❧❧② ❝♦♥✈✐♥❝✐♥❣ ❛❝❝♦✉♥t ♦❢ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♣❤❡♥♦♠❡♥❛✳
✹ ❈♦♥❝❧✉s✐♦♥
❙✉♠♠✐♥❣ ✉♣✱ ✐♥ t❤✐s ♣❛♣❡r ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t ❙❝✐❡♥t✐✜❝❛❧❧② ●✉✐❞❡❞ ■♥❢♦r♠❛✲
t✐♦♥❛❧ ❚❤❡♦r✐❡s ✭❙●■❚✮ ❢❛❝❡ ❢♦✉r ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠s✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡
❛r❣✉❡❞ t❤❛t t❤❡ r❡❛s♦♥s t❤❡② ❢❛✐❧ t♦ ♦✈❡r❝♦♠❡ t❤❡s❡ ❞✐✣❝✉❧t✐❡s ❛r❡ ❞❡❡♣✿ ❜❡✲
❝❛✉s❡ ♦❢ t❤❡✐r ❡①❝❧✉s✐✈❡ r❡❧✐❛♥❝❡ ♦♥ ✐♥❢♦r♠❛t✐♦♥✱ t❤❡② s✐♠♣❧② ❧❛❝❦ t❤❡ r❡s♦✉r❝❡s
❢♦r ♣r♦✈✐❞✐♥❣ s❛t✐s❢❛❝t♦r② s♦❧✉t✐♦♥s✳ ❚❤✉s✱ ♥❡✇✲✇❛✈❡ ✐♥❢♦r♠❛t✐♦♥❛❧ t❤❡♦r✐❡s ❛r❡
✉♥❧✐❦❡❧② t♦ s✉❝❝❡❡❞ ✐♥ t❤❡ ♣r♦❥❡❝t ♦❢ ♣r♦✈✐❞✐♥❣ ❛ t❤❡♦r② ♦❢ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ t❤❡
❝♦♥t❡①t ♦❢ ❝♦❣♥✐t✐✈❡ s❝✐❡♥❝❡✳
◆♦♥❡t❤❡❧❡ss✱ ✐t ✐s ✇♦rt❤ str❡ss✐♥❣ t❤❛t t❤❡ ❛r❣✉♠❡♥ts s✉❣❣❡st❡❞ ❤❡r❡ ❛r❡ ♥♦t
✐♥t❡♥❞❡❞ t♦ s❤♦✇ t❤❛t t❤❡ ♥♦t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✐s ✉s❡❧❡ss✳ ❈♦❣♥✐t✐✈❡ s❝✐❡♥t✐sts
❤❡❛✈✐❧② r❡❧② ♦♥ ✐♥❢♦r♠❛t✐♦♥❛❧ ♠❡❛s✉r❡s ❛♥❞ t❤❡ ✐♥t✉✐t✐♦♥ t❤❛t t❤✐s ♥♦t✐♦♥ ❝❛♣✲
t✉r❡s s♦♠❡t❤✐♥❣ ✐♠♣♦rt❛♥t ❛❜♦✉t ❝♦❣♥✐t✐♦♥ ✐s ❛ ♣♦✇❡r❢✉❧ ♦♥❡✳ ❖✉r ❛r❣✉♠❡♥ts
❛r❡ ♥♦t ✐♥t❡♥❞❡❞ t♦ s✉❣❣❡st t❤❛t t❤❡ ♥♦t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ s❤♦✉❧❞ ❜❡ ❡❧✐♠✐♥❛t❡❞✱
❜✉t r❛t❤❡r t❤❛t t❤❡ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐♥❢♦r♠❛t✐♦♥ ❛♥❞ r❡♣r❡s❡♥t❛t✐♦♥ ♥❡❡❞s t♦
❜❡ r❡❛ss❡ss❡❞✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛ttr✐❜✉t✐♦♥s ♦❢ r❡♣r❡s❡♥t❛t✐♦♥s ✐♥ s❝✐❡♥t✐✜❝ ♣r❛❝t✐❝❡s
♠✐❣❤t ♥♦t ❥✉st r❡❧② ♦♥ st❛t✐st✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ✭❛t ❧❡❛st ♥♦t ❛❧✇❛②s✱ ❛s s♦♠❡ ♦❢ t❤❡
❡①❛♠♣❧❡s ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ s❤♦✇✮✳ ❋✉rt❤❡r♠♦r❡✱ ♦t❤❡r ♥♦t✐♦♥s s✉❝❤ ❛s ✬❢✉♥❝✲
t✐♦♥✬ ♠✐❣❤t ♥❡❡❞ t♦ ❜❡ ❛❞❞❡❞ t♦ ❛❞❞r❡ss t❤❡ ♣r❡✈✐♦✉s ✇♦rr✐❡s✳ ❈♦♥s❡q✉❡♥t❧②✱
❛❞❞✐t✐♦♥❛❧ ❝♦♥s✐❞❡r❛t✐♦♥s ❤❛✈❡ t♦ ❜❡ ♠❛❞❡ ❡①♣❧✐❝✐t ❛♥❞ ♣❧❛❝❡❞ ♦♥ t❤❡ t❛❜❧❡ ❢♦r
❛ ♣r♦♣❡r ❛ss❡ss♠❡♥t✳ ❲❤❡t❤❡r t❤✐s s❤♦✉❧❞ ❧❡❛❞ t♦ ❛ r❡✈✐s✐♦♥ ♦❢ ♦✉r ❝✉rr❡♥t
s❝✐❡♥t✐✜❝ ♠❡t❤♦❞♦❧♦❣② ✐s ❛♥ ✐♠♣♦rt❛♥t q✉❡st✐♦♥ ✇❡ ❧❡❛✈❡ ❢♦r ❢✉t✉r❡ r❡s❡❛r❝❤✳
✶✹
❘❡❢❡r❡♥❝❡s
❇❛❤r✐❝❦✱ ▲✳ ❛♥❞ ▲✐❝❦❧✐t❡r✱ ❘✳ ✷✵✵✵✳ ■♥t❡rs❡♥s♦r② r❡❞✉♥❞❛♥❝② ❣✉✐❞❡s ❛tt❡♥✲
t✐♦♥❛❧ s❡❧❡❝t✐✈✐t② ❛♥❞ ♣❡r❝❡♣t✉❛❧ ❧❡❛r♥✐♥❣ ✐♥ ✐♥❢❛♥❝②✳ ❉❡✈❡❧♦♣♠❡♥t❛❧ Ps②❝❤♦❧✲
♦❣② ✸✻✱ ✶✺✸✕✶✽✻✳
❇❛❤r✐❝❦✱ ▲✳✱ ❋❧♦♠✱ ❘✳ ❛♥❞ ❘✳✱ ▲✳ ✷✵✵✷✳ ■♥t❡rs❡♥s♦r② r❡❞✉♥❞❛♥❝② ❢❛❝✐❧✐t❛t❡s ❞✐s✲
❝r✐♠✐♥❛t✐♦♥ ♦❢ t❡♠♣♦ ✐♥ ✸✵ ♠♦♥t❤ ♦❧❞ ✐♥❢❛♥ts✳ ❉❡✈❡❧♦♣♠❡♥t❛❧ Ps②❝❤♦❜✐♦❧♦❣②
✹✶✱ ✸✺✷✕✸✻✸✳
❇❛❤r✐❝❦✱ ▲✳✱ ▲✐❝❦❧✐t❡r✱ ❘✳ ❛♥❞ ❋❧♦♠✱ ❘✳ ✷✵✵✹✳ ■♥t❡rs❡♥s♦r② r❡❞✉♥❞❛♥❝② ❣✉✐❞❡s
✐♥❢❛♥ts s❡❧❡❝t✐✈❡ ❛tt❡♥t✐♦♥✱ ♣❡r❝❡♣t✐♦♥✱ ❛♥❞ ❝♦❣♥✐t✐♦♥ ✐♥ ✐♥❢❛♥❝②✳✳ ❈✉rr❡♥t ❉✐✲
r❡❝t✐♦♥s ✐♥ Ps②❝❤♦❧♦❣✐❝❛❧ ❙❝✐❡♥❝❡ ✶✸✱ ✾✾✕✶✵✷✳
❇✐r❝❤✱ ❏✳ ✷✵✶✹✳ Pr♦♣♦s✐t✐♦♥❛❧ ❝♦♥t❡♥t ✐♥ s✐❣♥❛❧❧✐♥❣ s②st❡♠s✳ P❤✐❧♦s♦♣❤✐❝❛❧ ❙t✉❞✐❡s
✶✼✶✲✸✱ ✹✾✸✕✺✶✷✳
❇r❡♠♥❡r✱ ❏✳ ●✳✱ ❙❧❛t❡r✱ ❆✳✱ ❏♦❤♥s♦♥✱ ❙✳ P✳✱ ▼❛s♦♥✱ ❯✳ ❈✳ ❛♥❞ ❙♣r✐♥❣✱ ❏✳ ✷✵✶✷✳ ❚❤❡
❡✛❡❝ts ♦❢ ❛✉❞✐t♦r② ✐♥❢♦r♠❛t✐♦♥ ♦♥ ✹ ♠♦♥t❤ ♦❧❞ ✐♥❢❛♥ts ♣❡r❝❡♣t✐♦♥ ♦❢ tr❛❥❡❝t♦r②
❝♦♥t✐♥✉✐t②✳ ❈❤✐❧❞r❡♥ ❞❡✈❡❧♦♣♠❡♥t ✽✸✭✸✮✱ ✾✺✹✕✾✻✹✳
❈❤✉♥✱ ◆✳ ❑✳ ❏✳ ▼✳ ▼✳ ✶✾✾✼✳ ❚❤❡ ❢✉s✐❢♦r♠ ❢❛❝❡ ❛r❡❛✿ ❆ ♠♦❞✉❧❡ ✐♥ ❤✉♠❛♥ ❡①✲
tr❛str✐❛t❡ ❝♦rt❡① s♣❡❝✐❛❧✐③❡❞ ❢♦r ❢❛❝❡ ♣❡r❝❡♣t✐♦♥✳ ❚❤❡ ❏♦✉r♥❛❧ ♦❢ ◆❡✉r♦s❝✐❡♥❝❡
✶✼✭✶✶✮✱ ✹✸✵✷✕✹✸✶✶✳
❉❛♠❛s✐♦✱ ❆✳✿ ✷✵✶✵✳
❡❞♥✳ P❛♥t❤❡♦♥✳
❙❡❧❢ ❈♦♠❡s t♦ ▼✐♥❞✿ ❈♦♥str✉❝t✐♥❣ t❤❡ ❈♦♥s❝✐♦✉s ❇r❛✐♥✳
✶
❉❡s✐♠♦♥❡✱ ❘✳ ✶✾✾✶✳ ❋❛❝❡✲s❡❧❡❝t✐✈❡ ❝❡❧❧s ✐♥ t❤❡ t❡♠♣♦r❛❧ ❝♦rt❡① ♦❢ ♠♦♥❦❡②s✳ ❏♦✉r✲
♥❛❧ ♦❢ ❈♦❣♥✐t✐✈❡ ◆❡✉r♦s❝✐❡♥❝❡ ✸✱ ✶✕✽✳
❉r❡ts❦❡✱ ❋✳✿ ✶✾✽✶❛✳
❑♥♦✇❧❡❞❣❡ ❛♥❞ t❤❡ ❋❧♦✇ ♦❢ ■♥❢♦r♠❛t✐♦♥✳
❉r❡ts❦❡✱ ❋✳✿ ✶✾✽✶❜✳
Pr❡ss✳
❑♥♦✇❧❡❞❣❡ ❛♥❞ t❤❡ ❋❧♦✇ ♦❢ ■♥❢♦r♠❛t✐♦♥✳
❚❤❡ ▼■❚ Pr❡ss✳
❈❛♠❜r✐❞❣❡✿ ▼■❚
❊❧✐❛s♠✐t❤✱ ❈✳✿ ✷✵✵✵✳ ❍♦✇ ♥❡✉r♦♥s ♠❡❛♥✿ ❆ ♥❡✉r♦❝♦♠♣✉t❛t✐♦♥❛❧ t❤❡♦r② ♦❢ r❡♣r❡✲
s❡♥t❛t✐♦♥❛❧ ❝♦♥t❡♥t✳ ❯♥♣✉❜❧✐s❤❡❞ ❉✐ss❡rt❛t✐♦♥✱ ❲❛s❤✐♥❣t♦♥ ❯♥✐✈❡rs✐t② ✐♥ ❙t✳
▲♦✉✐s✳
❊❧✐❛s♠✐t❤✱ ❈✳✿ ✷✵✵✺❛✳ ◆❡✉r♦s❡♠❛♥t✐❝s ❛♥ ❝❛t❡❣♦r✐❡s✳ ✐♥ ❍✳ ❈♦❤❡♥ ❛♥❞ ❈✳ ▲❛❢❡❜✲
✈r❡ ✭❡❞s✮✱ ❍❛♥❞❜♦♦❦ ♦❢ ❈❛t❡❣♦r✐③❛t✐♦♥ ✐♥ ❈♦❣♥✐t✐✈❡ ❙❝✐❡♥❝❡✳ ❊❧s❡✈✐❡r✳
❊❧✐❛s♠✐t❤✱ ❈✳ ✷✵✵✺❜✳ ❆ ♥❡✇ ♣❡rs♣❡❝t✐✈❡ ♦♥ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♣r♦❜❧❡♠ss✳
♦❢ ❈♦❣♥✐t✐✈❡ ❙❝✐❡♥❝❡ ✻✱ ✾✼✕✶✷✸✳
❊❧✐❛s♠✐t❤✱ ❈✳✿ ✷✵✶✸✳ ❍♦✇ t♦ ❜✉✐❧❞ ❛ ❜r❛✐♥✿
❝♦❣♥✐t✐♦♥✳ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✳
❆ ♥❡✉r❛❧ ❛r❝❤✐t❡❝t✉r❡ ❢♦r ❜✐♦❧♦❣✐❝❛❧
●♦❞❢r❡②✲❙♠✐t❤✱ P✳ ✶✾✾✶✳ ❙✐❣♥❛❧✱ ❞❡t❡❝t✐♦♥✱ ❛❝t✐♦♥✳
✽✽✭✶✷✮✱ ✼✵✾✕✼✷✷✳
✶✺
❏♦✉r♥❛❧
❏♦✉r♥❛❧ ♦❢ P❤✐❧♦s♦♣❤②
❍✉❜❡❧✱ ❉✳ ❍✳ ❛♥❞ ❲✐❡s❡❧✱ ❚✳ ◆✳ ✶✾✺✾✳ ❘❡❝❡♣t✐✈❡ ✜❡❧❞s ♦❢ s✐♥❣❧❡ ♥❡✉r♦♥❡s ✐♥ t❤❡
❝❛t str✐❛t❡ ❝♦rt❡①✳ ❏♦✉r♥❛❧ ♦❢ P❤②s✐♦❧♦❣② ✶✹✽✱ ✺✼✹✕✺✾■✳
❑r❛❡♠❡r✱ ❉✳ ✷✵✶✸✳ ❆❣❛✐♥st ✧s♦❢t✧ st❛t✐st✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥✳ P❤✐❧♦s♦♣❤✐❝❛❧
♦❣② ✷✽✭✶✮✱ ✶✸✾✕✶✹✼✳
❑r✐❡❣❡❧✱ ❯✳✿ ✷✵✵✾✳ ❙✉❜❥❡❝t✐✈❡ ❈♦♥s❝✐♦✉s♥❡ss✿
❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❯❙❆✳
Ps②❝❤♦❧✲
❆ ❙❡❧❢✲❘❡♣r❡s❡♥t❛t✐♦♥❛❧ ❚❤❡♦r②✳
▲❡❉♦✉①✱ ❏✳ ✷✵✵✸✳ ❚❤❡ ❡♠♦t✐♦♥❛❧ ❜r❛✐♥✱ ❢❡❛r✱ ❛♥❞ t❤❡ ❛♠②❣❞❛❧❛✳
▼♦❧❧❡❝✉❧❛r ◆❡✉r♦❜✐♦❧♦❣② ✷✸✭✶✮✱ ✼✷✼✕✼✸✽✳
❈❡❧❧✉❧❛r ❛♥❞
▼❡♥❞❡❧♦✈✐❝✐✱ ❆✳ ✷✵✶✸✳ ❘❡❧✐❛❜❧❡ ▼✐sr❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ❚r❛❝❦✐♥❣ ❚❤❡♦r✐❡s ♦❢ ▼❡♥✲
t❛❧ ❘❡♣r❡s❡♥t❛t✐♦♥✳ P❤✐❧♦s♦♣❤✐❝❛❧ ❙t✉❞✐❡s ✶✻✺ ✭✷✮✿✹✷✶✲✹✹✸✳
▼❡♥❞❡❧♦✈✐❝✐✱ ❆✳ ✷✵✶✻✳ ❲❤② ❚r❛❝❦✐♥❣ ❚❤❡♦r✐❡s ❙❤♦✉❧❞ ❆❧❧♦✇ ❢♦r ❈❧❡❛♥ ❈❛s❡s ♦❢
❘❡❧✐❛❜❧❡ ▼✐sr❡♣r❡s❡♥t❛t✐♦♥✳ ❉✐s♣✉t❛t✐♦ ✽ ✭✹✷✮✿✺✼✲✾✷✳
❚❤❡ ❏♦✉r♥❛❧ ♦❢ P❤✐❧♦s♦♣❤② ✽✻✱ ✷✽✶✕✷✾✼✳
❖♥ ❈❧❡❛r ❛♥❞ ❈♦♥❢✉s❡❞ ■❞❡❛s✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t②
▼✐❧❧✐❦❛♥✱ ❘✳ ●✳ ✶✾✽✾✳ ❇✐♦s❡♠❛♥t✐❝s✳
▼✐❧❧✐❦❛♥✱ ❘✳ ●✳✿ ✷✵✵✵✳
Pr❡ss✳
◆❡❛♥❞❡r✱ ❑✳✿ ✷✵✶✼✳ ❆ ▼❛r❦ ♦❢ t❤❡ ▼❡♥t❛❧✿
♠❛♥t✐❝s✳ ❈❛♠❜r✐❞❣❡✱ ▼■❚ Pr❡ss✳
■♥ ❉❡❢❡♥s❡ ♦❢ ■♥❢♦r♠❛t✐♦♥❛❧ ❚❡❧❡♦s❡✲
❖❡❤♠❛♥✱ ❆✳ ❛♥❞ ▼✐♥❡❦❛✱ ❙✳ ✷✵✵✶✳ ❋❡❛rs✱ ♣❤♦❜✐❛s ❛♥❞ ♣r❡♣❛r❡❞♥❡ss✿ ❚♦✇❛r❞ ❛♥
❡✈♦❧✈❡❞ ♠♦❞✉❧❡ ♦❢ ❢❡❛r ❛♥❞ ❢❡❛r ❧❡❛r♥✐♥❣✳ ❈✉rr❡♥t ❇✐♦❧♦❣② ✶✼✭✶✸✮✱ ✶✷✾✕✸✸✳
Pr✐♥③✱ ❏✳✿ ✷✵✵✹✳
✈❡rs✐t② Pr❡ss✳
●✉t ❘❡❛❝t✐♦♥s✿ ❆ P❡r❝❡♣t✉❛❧ ❚❤❡♦r② ♦❢ ❊♠♦t✐♦♥✳ ❖①❢♦r❞ ❯♥✐✲
❈♦♥s❝✐♦✉s♥❡ss ❛♥❞ ♠✐♥❞✳ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✳
❘✉♣❡rt✱ ❘✳ ✶✾✾✾✳ ❚❤❡ ❜❡st t❡st t❤❡♦r② ♦❢ ❡①t❡♥s✐♦♥✿ ❋✐rst ♣r✐♥❝✐♣❧❡✭s✮✳ ▼✐♥❞ ❛♥❞
▲❛♥❣✉❛❣❡ ✶✹✭✸✮✱ ✸✷✶✕✸✺✺✳
❘♦s❡♥t❤❛❧✱ ❉✳ ▼✳✿ ✷✵✵✺✳
❙❝❛r❛♥t✐♥♦✱ ●✳ P✳ ✳ ❆✳ ✷✵✶✵✳ ■♥❢♦r♠❛t✐♦♥ ♣r♦❝❡ss✐♥❣✱ ❝♦♠♣✉t❛t✐♦♥✱ ❛♥❞ ❝♦❣♥✐t✐♦♥✳
❏♦✉r♥❛❧ ♦❢ ❇✐♦❧♦❣✐❝❛❧ P❤②s✐❝s✳
❙❝❤✉❧t❡✱ P✳ ✷✵✶✽✳ P❡r❝❡✐✈✐♥❣ t❤❡ ❲♦r❧❞ ❖✉ts✐❞❡✿ ❍♦✇ t♦ ❙♦❧✈❡ t❤❡ ❉✐st❛❧✐t② Pr♦❜✲
❧❡♠ ❢♦r ■♥❢♦r♠❛t✐♦♥❛❧ ❚❡❧❡♦s❡♠❛♥t✐❝s✳ P❤✐❧♦s♦♣❤✐❝❛❧ ◗✉❛rt❡r❧②✳ ✻✽✭✷✼✶✮✱ ✸✹✾✕
✸✻✾✳
❙❦②r♠s✱ ❇✳✿ ✷✵✶✵❛✳ ❙✐❣♥❛❧s✿
✈❡rs✐t② Pr❡ss✳ ❖①❢♦r❞✳
❊✈♦❧✉t✐♦♥✱ ▲❡❛r♥✐♥❣✱ ✫ ■♥❢♦r♠❛t✐♦♥✳ ❖①❢♦r❞ ❯♥✐✲
❙❦②r♠s✱ ❇✳✿ ✷✵✶✵❜✳ ❙✐❣♥❛❧s✿
✈❡rs✐t② Pr❡ss✱ ❖①❢♦r❞✳
❊✈♦❧✉t✐♦♥✱ ▲❡❛r♥✐♥❣✱ ❛♥❞ ■♥❢♦r♠❛t✐♦♥✳ ❖①❢♦r❞ ❯♥✐✲
✶✻
❙t❡❣♠❛♥♥✱ ❯✳✿ ✷✵✶✸✳ ❆ ♣r✐♠❡r ♦♥ ✐♥❢♦r♠❛t✐♦♥ ❛♥❞ ✐♥✢✉❡♥❝❡ ✐♥ ❛♥✐♠❛❧ ❝♦♠♠✉✲
♥✐❝❛t✐♦♥✳ ✐♥ ❯✳ ❙t❡❣♠❛♥♥ ✭❡❞✳✮✱ ❆♥✐♠❛❧ ❈♦♠♠✉♥✐❝❛t✐♦♥ ❚❤❡♦r②✳ ■♥❢♦r♠❛t✐♦♥
❛♥❞ ■♥✢✉❡♥❝❡✳ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✳
❙t❡❣♠❛♥♥✱ ❯✳ ✷✵✶✺✳ Pr♦s♣❡❝ts ❢♦r ♣r♦❜❛❜✐❧✐st✐❝ t❤❡♦r✐❡s ♦❢ ♥❛t✉r❛❧ ✐♥❢♦r♠❛t✐♦♥✳
❊r❦❡♥♥t♥✐s ✽✵✱ ✽✻✾✕✽✾✸✳
❯s❤❡r✱ ▼✳ ✷✵✵✶✳ ❆ st❛t✐st✐❝❛❧ r❡❢❡r❡♥t✐❛❧ t❤❡♦r② ♦❢ ❝♦♥t❡♥t✿ ❯s✐♥❣ ✐♥❢♦r♠❛t✐♦♥
t❤❡♦r② t♦ ❛❝❝♦✉♥t ❢♦r ♠✐sr❡♣r❡s❡♥t❛t✐♦♥✳ ▼✐♥❞ ❛♥❞ ▲❛♥❣✉❛❣❡ ✶✻✭✸✮✱ ✸✸✶✕✸✸✹✳
✶✼
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