Must probabilities be completely unknown?

No. The approach is designed to use partial knowledge such as intervals and relative likelihoods.

Does it select one true distribution?

No. The feasible set preserves multiple possibilities unless the constraints happen to determine one distribution.

Does maxEmin guarantee the stated payoff on every occasion?

No. It compares lower bounds on expected payoffs under the allowed distributions, not guaranteed realized outcomes.

AI Glossary / Graphs and symbolic AI / Intermediate

Linear partial information

Definition

Linear partial information is a way to represent incomplete knowledge through linear restrictions, especially restrictions on probabilities or normalized weights. Instead of claiming one precisely known probability distribution, a decision model retains all distributions satisfying the stated information and evaluates choices against that set.

What it means

An analyst may know that an event is at least as likely as another event, or that a probability lies between two bounds, without knowing its exact value. Such statements can still guide a decision. For example, pA >= pB is useful information even though it does not identify either probability.
The term belongs to decision theory and uncertainty modeling. This entry focuses on its probability-based form. It is not a synonym for missing spreadsheet cells, partial access to an AI model, or linear regression. Other treatments also use constraints on normalized weights, so the modeled quantities should always be named.

How it works

List the possible states and introduce their probabilities. Require nonnegative probabilities summing to one, then add the available linear restrictions. Together these conditions define a feasible set of distributions.
For each action, calculate expected payoff as a function of those probabilities. One associated decision principle, maxEmin, chooses the action with the largest minimum expected payoff across the allowed distributions. This is a choice criterion, not a discovery of the true distribution. Other questions include whether one action dominates another throughout the feasible set. In larger finite-state models, linear optimization can find bounds when the payoffs are fixed.

Three practical examples

A product-planning team can compare options when demand probabilities are known only within defensible bounds.
An operations analyst can represent a judgment that one disruption scenario is no more likely than another.
A decision-support system can show whether an action remains preferable across multiple probability distributions consistent with the evidence.

Think of it this way

Think of a map marking all locations where a missing item could be, rather than placing a pin at an unsupported exact location. Each additional reliable clue narrows the region. The analogy has limits: here the points represent complete probability distributions, and decision quality depends on a specified payoff model and decision rule.

Worked example

Consider an invented planning exercise with two states: high demand H and low demand L. Let p be the probability of H, with 0.3 <= p <= 0.6. Then the probability of L is 1 - p. No single value within that interval is assumed correct.
Action A earns 100 planning units in H and 20 in L. Its expected payoff is 100p + 20(1 - p) = 20 + 80p. At p = 0.3 this is 44; at p = 0.6 it is 68. Because it increases with p, its minimum over the allowed interval is 44.
Action B earns 55 units in either state, so its expected payoff is 55 for every allowed p. Under maxEmin, B is selected because its minimum expected payoff, 55, exceeds A's 44.
This does not mean B has the greater expected payoff at every allowed distribution. A overtakes B when 20 + 80p > 55, or p > 0.4375. At p = 0.6, A's 68 exceeds B's 55. Nor does A's lower expected bound of 44 prevent an actual low-demand outcome of 20. The guarantee concerns an expectation conditional on the model, not each realized outcome.
If reliable new evidence raises the lower bound to 0.5, A's minimum becomes 60 and maxEmin selects A. Changing an evidence-based restriction can therefore change the preferred action without inventing an exact probability.

Illustrative case study

In a fictional project, Rowan Planning prepares a choice between two service-capacity options. The first presentation quietly replaces an uncertain demand interval with its midpoint. That produces a single attractive expected return but hides how the recommendation changes across plausible probabilities.
The team revises the presentation to show the interval, the payoff assumptions, and the result under its chosen cautious criterion. Managers can now distinguish disagreement about evidence from disagreement about decision preferences. This fictional case illustrates transparency in a model; it does not claim that the method improved revenue or identified a uniquely correct choice.

Common misconceptions and mistakes

Partial information is not absence of information. Bounds and orderings can exclude distributions and materially affect a decision. Conversely, a narrow feasible set is not automatically well supported: unjustified restrictions can create misleading confidence.
Do not confuse the minimum expected payoff with the worst individual outcome. Also, maxEmin is not the only possible attitude toward uncertainty, and choosing it should be explicit. Inconsistent restrictions can leave no feasible distribution at all. Finally, the word linear refers to the form of the restrictions and calculations; it does not mean the real world responds linearly in every respect.

Business use

Use the approach when stakeholders can justify some probability information but cannot responsibly supply precise probabilities. Record the origin of each restriction: observed data, a stated expert judgment, or an assumption made for exploration.
Check whether all restrictions can hold together. Present sensitivity results when a small change in a bound changes the decision. Separate uncertainty in probabilities from uncertainty in payoffs, since fixing uncertain payoffs can hide another important source of ambiguity. For practical adoption, explain the selected decision criterion in everyday terms and make its implications visible with a small example before presenting a large optimization result.

Check your understanding

Why does the worked example choose B under maxEmin even though A is better at p = 0.6?

Answer: The rule compares the minimum expected payoff of each action across the entire permitted interval. A's minimum is 44 and B's is 55; superiority at one permitted distribution does not determine this criterion.

Related terms

Expected utility
Linear constraints
Maximin
Sensitivity analysis
Probability distribution
Imprecise probability
robust decision-making
dominance

Learn next

Repeat the example with a lower probability bound of 0.4, then 0.45. Calculate A's minimum in each case and compare it with 55. Next, distinguish a statement about which action has the best worst-case expectation from a statement about dominance at every feasible distribution.

Sources

Edward Kofler, Linear partial information with applications, publisher record and indexed abstract: https://doi.org/10.1016/S0165-0114(99)00088-3 Peter Zweifel, Bridging the gap between risk and uncertainty in insurance: https://link.springer.com/article/10.1057/s41288-021-00220-y

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