The value of information

Before spending money on a study, an inspection, a pilot programme, or an underwriting survey, a decision-maker should ask: what is that information actually worth? Decision theory gives an exact answer.

Expected Value of Perfect Information (EVPI). Compare (a) the expected payoff if you could act knowing the true state in advance, choosing the best action for each state and averaging over the prior, against (b) the expected payoff of the single best action chosen without that knowledge:

$\text{EVPI} = \underbrace{E_p\!\left[\max_a U(a,s)\right]}_{\text{decide with hindsight, then average}} \;-\; \underbrace{\max_a E_p[U(a,s)]}_{\text{best action, averaged over states}}.$

EVPI is always $\ge 0$ (choosing optimally per state can never do worse than committing to one action in advance) and it is a hard upper bound on what any real, imperfect information source could be worth. If a proposed study costs more than EVPI, it can be rejected without further analysis — a fast, decisive screening tool.

Worked example. A carrier chooses between "Insure" (cost $10 regardless of state) and "Retain" (cost $100 if a large loss occurs, $0 otherwise), with $P(\text{loss})=0.4$. With perfect foresight: insure when a loss is coming (avoiding $100, paying only $10), retain when it is not (paying $0): $E[\max] = 0.4(-10)+0.6(0) = -4$. Without foresight, the best single action is "Insure" always, at $E[U]=0.4(-10)+0.6(-10)=-10$. So $\text{EVPI} = -4-(-10) = 6$: perfect foresight about this one binary event is worth at most $6 per policy — a useful bound before commissioning any predictive model that claims to forecast large losses.

Expected Value of Sample Information (EVSI) — preposterior analysis. Real information sources (a survey, an inspection, a diagnostic test, a pilot cohort) are imperfect: they generate evidence $e$ with a likelihood $L(e\mid s)$, not the true state itself. Before collecting the evidence, we do not know which $e$ we will observe, so we must average over its possible values, weighting by how likely each is under the prior (the marginal $P(e)=\sum_s L(e\mid s)p(s)$), and for each possible $e$ take the best action under the resulting posterior:

$\text{EVSI} = E_e\!\left[\max_a E_{s\mid e}[U(a,s)]\right] - \max_a E_p[U(a,s)].$

This is called preposterior analysis because it evaluates, before seeing any data, how much the (as yet unknown) posterior-informed decisions would be worth on average. EVSI is always $\le$ EVPI (sample information can never be worth more than perfect information) and $\ge 0$. The decision to gather information is then a simple cost-benefit test: gather it if $\text{EVSI} > \text{cost of the study}$.

Practical use. ADE computes both automatically whenever a likelihood matrix is supplied (e.g., "if we commission an underwriting survey with these true/false-positive rates, is it worth the $X it costs?"), turning "should we investigate further?" into the same quantitative decision as the underlying choice itself, rather than a separate judgement call.

Limits. EVPI/EVSI depend on the same prior and payoff structure as the rest of ADE — they answer "is more information worth its cost given what we currently believe," not "should we trust our current beliefs." Sensitivity analysis on the prior (or on $Z$, the credibility weight) should typically accompany any information-value calculation used to justify a real expenditure.