Lesson 5 of 8
Bayes by hand
One prior, one catch rate, one measured flag rate, one division. The update takes thirty seconds and it changes what a screening result is worth.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- Bayes is one multiplication and one division: prior times catch rate, divided by the overall flag rate.
- The posterior is dragged towards the prior, which is why the prior is the input that matters most.
- A weak prior and a moderately accurate test produce a posterior that is still small, and no amount of test accuracy fixes it.
The long table in the last lesson can be done in one line. The line has a name and a reputation for difficulty that it does not deserve, because with a measured flag rate in hand it is a single division.
What a flag is worth, at three assumed starting rates
- If 1 in 100 are really misstating4.79%
- If 2 in 100 are9.59%
- If 5 in 100 are23.97%
The one line
Posterior equals Prior multiplied by the catch rate divided by Overall flag rate.
Example0.02 x 0.80 / 0.1669 = 9.59%
Three inputs. The prior is how common the thing is before you look. The catch rate is how often the test flags a real case. The flag rate is how often the test flags anything at all, which in this case was measured directly: 687 of 4,117 listings, or 16.69 percent.
Most textbook versions expand the denominator into two terms because the flag rate is unknown. When you can count it, you do not have to.
Working one
A flagged company, at a 2 percent prior
- Prior: assume 2 in 100 US listings are misstating in a given year. Write 0.02.
- Catch rate: assume the score flags 80 percent of real cases. Write 0.80.
- Multiply: 0.02 times 0.80 is 0.016.
- Flag rate, measured on 13 September 2026: 687 of 4,117, which is 0.1669.
- Divide: 0.016 divided by 0.1669 is 0.0959.
- The flag moved the probability from 2 percent to 9.6 percent.
That is a real update. The flag multiplied the probability by nearly five. It is also still under ten percent, which is the part that surprises people who expect a flag to be close to a finding.
What the prior does
Change only the prior and watch the answer move proportionally: at 1 percent the posterior is 4.8 percent, at 5 percent it is 24 percent. The posterior is directly proportional to the prior, because the prior appears once in the numerator and nowhere else.
That is why arguing about the prior is the productive argument and arguing about the test's accuracy is usually not. A prior is a claim about the world, it can be researched, and it is the input with the largest effect.
Updating twice
If a second, genuinely different piece of evidence arrives, yesterday's posterior becomes today's prior and you run the same line again. The word "different" carries the weight. Two forensic scores built from overlapping line items are not two pieces of evidence, and treating them as two updates compounds one signal into what looks like a conclusion.
That is the same independence condition as two lessons ago, in its most expensive form.
Check yourself
A test flags 16.69 percent of a population and catches 80 percent of real cases. By what multiple does one flag change the starting probability?
- By 0.80
- By about 4.8
- By 16.69
Show the answer
Answer: B.
The multiple is the catch rate divided by the flag rate: 0.80 divided by 0.1669 is 4.79. It is the same whatever the prior, which is why the prior sets the level of the answer.Two forensic scores built from overlapping line items both flag a company. What is wrong with updating twice?
- Nothing, two flags are stronger than one
- The second update assumes the evidence is independent of the first, and it is not
- The order of the updates would change the answer
Show the answer
Answer: B.
Bayes requires each new piece of evidence to add something the previous one did not. Scores sharing inputs largely repeat a single signal, and updating twice on it overstates the result.Which input has the largest effect on the posterior?
- The catch rate
- The prior
- The number of companies in the population
Show the answer
Answer: B.
The posterior is directly proportional to the prior. Doubling the prior doubles the answer, while changes to the catch rate move it far less, and the population size does not enter the formula at all.Sources
- M-Score values computed by ValueScreener across 4,117 active US equity listings, read on 13 September 2026: 687 above the published cutoff, a flag rate of 16.69 percent.
- Beneish, Financial Analysts Journal, 1999, "The Detection of Earnings Manipulation": the score and its cutoff of minus 1.78.
- Kahneman and Lovallo, 'Timid Choices and Bold Forecasts: A Cognitive Perspective on Risk Taking', Management Science, 1993, on the outside view, which is what a prior is.
What this leaves out
- Only the flag rate is measured. The prior and the catch rate are assumptions, labelled as such in every table, and the lesson is arranged so a reader can substitute their own.
- We hold no measurement of how many US listed companies misstate accounts in a given year. That is why three priors are shown rather than one.
- The posterior is a probability about a category, not a finding about any company. A flagged company has been selected for reading and nothing more.
- Nothing here is advice about any security.