When is another question worth asking?

The prototype glider has crossed the water three times and fallen into it once. The workshop can afford another test. Tomorrow, someone has to choose which glider will carry the island’s deliveries.

There is always another test to imagine. A longer crossing. A windier afternoon. A heavier parcel. Each might teach us something. Each also consumes time and supplies.

Now change one detail: the chosen design will make twelve deliveries. Then change it again: four hundred deliveries across an island network.

The observations have stayed exactly the same. The value of another observation has changed.

Epistemic Physics lets us carry a belief as a ledger of trials, with a direction and an amount behind it. To decide when to stop gathering trials, we need to bring something else to the ledger: what the answer will be used for.

Put a price on a clue

Our workshop has two gliders. The familiar glider succeeds on 60% of deliveries. Treat that rate as known. The prototype has one fixed but unknown success rate, called $p$.

Every successful future delivery earns one token. Testing the prototype costs $c$ tokens per trial and earns no delivery reward. After testing, we choose one glider for all $U$ future deliveries. Here $U$ measures the stake: how many times this one decision will be used.

The workshop begins with a uniform belief over the prototype’s possible rates. A rate of $0.2$ starts on equal footing with a rate of $0.8$. In Beta notation, this is

$$p\sim \mathrm{Beta}(1,1).$$

The symbol $\sim$ means “has this distribution.” The Beta family describes uncertainty about a probability. Its two parameters are weights for success and failure. The initial weights of one are our chosen prior; they are not two observed test flights.

After $s$ successful tests and $f$ failures, the weights become

$$a=1+s,b=1+f.$$

The predictive mean, our probability that the next test succeeds, is

$$\mu =\frac{a}{a+b}.$$

We assume the prototype’s tests and deliveries share the same rate, with independent outcomes once that rate is fixed. Each purchased test is a fresh trial. Watching the same flight’s video twice supplies no second flight.

Before opening the value desk in the experiment, make your own decision. Would you buy another test, commit to the familiar glider, or try the prototype? Then replay the same prototype with a different stake.

Every clue has a price

Run the courier workshop

Replay this prototype

Choose a glider for an island's deliveries. Each successful delivery earns one token. The familiar glider has a known success rate; the prototype's rate is hidden. You can buy up to eight prototype tests before committing to either glider.

Future deliveries 12 · a short trip100 · a season400 · an island network Price of each test 1 token4 tokens10 tokens Familiar success rate 60%80%

Changing the brief replays the same hidden prototype and test sequence.

Familiar glider 60.0% Known success rate

Commit to familiar

Prototype glider 50.0% Predictive mean from your notebook

Commit to prototype

1. Test 1 · unopened
2. Test 2 · unopened
3. Test 3 · unopened
4. Test 4 · unopened
5. Test 5 · unopened
6. Test 6 · unopened
7. Test 7 · unopened
8. Test 8 · unopened

Buy the next test · 1 token Test bill: 0 tokens

No observations yet. Notebook: Beta(1, 1).

Open the value desk: is another test worth buying?

These are expected tokens from now. The 0 tokens already spent are the same for every next choice, so they do not change this comparison.

Best choice now

60.00

One test, then commit

62.33

One test, then adapt

63.59

The model favors: Buy a test. Buying the next test, with later choices kept open, changes expected value by +3.59 tokens.

If the next test succeeds 50.0% predictive probability Then: buy a test

If the next test fails 50.0% predictive probability Then: choose familiar

Same notebook, different stakes

Keep these observations, this test price and the familiar rate fixed.

Future deliveries	Next test's net value	Model's choice
12	-0.60	Choose familiar
100	+3.59	Buy a test
400	+22.34	Buy a test

Exact finite-horizon calculation for this toy model: the prototype starts with a uniform Beta(1,1) prior, its rate stays fixed, and tests are independent conditional on that rate. A test earns no delivery token. After committing, all future deliveries use the chosen glider. The seeded simulation and the model use the same prior.

Open the courier workshop in the playground.

The game allows at most eight tests. This makes the entire decision tree small enough to calculate exactly. Its value desk averages over every possible future test sequence under the model, including sequences you did not happen to see.

A question is valuable when its answer can change a choice

Before any tests, the prototype’s predictive mean is one half. For a hundred deliveries, choosing it immediately earns an expected 50 tokens. The familiar glider earns an expected 60.

“Expected” means a probability-weighted average, not a promise about the next hundred flights.

Suppose a test costs one token, and we buy exactly one before deciding. Its two possible outcomes give us two different choices:

Average those branches and subtract the test price:

$$\underset{\text{choose after seeing the answer}}{\underbrace{\frac{1}{2}(100\times \frac{2}{3})+\frac{1}{2}(60)}}-\underset{\text{test price}}{\underbrace{1}}=62\frac{1}{3}.$$

That is $2\frac{1}{3}$ tokens better than choosing the familiar glider immediately.

The information has value because we can respond differently to its two answers. If the answer were sealed in an envelope until after we had committed, that benefit would disappear.

Now repeat the calculation for twelve deliveries. The familiar glider is worth $7.2$ tokens. A successful test makes the prototype worth $8$; a failure leaves the familiar glider preferable. The single test gives

$$\frac{1}{2}(8)+\frac{1}{2}(7.2)-1=6.6.$$

The same question, the same possible answers, and the same learning rule have become a poor purchase. Its price has not shrunk with the trip.

This is the central idea of value of information: evaluate an observation through the decisions it enables and their consequences. Ronald A. Howard’s “Information Value Theory” develops this decision-based viewpoint.

Leave room to change the plan

The workshop has more than one test available. Buying the next test need not mean committing to a whole batch. After a failure we might stop. After a success we might become curious enough to buy another.

We can put that flexibility into an equation.

Let $p_0$ be the familiar glider’s known rate. At notebook state $(a,b)$, the best expected reward from deciding immediately is

$$R(a,b)=Umax(p_0,\frac{a}{a+b}).$$

The function $max$ selects the larger number. We are choosing whichever glider has the higher expected reward.

Let $J_r(a,b)$ be the best expected reward available when we still have room for $r$ tests, counting any future test costs. With no tests left, we must decide:

$$J_0(a,b)=R(a,b).$$

With at least one test available, the value of buying the next one is

$$C_r(a,b)=-c+\mu J_{r-1}(a+1,b)+(1-\mu )J_{r-1}(a,b+1),\mu =\frac{a}{a+b}.$$

Read it as a little itinerary: pay for a test; follow the success branch with probability $\mu$; follow the failure branch with probability $1-\mu$; make the best remaining choices in whichever state you reach.

Finally, compare asking with stopping:

$$\boxed{J_r(a,b)=max\{R(a,b),C_r(a,b)\}.}$$

Buy the test when . Stop when deciding now is at least as valuable. At a tie, the game chooses to stop.

This method is called backward induction. Start at the end, where the decision is simple, then work backward through the possible answers. It is a standard tool of Bayesian sequential decision theory. The EP ledger supplies the state and its predictive branches; the specified rewards and prices tell us which branches are worth exploring.

With the default hundred deliveries, 60% familiar rate, and one-token test price, the workshop’s values before observing anything are:

Flexibility is worth something. The last row can stop early on unpromising paths and continue on useful ones.

The test bill already paid does not appear in the comparison. It is the same subtraction whichever next action we choose. Continuing cannot recover those tokens merely by making the bill larger.

The first question can open a door that it cannot cross

Try the workshop with 400 deliveries, a one-token test, and a familiar glider that succeeds 80% of the time. Start from the empty notebook.

One successful test moves the prototype mean from $1/2$ to $2/3$. It still trails the familiar glider. One failed test moves it to $1/3$, also behind. If we insist on deciding immediately after that one test, we will choose the familiar glider either way.

Its value is therefore $320-1=319$ tokens, compared with 320 for deciding now.

Yet the adaptive eight-test calculation gives about 321.85 tokens. Starting is worth about 1.85 tokens, even though a plan that sees only one test ahead would refuse to begin.

Why? Several successes can move the mean far enough. Four consecutive successes take it to $5/6$, above $0.8$. We do not have to pay for all four in advance. In this example, an immediate failure makes the model stop; an immediate success makes it continue.

The first test buys access to a more informative second decision. That option has value even when the first answer cannot change the final choice by itself.

This is a useful possibility to remember when a problem has a sequence of clues. A first measurement may locate the right region of a map, identify which instrument to use, or reveal which question to ask next. Its worth includes the inquiry it makes possible.

The same mean can hide very different prospects for learning

The predictive mean is only part of the notebook.

Compare two starting beliefs about the prototype: $\mathrm{Beta}(1,1)$ and $\mathrm{Beta}(25,25)$. Both have mean $1/2$. The first spreads its weight uniformly over possible rates. The second concentrates it near one half.

Their different malleability, or ease of changing the belief, is visible in the update. If the next result is $z=1$ for success or $z=0$ for failure, then

$${\mu }^{+}=\mu +\frac{z-\mu }{a+b+1}.$$

The superscript $+$ means “after the new trial.” A larger starting trial amount $a+b$ makes one extra result move the mean less.

With a 60% familiar glider and only eight tests available, even eight successes cannot make the second notebook favor the prototype. Under this model, no possible answer sequence can change the final choice. Paying a positive price to test has no decision benefit.

The heavier notebook could represent earlier independent trials, a concentrated prior, or a mixture of the two. Its origin matters. Inflating that amount by counting copied reports as fresh observations could make a learner prematurely stop exploring.

We can now see why “ask until the mean settles down” misses something. A stopping decision depends on the distribution of possible answers, their ability to change actions, the price of asking, the number of opportunities left, and the consequences of the final choice.

What happened to the repeated-checking guarantee?

When the numbers refuse to fit explores a different part of the problem: checking evidence repeatedly while controlling false alarms. Its evidence ledger can cross a chosen threshold with a bounded probability under a specified reference model.

Keep both instruments on the desk:

A fixed evidence threshold has no input for “twelve deliveries” versus “four hundred deliveries.” Our decision calculation does. Conversely, an expected-reward optimum does not automatically give a chosen bound on false alarms across all possible true rates.

An inquiry can use both. One instrument governs a statistical claim; another helps allocate the inquiry’s resources. They can also disagree about what to do next, because they answer different questions.

Let the questions have a future

The workshop ends once a glider is chosen. A real workshop will probably design another glider next month.

That changes the stakes again. Today’s test might improve this design decision and teach us how promising future prototypes tend to be. A question with little immediate value could help many later inquiries. To price that benefit, the model must represent those future inquiries too.

There is a particularly interesting trap here: a workshop that concludes “new designs are never worth testing” can stop collecting the observations that would challenge that conclusion. Exploring a new family of possibilities requires a model of what that exploration might unlock.

Our eight-test game solves its stated, fixed-rate problem exactly. Extending it to changing conditions, uncertain familiar rates, different kinds of tests, or learning across many projects means enlarging the state and following the new branches. The general disclaimer covers the wider limits of these exploratory models.

For another route into this idea, visit Move to ask a better question: sometimes the next decision is where to point the sensor. Here it was whether to buy the next measurement at all. In both cases, a belief becomes useful through the actions it makes possible.