Think in Likelihood Ratios

Teach me to reason about evidence the way a Bayesian does, and make it usable in my head, not on paper. Work through this in order, with me answering before each reveal.

GATE 1: THE FAILURE. Give me the test problem cold, before any theory. A disease affects 1 in 1000 people. A test catches 99% of those who have it and gives a false positive 5% of the time. I test positive. Ask what I think the odds are that I have it, and make me commit to a number. Then work it out: the answer is roughly 2%. When this exact problem was put to physicians and staff at a teaching hospital in the 1970s, most answered around 95%. Show me precisely where the intuition failed rather than just correcting the number.

GATE 2: COUNT PEOPLE, NOT PERCENTAGES. Before any algebra, redo gate 1 as a table of a thousand actual people: how many have it, how many of those test positive, how many of the healthy also test positive. This reframing into natural frequencies is the single best-evidenced fix in the entire literature on this, and it works on experts who get the percentage version wrong. Note the catch: the improvement mostly disappears if the question is asked in frequencies but answered in probabilities, so both ends have to be counts.

GATE 3: THE FORM THAT WORKS IN YOUR HEAD. Skip the version of Bayes' theorem with the fraction. Teach the odds form: prior odds times the likelihood ratio gives posterior odds. Explain why this version is tractable, which is that the awkward denominator cancels completely, so I never have to ask how likely the evidence was overall. I only ask how much better one hypothesis predicted it than the other. Rerun gate 1 in odds form and show me it takes one multiplication.

GATE 3: LIKELIHOOD RATIOS AS THE UNIT OF EVIDENCE. Drill the central question until it is automatic: how much more likely is what I just saw if the claim is true than if it is false? Give me five quick cases to rate, one at a time, including at least one where the evidence feels damning but is roughly as likely under both hypotheses and therefore proves nothing. That case is the most useful thing in this whole conversation.

GATE 4: THE MENTAL ARITHMETIC. Teach the logarithmic trick, which turns multiplication into addition so evidence can be accumulated in the head. Alan Turing invented a unit for this at Bletchley Park, the deciban, chosen so that one unit is roughly the smallest change in evidence a person can actually perceive. Give me the handful of round numbers worth memorizing, and have me add up three pieces of evidence on a real question without touching a calculator.

GATE 5: WHERE PRIORS COME FROM. The hardest honest part. Cover base rates and how to find them, the reference class problem and why reasonable people can pick different classes, and what to do when no base rate exists. Be straight that this is where Bayesian reasoning gets genuinely contested rather than pretending priors fall from the sky.

GATE 6: THE PARADOXES AS EXERCISES. Now use the machinery on the classics, having me commit first each time.

Monty Hall, and then the variant that exposes whether I actually understand it: suppose the host trips and knocks a door open at random, revealing a goat. Same doors, same visible information, but now switching gains nothing. If my explanation of the original cannot account for that, my explanation was wrong even though my answer was right. What does the work is conditioning on the host's behavior, not on the doors.

Then Simpson's paradox, where a treatment looks better for men, better for women, and worse for everyone combined. Make the point that no amount of statistics resolves it and only a causal model does: you have to know what causes what to know which table to trust, and 'always check the subgroups' is not the lesson, since sometimes the combined table is the correct one.

GATE 7: THE LIMITS. Close with the honest critique. Numbers can dress up guesses, likelihood ratios can be invented to fit a conclusion, and evidence is noisier than beginners expect: a well-designed test still misleads you sometimes, and that failure rate is itself calculable. Tell me when this framework helps and when reaching for it is theater.

Finish by having me state one belief I hold, its current odds, and precisely what evidence would move it by a factor of ten in each direction. If I cannot answer the second part, say so plainly.