Bayes' theorem and applications: Visual walkthrough

This page is the visual companion to the parent note. It leans on one idea from 1.3.01-Conditional-probability-and-independence (what "given" means) and one from 1.3.02-Law-of-total-probability (splitting a total into pieces). We rebuild both from zero as we need them.

Step 1 — Draw the whole world as one square

WHAT. We start with everything that could possibly happen. Draw a square. Its total area is exactly $1$ , because the probability that something happens is certain, and certain means $1$ .

WHY a square of area 1? Probability is just "what fraction of the possibilities is this?" A fraction is an area compared to the whole. If the whole square is $1$ , then any region inside is automatically a probability — you never have to divide by anything to read it off. Area = probability. That is the whole trick.

PICTURE. The full cream square is "the world". We will keep slicing it.

Figure — Bayes' theorem and applications

Step 2 — Cut the square by $A$ : the prior

WHAT. Slice the square with a vertical line. Everything left of the line is $A$ (the fact is true); everything right is $\neg A$ , read "not $A$ " (the fact is false).

WHY vertical, why now? We split by $A$ first because $A$ is our belief before any evidence — this is the prior. Its width is how much of the world $A$ takes up on its own.

area of left strip P (A) + area of right strip P (\neg A) = 1

$P (A)$ — width of the left strip. The two strips fill the whole square, so their areas must add to $1$ .
$P (\neg A)$ — width of the right strip. Nothing is left over: every outcome is either $A$ or not.

PICTURE. Teal strip = $A$ , plum strip = $\neg A$ . In the medical story $A$ is thin (disease is rare, $P (A) = 0.01$ ) — remember that thinness, it is the whole reason the final answer surprises people.

Step 3 — Inside each strip, how often does the evidence fire? The likelihood

WHAT. Now bring in the evidence $B$ . Inside the $A$ strip, shade the fraction where $B$ also happens. Do the same inside the $\neg A$ strip. These shaded fractions are the likelihoods.

WHY inside each strip separately? Because the evidence behaves differently depending on whether $A$ is true. A test fires often on sick people and rarely on healthy people. "How often $B$ fires given we are inside strip $A$ " is exactly what $P (B ∣ A)$ means — the fraction of the $A$ strip that is shaded.

P (B ∣ A) = \frac{shaded area inside A}{whole area of A} ⟹ shaded area inside A P (A \cap B) = P (B ∣ A) \cdot P (A)

$P (B ∣ A)$ — the fraction of the $A$ strip that is shaded. A ratio, so it lives between $0$ and $1$ .
$P (A)$ — the width of the $A$ strip (from Step 2).
$P (A \cap B)$ — " $A$ and $B$ together", read as the joint. Multiply fraction × width to get the actual area of the shaded block. This is the key move: area of a sub-block = its fraction of the strip, times the strip's size.

PICTURE. The dark teal block (top of the $A$ strip) is $P (A \cap B)$ . The dark plum block (top of the $\neg A$ strip) is $P (\neg A \cap B)$ — the false alarms.

Step 4 — Collect all the evidence: the law of total probability

WHAT. The evidence $B$ appears in two places: the shaded block inside $A$ , and the shaded block inside $\neg A$ . Add them to get the total shaded area — the total chance of seeing $B$ at all.

WHY add these two and nothing else? Because $A$ and $\neg A$ cover the whole square with no overlap. Any point where $B$ happens is either in $A$ or in $\neg A$ — there is no third place. This is the law of total probability: split a total into non-overlapping cases and sum.

P (B) = shaded in A (true positives) P (B ∣ A) P (A) + shaded in \neg A (false positives) P (B ∣ \neg A) P (\neg A)

First term — the teal block from Step 3: the times $B$ fires and $A$ is genuinely true.
Second term — the plum block: the times $B$ fires while $A$ is false. These are the false alarms.
$P (B)$ — the two blocks stacked together: the evidence, the total area shaded anywhere.

PICTURE. We slide both shaded blocks side by side into one bar. That bar's length is $P (B)$ .

Step 5 — The reversal: from $P (B ∣ A)$ to $P (A ∣ B)$

WHAT. Here is the question Bayes actually answers. We saw the evidence — we are now standing inside the shaded region only. Of that shaded region, what fraction is the teal part (truly $A$ )? That fraction is the posterior, $P (A ∣ B)$ .

WHY is this the reversal? In Steps 3–4 we asked "given the strip, how much is shaded?" Now we ask the opposite: "given the shaded part, how much is the strip?" We flipped what is known and what is asked. Same two blocks — we just divide by the other total.

P (A ∣ B) = \frac{teal shaded block}{whole shaded bar} = \frac{P ( A \cap B )}{P ( B )} = \frac{P ( B ∣ A ) P ( A ) teal block}{whole shaded bar P ( B ∣ A ) P ( A ) + P ( B ∣ \neg A ) P ( \neg A )}

Numerator — the teal true-positive block (Step 3).
Denominator — the full shaded bar (Step 4): teal plus plum.
The ratio — what fraction of "everyone who saw the evidence" is genuinely in $A$ .

This is Bayes' theorem. We never memorised it; we counted areas and divided.

PICTURE. We discard the unshaded square entirely (grey it out) and keep only the shaded bar. Bayes = teal length ÷ total bar length.

Step 6 — Watch the numbers: why a 95%-accurate test still says 16%

WHAT. Plug the medical story into the picture. Disease is rare: $P (A) = 0.01$ . Test fires on $95%$ of sick people ( $P (B ∣ A) = 0.95$ ) and $5%$ of healthy people ( $P (B ∣ \neg A) = 0.05$ ).

WHY does this feel wrong? Because the teal strip is tiny to start with. Even at $95%$ hit-rate, $0.95 \times 0.01$ is a sliver. But $5%$ of the huge healthy population, $0.05 \times 0.99$ , is a bigger sliver. The plum false-alarm block out-areas the teal true-positive block.

P (A ∣ B) = \frac{0.95 \times 0.01}{0.95 \times 0.01 + 0.05 \times 0.99} = \frac{0.0095}{0.0095 + 0.0495} = \frac{0.0095}{0.0590} \approx 0.161

$0.0095$ — teal block, true positives.
$0.0495$ — plum block, false positives — five times larger.
$0.161$ — teal share of the total: only about $16%$ .

PICTURE. Draw the two blocks to scale. The plum false-alarm block visibly dwarfs the teal one — you see why the answer is $16%$ , not $95%$ .

Step 7 — Edge case: evidence that can never happen

WHAT. What if $P (B) = 0$ — the shaded bar has zero length? Then Bayes divides by zero and is undefined.

WHY is that the right behaviour? $P (B) = 0$ means the evidence never occurs. Asking "given something that never happens…" is a question with no meaning — there is nothing to zoom into. The formula refusing to answer is correct, not a bug. This is why the parent's definition demands $P (B) \neq = 0$ .

PICTURE. The shaded bar collapses to a line of zero width. There is no region to take a fraction of.

Step 8 — Edge case: certain and impossible priors

WHAT. Two more corners.

If $P (A) = 0$ (we were sure $A$ is impossible), the teal strip has zero width, teal block $= 0$ , so $P (A ∣ B) = 0$ . No evidence can revive it.
If $P (A) = 1$ (we were sure $A$ is certain), the plum strip vanishes, the shaded bar is all teal, so $P (A ∣ B) = 1$ .

WHY does this matter? Bayes can only rescale a belief, never resurrect a $0$ or dethrone a $1$ . A prior of exactly $0$ or $1$ is a locked door — evidence cannot open it. This is a real modelling warning: never assign a hard $0$ to something you might later want to reconsider.

PICTURE. Left panel: $A$ -strip width $0$ → posterior pinned at $0$ . Right panel: $\neg A$ -strip width $0$ → posterior pinned at $1$ .

The one-picture summary

Everything above collapses into a single frame: the unit square, sliced vertically by the prior, shaded by the likelihoods, then read sideways to give the posterior.

Recall Feynman retelling — say it in plain words

Draw a square; it's the whole world and its area is one. Cut it left/right: left is "the thing is true" (that width is the prior), right is "the thing is false". Inside each side, shade how often the evidence shows up — often on the true side, rarely on the false side (those shaded fractions are the likelihoods). Now real life happens: you see the evidence, so throw away everything unshaded and stand inside the shaded region only. Ask: of this shaded region, how much is the true side? That fraction is your updated belief, the posterior. That's the whole theorem — it's just "true-shaded ÷ all-shaded". And the famous shock (a 95%-accurate test giving 16%) is simply that the true side started as a sliver, so even a small slip on the enormous false side leaves more false-shading than true-shading. Watch the two edge doors too: if the evidence never happens the shaded bar has zero width and the question is meaningless; and if your prior was a hard 0 or 1, that side has zero or full width forever — no evidence can move it.

Recall

What is the area of the whole square, and why? ::: It is $1$ — the probability that something happens is certain, so every event's probability can be read directly as an area with no division. In the square picture, what is the prior $P (A)$ ? ::: The width of the left strip ( $A$ -true) before any evidence. What is the posterior $P (A ∣ B)$ geometrically? ::: The teal (true-and-shaded) block divided by the whole shaded bar. Why is the medical answer only 16%? ::: The disease strip is tiny, so $5%$ false positives from the huge healthy strip out-area the $95%$ true positives from the tiny sick strip. What happens to Bayes when $P (B) = 0$ ? ::: It is undefined — the shaded bar has zero width, so there is no region to take a fraction of. Can evidence move a prior of exactly $0$ or $1$ ? ::: No — those strips have zero or full width, so the posterior stays pinned at $0$ or $1$ .

Connections

Bayes' theorem and applications — the parent note this page visualises
1.3.01-Conditional-probability-and-independence — "given $A$ " = zoom into the $A$ strip
1.3.02-Law-of-total-probability — stacking the teal and plum blocks into $P (B)$
2.1.05-Naive-Bayes-classifier — where this reversal becomes a classifier
4.2.03-Bayesian-inference — posterior-becomes-next-prior, repeated
3.4.01-Maximum-likelihood-estimation — the prior-free alternative
1.3.04-Random-variables-and-distributions — the same picture with continuous strips