≻, ≺

preference relations

2

~

equivalence relation

2

Z

set of outcomes, rewards

3

z

outcome, generic element of Z

3

z⁰

best outcome

3

z₀

worst outcome

3

Z_i1i2...is0k

outcome from actions when the state of nature is

12

Z_i1i2...iSk

outcome from actions when the state of nature is

12

Θ

set of states of nature, parameter space

3

θ

event, state of nature, parameter, generic element of Θ, scalar

2

θ

state of nature, parameter, generic element of Θ, vector

7

generic state of nature upon taking stopping action at stage s

12

generic state of nature at last stage S upon taking action

12

π

subjective probability

2

π_θ

price, relative to stake S, for a bet on θ

2

π_θ1|θ2

price of a bet on θ₁ called off if θ₂ does not occur

2

current probability assessment of θ

2

future probability assessment of θ

2

π^M

least favorable prior

7

S_θ

stake associated with event θ

2

g_θ

net gains if event θ occurs

2

A

action space

3

a

action, generic element of A

3

a(θ)

VNM action, function from states to outcomes

3

expected value of the rewards for a lottery a

4

z*(a)

certainty equivalent of lottery a

4

λ(z)

Arrow–Pratt (local) measure of risk aversion at z

4

P

set of probability functions on Z

6

a

AA action, horse lottery, list of VNM lotteries a(θ) in P for θ ∊ Θ

6

a(θ, z)

probability that lottery a(θ) assigns to outcome z

6

a_S,θ

action of betting stake S on event θ

2

a_θ

action that would maximize decision maker’s utility if θ was known

13

a

action, vector

7

generic action, indexed by i_s, at stage s of a multistage decision problem

12

stopping action at stage s of a multistage decision problem

12

a^M

minimax action

7

a*

Bayes action

3

p(z)

probability of outcome z

3

p

lottery or gamble, probability distribution over Z

3

χ_z

degenerate action with mass 1 at reward z

3

u

utility

3

u(z)

utility of outcome z

3

u(a(θ))

utility of outcome a(θ)

3

u_θ(z)

state-dependent utility of outcome z

6

S(q)

expected score of the forecast probability q

10

s(θ, q)

scoring rule for the distribution q and event θ

10

RP(a)

risk premium associated with action a

4

X

sample space

7

x

random variable or observed outcome

7

xⁿ

random sample (x₁, ..., x_n)

14

x

multivariate random sample

7

random variable with possible values x_i1, ..., x_iJs observed at stage s, upon taking continuation action

12

f (x|θ)

probability density function of x or likelihood function

7

m(x)

marginal density function of x

7

F(x|θ)

distribution function of x

7

π(θ)

prior probability of θ; may indicate a density if θ is continuous

7

π(θ|x), π_x

posterior density of θ given x

7

E_x[g(x)]

expectation of the function g(x) with respect to m(x)

7

E_x|θ[g(x)], E[g(x)|θ]

expectation of the function g(x) with respect to f (x|θ)

7

u_π(a)

expected utility of action a, using prior π

3

u_π(d)

expected utility of the sequential decision procedure d

15

δ

decision rule, function with domain X and range A

7

δ

multivariate decision rule

7

δ^M

minimax decision rule

7

δ*

Bayes decision rule

7

δ^R(x, ·)

randomized decision rule for a given x

7

δ_n

terminal decision rule after n observations xⁿ

15

δ

sequence of decision rules δ₁(x¹), δ₂(x²), ...

15

ζ_n

stopping rule after n observations xⁿ

15

ζ

sequence of stopping rules ζ₀, ζ₁(x¹), ...

15

d

sequential decision rule: pair (δ, ζ)

15

L(θ, a)

loss function (in regret form)

7

L_u(θ, a)

loss function as the negative of the utility function

7

L(θ, a, n)

loss function for n observations

14

u(θ, a, n)

utility function for n observations

14

L(θ, δ^R(x))

loss function for randomized decision rule δ^R

7

L_π(a)

prior expected loss for action a

7

L_πx(a)

posterior expected loss for action a

7

U(θ, d)

average utility of the sequential procedure d for given θ

15

R(θ, δ)

risk function of decision rule δ

7

V(π)

maximum expected utility with respect to prior π

13

W₀(π_xn, n)

posterior expected utility, at time n, of colleting 0 additional observations

15

W_k–n(π_xn, n)

posterior expected utility, at time n, of continuing for at most an additional k – n steps

15

r(π, δ)

Bayes risk associated with prior π and decision δ

7

r(π, n)

Bayes risk adopting the optimal terminal decision with n observations

14

D

class of decision rules

7

D^R

class of all (including randomized) decision rules

7

ɛ^∞

perfect experiment

13

ɛ

generic statistical experiment

13

ɛ₁₂

statistical experiment consisting of observing both variables x₁ and x₂

13

ɛ⁽n)

statistical experiment consisting of experiments ɛ₁, ɛ₂, ..., ɛ_n where ɛ₂, ..., ɛ_n are conditionally i.i.d. repetitions of ɛ₁

13

V^θ(^ɛ∞)

conditional value of perfect information for a given θ

13

V(ɛ^∞)

expected value of perfect information

13

V_x(ɛ)

observed value of information for a given x in experiment ɛ

13

V(ɛ)

expected value of information in the experiment ɛ

13

V(ɛ₁₂

expected value of information in the experiment ɛ₁₂

13

V(ɛ₂|ɛ₁)

expected information of x₂ conditional on observed x₁

13

I_x(ɛ)

observed (Lindley) information provided by observing x in experiment ɛ

13

I(ɛ)

expected (Lindley) information provided by the experiment ɛ

13

I_x(ɛ₁₂)

expected (Lindley) information provided by the experiment ɛ₁₂

13

I(ɛ₂|ɛ₁)

expected (Lindley) information provided by E₂ conditional on E₁

13

c

cost per observation

14

C(n)

cost function

14

Bayes rule based on n observations

14

U_π (n)

expected utility of making n observations and adopting the optimal terminal decision

14

n*

Bayesian optimal sample size

14

n^M

Minimax optimal sample size

14

η

nuisance parameter

7

Φ(.)

cumulative distribution function of the N(0,1)

8

M

class of parametric models in a decision problem

11

M

generic model within M

11

H_s

history set at stage s of a multistage decision problem

12