While most patients learn of hospice through their clinician, such referrals often come too late to optimize hospice benefits [63]. We know little about patients’ willingness to consider hospice prior to receiving this referral and the majority of hospice research has been conducted after hospice referral or enrollment. Critics of the late referrals often site clinician difficulties in prognosing patient life span or accepting their patient is dying, or the belief that the patient is not ready for hospice as key barriers to earlier referral and hospice enrollment [94,95]. However, patients may be willing to consider such alternatives earlier than clinicians assume [96].

• Decision theory is concerned with the reasoning underlying an
  agent’s choices, whether this is a mundane choice between taking
  the bus or getting a taxi, or a more far-reaching choice about whether
  to pursue a demanding political career.
• Since states
  may be probabilistically dependent on acts, an agent can be
  represented as maximising the value of Jeffrey’s desirability
  function while violating the STP.
• Slovic
  & Tversky (1974), for example, report that 17 out of 29 (59%) of
  subjects in their study exhibit Allais preferences in their
  investigation of the Common Consequence problem.
• David Lewis (1988, 1996) famously employed EU theory to argue
  against anti-Humeanism, the position that we are sometimes
  moved entirely by our beliefs about what would be good, rather than by
  our desires as the Humean claims.
• In novel conditions, the case-based estimate represents a kind of half-educated guess.
• We first describe the
  prospects or decision set-up and the resultant expected utility rule,
  before turning to the pertinent rationality constraints on preferences
  and the corresponding theorem.

Theorem 2 (von Neumann-Morgenstern)

Let \(\bO\) be a finite set of outcomes, \(\bL\) a set of
corresponding lotteries that is closed under probability mixture and
\(\preceq\) a weak preference relation on \(\bL\). Then \(\preceq\)
satisfies axioms 1–4 if and only if there exists a function
\(u\), from \(\bO\) into the set of real numbers, that is unique up to
positive linear transformation, and relative to which \(\preceq\) can
be represented as maximising expected utility. Those who are less inclined towards behaviourism might, however, not
find this lack of uniqueness in Bolker’s theorem to be a
problem. James Joyce (1999), for instance, thinks that Jeffrey’s
theory gets things exactly right in this regard, since one should not
expect that reasonable conditions imposed on a person’s
preferences would suffice to determine a unique probability function
representing the person’s beliefs.

2 Decision Theories

When the person goes inside, the situation (context) changes and the outdoor temperature may then have a very small importance for the comfort level. Similarly, both a very cold and a very warm outdoor temperature might have a low utility for a person’s comfort level but the level of utility can be modified by adding or removing clothes. A good decision is a good choice given the alternatives at the time when a commitment and resources are pledged. The chronological separation, between commitment and outcomes, permits uncertainty to intervene, aleatory unpredictable conditions that can generate an unintended outcome. The
interested reader is referred to the recent general treatment given by
Galaabaatar & Karni (2013), who relate their results to important
earlier work by the likes of Bewley (1986), Seidenfeld et al.
(1995), Ok et al. (2012), and Nau (2006), among others.

• There are many false starts and punctuated by what Kuhn (2012) calls paradigm shifts.
• In any case, it turns out that when a person’s
  preferences satisfy Savage’s axioms, we can read off her
  preferences a comparative belief relation that can be represented by a
  (unique) probability function.
• In some special cases, however, like the economic theory of perfect competition, these complexities are absent.
• A number of experimental studies in the 1960s and 1970s subsequently
  confirmed the robustness of the effects uncovered by Allais.
• See MacCrimmon &
  Larson (1979) for a helpful summary of this and other early work and
  further data of their own.

In most ordinary choice situations, the objects of choice, over which
we must have or form preferences, are not like this. Rather,
decision-makers must consult their own probabilistic beliefs
about whether one outcome or another will result from a specified
option. Decisions in such circumstances are often described as
“choices under uncertainty” (Knight 1921). For example,
consider the predicament of a mountaineer deciding whether or not to
attempt a dangerous summit ascent, where the key factor for her is the
weather. If she is lucky, she may have access to comprehensive weather
statistics for the region.

More generally, we can
catalogue theories in terms of the kinds of properties (whether
intrinsic or in some sense relational) that distinguish acts/outcomes
and also in terms of the nature of the ranking of acts/outcomes that
they yield (whether transitive, complete, continuous and so on). So under what conditions can a preference relation \(\preceq\) on the
set \(\Omega\) be represented as maximising desirability? Some of the
required conditions on preference should be familiar by now and will
not be discussed further. In particular, \(\preceq\) has to be
transitive, complete and continuous (recall our discussion in
Section 2.3
of vNM’s Continuity preference axiom). Another important thing to notice about Jeffrey’s way of
calculating desirability, is that it does not assume probabilistic
independence between the alternative that is being evaluated, \(p\),
and the possible ways, the \(p_i\)s, that the alternative may be
realised.

Decision Theory: An Introduction

Therefore,
in that case many people do think that the slight extra risk of $0 is
worth the chance of a better prize. Despite all research efforts on explainability of AI systems since decades, the emergence of a new name (XAI) for the domain as recently as 2016 is an indication that XAI is still quite immature. Current XAI research notably on outcome explanation also seems to ignore the wealth of knowledge accumulated also by closely related domains for decades. This paper proposes extending the traditional MCDM concepts of importance and utility from the linear models towards the non-linear models produced by ML techniques. Figure 3 shows what non-linear classification models could look like for an ‘AND’/‘not AND’ classifier, with two inputs x, y and two outputs.

The first output corresponds to the class ‘not AND’ and the second output corresponds to the class ‘AND’. 3 that modifying x will not affect the result z much and the CI of x is indeed only 0.07 for both classes, whereas the CU of x is 0.50 for both classes, which is expected. However, modifying y will modify the result z much more, which is also reflected by a CI of y of 0.50 for both classes. The CU of y is 0.93 for the class ‘not AND’ and 0.07 for the class ‘AND’, which is also expected. It seems like the term Explainable AI (XAI) dates back to a presentation by David Gunning in 2016 [13] and much recent work tends not to look at or cite research papers that are older than so.

Decision Theory Meets Explainable AI

Some of these branches lead to further choice points, often
after the resolution of some uncertainty due to new evidence. When the above holds, we say that there is an expected utility
function that represents the agent’s preferences; in other
words, the agent can be represented as maximising expected
utility. Then
there is an ordinal utility function that represents \(\preceq\) just
in case \(\preceq\) is complete and transitive. In our continuing investigation of rational preferences over
prospects, the numerical representation (or
measurement) of preference orderings will become important. The two main types of utility function that will play
a role are the ordinal utility function and the more
information-rich interval-valued (or cardinal)
utility function.

It would have been
better were he able to sail unconstrained and continue on home to
Ithaca. This sequence could have been achieved if Ulysses were
continuously rational over the extended time period; say, if
at all times he were to act as an EU maximiser, and change his beliefs
and desires only in accordance with Bayesian norms (variants of
standard conditionalisation). On this reading, sequential
decision models introduce considerations of rationality-over-time. From the perspective of decision-making, unawareness of unawareness is
not of much interest. After all, if one is not even aware of the
possibility that one is unaware of some state or outcome, then that
unawareness cannot play any role in one’s reasoning about what
to do. However, decision-theoretic models have been proposed for how a
rational person responds to growth in awareness (that is
meant to apply even to people who previously were unaware of their
unawareness).

3 The von Neumann and Morgenstern (vNM) representation theorem

Then since \(p\cup q\) is compatible
with the truth of either the more or the less desirable of the two,
\(p\cup q\)’s desirability should fall strictly between that of
\(p\) and that of \(q\). However, if \(p\) and \(q\) are equally
desirable, then \(p\cup q\) should be as desirable as each of the
two. As noted, a special case is when the content of
\(p\) is such that it is recognisably something the agent can choose
to make true, i.e., an act.

Examples of decision theory in a Sentence

Instead of adding specific belief-postulates to Jeffrey’s
theory, as Joyce suggests, one can get the same uniqueness result by
enriching the set of prospects. The above result may seem remarkable; in particular, the fact that a
person’s preferences can determine a unique probability function
that represents her beliefs. On a closer look, however, it is evident
that some of our beliefs can be determined by examining our
preferences. Suppose you are offered a choice between two lotteries,
one that results in you winning a nice prize if a coin comes up heads
but getting nothing if the coin comes up tails, another that results
in you winning the same prize if the coin comes up tails but getting
nothing if the coin comes up heads.

The aim is to characterise the attitudes of agents who
are practically rational, and various (static and sequential)
arguments are typically made to show that certain practical
catastrophes befall agents who do not satisfy standard
decision-theoretic constraints. In particular, normative
decision theory requires that agents’ degrees of beliefs satisfy
the probability axioms and that they respond to new information by
conditionalisation. Therefore, decision https://1investing.in/ theory has great implications
for debates in epistemology and philosophy of science; that is, for
theories of epistemic rationality. Defenders of resolute choice may have in mind a different
interpretation of sequential decision models, whereby future
“choice points” are not really points at which an agent is
free to choose according to her preferences at the time. If so, this
would amount to a subtle shift in the question or problem of interest.

The Standard Model: Subjective Expected Utility

But unlike Buchak,
they suggest that what explains Allais’ preferences is that the
value of wining nothing from a chosen lottery partly depends on what
would have happened had one chosen differently. To accommodate this,
they extend the Boolean algebra in Jeffrey’s decision theory to
counterfactual propositions, and show that Jeffrey’s
extended theory can represent the value-dependencies one often finds
between counterfactual and actual outcomes. In particular, their
theory can capture the intuition that the (un)desirability of winning
nothing partly depends on whether or not one was guaranteed to win
something had one chosen differently.