Signal · Chapter 02 · Reasoning under uncertainty

Question the signal.

A robot helper thinks it knows where the beacon is. You get the final say.

One hidden rescue beacon, three possible stations, and a sensor that isn't perfect. Scan for clues, make your own guess before the AI answers, and decide when you've learned enough to call in the rescue.

Mission briefing

A hiking group got caught out on the mountain when a storm rolled in, and their emergency beacon is the only way to find them. It's transmitting from exactly one of three ranger stations — Aster, Boreal, or Cygnus — but nobody knows which. You're running the rescue control room, with a robot helper that can scan any station for a signal. The trouble is, its sensor isn't perfect: sometimes it “hears” a beacon that isn't really there, and sometimes it misses one that is. Every scan also burns time (and points), so you can't just check everywhere. Gather clues, trust your own judgment, and call in the rescue before you run out of scans.

How to play

Pick a stationTap Aster, Boreal, or Cygnus below — that's the spot you want your robot to check next.

Send a scanHit Scan. It costs a few points, but you'll get a real clue back: a ping, or silence.

Make your guessBefore the AI does its math, slide to how sure YOU are the beacon's there — then reveal and compare.

Scan again, or call it inNot sure yet? Scan another station. Confident? Hit Rescue — that's your final answer.

Tip: the “AI's advice” panel on the right will always suggest a move — but the decision is yours. Try disagreeing with it once and see what happens!

Investigation 01

The false alarm

4 scans left

Most rescuers would guess the beacon isn't at Aster — it starts out the least likely spot. But scan it anyway, and watch: the detector can still "hear" a ping there even when nothing's there. Make your own guess first, then see what the math actually says.

True signal detected 80.0%False alarm rate 20.0%Cost per scan 8 pts

ONE SIGNAL · THREE POSSIBLE WORLDS

Belief distributionUncertainty: 1.369 bits

Aster 10.0%Boreal 45.0%Cygnus 45.0%

Ready when you are — tap a station below, then hit Scan to send your first clue.

You can only rescue once — that's your final answer: +100 points if you're right, +0 if you're wrong. Every new scan costs a few points too, so the fastest, smartest detective wins. Replaying an old clue is free, but it won't teach the AI anything it doesn't already know.

Evidence trail

Every reading will leave a trace here. Select one to inspect its calculation.

The AI’s advice

You decide

Expected-score planner

Scan Boreal.

That scan has the highest expected future score: 64.4 points, versus 45.0 for rescuing now. These values exclude points already spent.

Curiosity asks a different question

Which scan reduces uncertainty most?

Aster

0.105 bits

Boreal ★

0.275 bits

Cygnus

0.275 bits

Expected information gain averages both possible readings. Reducing uncertainty and maximising reward are different objectives.

Curious exactly how the AI thinks? Open any panel below to see the real math behind it — you don't need any of it to play.

01 · Bayes, with your actual numbers

Scan and reveal a reading to see its full calculation. Bayes’ theorem follows by equating P(H,E) = P(E|H)P(H) = P(H|E)P(E), then dividing by P(E) > 0.

02 · Why an echo teaches us nothing

For a repeated copy E of the same packet, P(E,E|H) = P(E|H), because E ∩ E = E. Therefore P(H|E,E) = P(H|E).

Multiplying by the likelihood again assumes a new, conditionally independent measurement. Try “Replay last packet” to see that mistake. The careful AI deduplicates packet IDs.

03 · Can evidence make us less certain?

Entropy H(p) = −Σ p log₂ p measures uncertainty. A surprising outcome can increase it. But expected information gain is H(before) − Σ P(reading)H(after reading).

This equals a weighted average of KL divergences between posterior and prior. Each is nonnegative (Gibbs’ inequality), so the average is nonnegative. The theorem is about the average, not every reading.

04 · When should the AI stop?

The planner evaluates every scan and both outcomes for up to 4 remaining scans. It may rescue at any step.

V(b, k) = max [100 max(b),
max scan (−cost + Σ P(e) V(b′, k−1))]

At k = 0, rescue the most likely station. Working backward gives the best expected score under these stated probabilities and rewards. This is Bellman’s principle of optimality: after any observation, the remaining plan must itself be optimal for the updated belief and budget.

Belief states and decisions · CMU research

The idea to take with you

A positive signal is not an 80% certainty.

Real AI systems make decisions from incomplete evidence. Ask where the prior came from, whether observations are independent, and which objective the system is actually optimising.