Don’t Guess, Update: Bayes’ Theorem for Managers

Managers make decisions without ever having complete information. If a candidate performs well in an interview, does that mean they will definitely be a good employee? If the first print run of a new book sells out in two weeks, is that a sign of sustained demand or just launch hype? If sales rise by 15% after an advertising campaign, will increasing the budget lead to further growth?

Imagine a publisher preparing to release a new book. The author is well known, the subject feels timely, and the editor is confident in the project. But what does it actually mean to be confident in a project? Can that confidence be expressed as a number?

Before publication, all we know is the author, the subject, the cover, the planned price, and the performance of similar projects. Then we can collect pre-orders. After publication, we get the first sales figures. Later come repeat orders and returns. Every new piece of information can change how confident we are.

Being able to revise decisions as new information appears is an important management skill. And Bayes’ theorem formalizes exactly this process.

How to become more certain about something

In our book example, the editor’s confidence in its success did not come from nowhere. We already know something about the book market — for example, how books in this genre tend to sell or how the author’s previous books performed. This knowledge gives us an initial belief about the book’s future success or, if we translate that belief into something measurable, an initial estimate of its probability of success. In Bayesian terms, this is the prior probability — our best estimate before receiving new information.

New information proves nothing by itself. Suppose the book gets strong pre-orders. Intuitively, we may want to treat this as evidence that it will succeed after publication. From a Bayesian perspective, however, we first need to ask two questions:

• how common are strong pre-orders among successful books?
• how often do unsuccessful books also get strong pre-orders?

If almost every future bestseller has strong pre-orders while unsuccessful books almost never do, this is a strong signal in favor of success. If strong pre-orders are common among both groups, the signal is almost useless. What matters is not only how characteristic the observation is of successful books, but also how much less often it occurs among unsuccessful ones. In Bayesian terminology, this is likelihood.

Suppose we estimated the prior probability of success for our new book at 20%. Before sending it to print, we collected pre-orders because we were uncertain about the print run and saw order volumes comparable to those of past hits. We now consider success more likely. In Bayesian terminology, this updated estimate is the posterior probability. It then becomes the starting point for the next batch of information — for example, the first shipments.

In other words, the probability of a hypothesis depends on how strongly we believed in it before and how typical the new observation is of a world in which that hypothesis is true.

$P(A)$ is the prior probability, $P(B|A)$ is the likelihood of observing $B$ if the hypothesis is true, $P(B)$ is the overall probability of observing $B$, and $P(A|B)$ is the posterior probability of event $A$ given observation $B$.

Of course, calculating all of this every time we need to make any decision is impossible. For a manager, the main value of Bayes’ theorem is the discipline it imposes on thinking: when new information appears, first, do not be afraid to revise your original belief; second, ask how much that new information should actually change it.

This raises another important question: where did the original belief come from? Why did we estimate the probability of success at 20% in the first place? Let’s look at a simple example.

Suppose historical data tells us that only 20% of comparable new releases become successful. Imagine 100 similar books. On average, 20 of them will succeed and 80 will not. This is our prior probability: $P(\text{success}) = 0{.}2$.

Now we receive new information: our book sells at least 1,000 copies in its first week. Historical data tells us that 80% of successful books have starts like this, while only 10% of unsuccessful books do. This means that among the 20 successful books, $20 \times 0{.}8 = 16$ would have such a start, while among the 80 unsuccessful books, $80 \times 0{.}1 = 8$ would.

In total, 24 books would sell more than 1,000 copies in their first week: 16 would eventually become successful, while 8 would not. So if we have already observed such a first-week result, the probability that our new book will succeed is:

The same calculation can be written using Bayes’ theorem:

Where does the prior probability come from?

A prior probability is not a number we can simply make up. We need to derive it from accumulated experience: what usually happened in similar cases?

Suppose the author of our new release has published five books before and one of them became a hit. Based on that experience, we might estimate the probability that the next book succeeds at 20%. Or suppose the publisher released 100 comparable new titles this year and only 20 of them reached a predefined level of success. In that case, 20% is no longer the editor’s intuition but a base rate: if we know nothing else about the particular book, it is reasonable to expect it to behave roughly like similar titles.

Before seeing the first sales figures, we estimated the probability of success at 20%. One sufficiently strong signal raised that estimate to about 67%. But the probability did not become 100%: among books with the same strong start in our example, one in three still fails to become successful.

What counts as a comparable title is, of course, another difficult question. If we take every book published by the company, the success rate might be 20%. Among books by this particular author, it might already be 30%. Among books by this author, in this genre, and in a particular price segment, it might be as high as 50%. In other words, the prior probability depends on the category we assign the new release to.

This brings us to the first management problem: the narrower the comparison group, the more relevant it is to the specific book, but the fewer observations it contains. Take 10,000 books from the publisher and we get plenty of observations, but they are too different. Take two almost identical comparable titles and the books are highly relevant, but two observations make for a very unreliable baseline.

The basis for a prior probability needs to be both relevant and reliable. In publishing, deciding what should count as a comparable title is particularly difficult. Beyond obvious marketing characteristics such as price segment, a book can belong to a huge number of genres and subgenres, each containing different subjects, while the natural monopoly over authors and their works makes truly identical comparables impossible.

If we focus only on books by the same author, we may hear two kinds of arguments — and I encounter these in actual work. When approving a print run, an editor may point to the author’s previous book, which sold 30,000 copies. Formally, the editor is using historical data, but they have selected the most noticeable case. Meanwhile, the author has published ten books in total. The latest one really did sell 30,000 copies. The two before it sold 10,000 each, while the rest sold only 2,000.

This is where we can run into a common cognitive error: base-rate neglect. We tend to give too much weight to vivid information about a specific case and too little to boring statistics. To distance ourselves slightly from our own confidence, a useful Bayesian question is:

what usually happens in similar cases when I am not the one making the decision, or when the project in question is not mine?

Historical data is useful not only for reporting on the past. It gives us starting probabilities for decisions about the future. Still, good historical data is only the beginning. Then come pre-orders, first-month sales, returns, reviews, and repeat-order dynamics — and not every one of these metrics is equally useful when deciding, for example, on the size of an additional print run. How do we distinguish a strong signal from a weak one, and why should we avoid changing our confidence based on noisy data?

Not all new information is equally useful

A Bayesian approach requires us to evaluate not only the direction of a new signal, but also its diagnostic strength: how likely are we to see this data if our hypothesis is true, and how likely are we to see the same data if it is false?

Suppose our book has been published and first-week sales are above expectations. Intuitively, we want to raise our estimate of its future success. But first we need to ask how well first-week sales predict long-term performance in the first place. Our new release sold 1,000 copies in its first week. Historical data tells us that 80% of our bestsellers sold at least 1,000 copies during their first week, while only 10% of unsuccessful books did. Now we understand not only the first-week sales figure itself, but also its ability to distinguish successful books from unsuccessful ones.

A vivid signal is not necessarily a useful signal. A first print run that sells out in two weeks looks impressive on its own. But if this happens regularly because initial print runs are too small, the signal is not particularly strong. At the same time, bookstores placing modest repeat orders over several weeks after launch may tell us much more about sustained demand, even though the signal looks less dramatic. And, of course, we cannot separate the size of an effect from the amount of data behind it: a 40% increase in sales can mean growth from 5,000 to 7,000 copies or from 50 to 70.

Now it is time to decide whether the book in our example needs an additional print run. We have the following information:

• 70% of the first print run has sold;
• first-week sales were strong;
• bookstores are placing repeat orders consistently;
• the book has entered bestseller rankings;
• several bloggers have written about it.

All of these signals may persuade us to reprint the book. But they can carry very different weights:

• 70% of the first print run means something very different when the original run was 2,000 copies versus 5,000;
• strong first-week sales may reflect the author’s existing audience;
• steady repeat orders may genuinely be an important signal of sustained demand;
• a bestseller ranking can mean very different things depending on how the ranking is constructed;
• posts by bloggers may explain a one-off spike and may even reduce our confidence that sales will remain at the same level.

Several positive indicators do not necessarily amount to several independent confirmations of the same hypothesis. If a book gets more social-media mentions, sales rise, and it enters bestseller rankings, we may mistakenly treat these as separate pieces of evidence. But all three observations may be consequences of the same event — for example, an advertising campaign.

When new data arrives, a manager should ask three questions:

1. How often do I see this result when my hypothesis is true?
2. How often do I see the same result when my hypothesis is false?
3. Could several of the signals I am observing be consequences of the same underlying cause?

A high probability is still not a decision. Even if, after all this, we estimate the probability of a successful reprint at, say, 70%, it does not automatically follow that we should reprint the book. The final decision also depends on the size of the print run, printing costs, potential profit, and the cost of being wrong.

A high probability is not a decision

At 70% confidence, it may seem obvious that we should reprint the book. But probability alone tells us nothing about how good the decision actually is.

Suppose we are considering an additional print run of 3,000 copies. If demand persists, we will sell the entire run and, after accounting for printing, logistics, discounts, and other variable costs, earn an additional 450,000 rubles from the decision. If we are wrong and demand quickly disappears, part of the print run will remain in inventory. After accounting for the copies we do sell, printing costs, and the cost of unsold stock, the result of the decision will be a loss of 250,000 rubles.

We now have two possible outcomes:

• with 70% probability — $+450\,000$ rubles;
• with 30% probability — $-250\,000$ rubles.

We can now calculate the expected value of the decision:

This does not mean that the publisher will actually earn 240,000 rubles. In reality, the result will be closer to one of the two scenarios: we will either earn around 450,000 or lose around 250,000 rubles. Expected value answers a different question: how attractive would it be to make this decision repeatedly under similar conditions?

Under our assumptions, the additional print run has a positive expected value, so the decision looks rational. But it becomes more interesting if we change the probabilities. What is the minimum confidence in sustained demand at which the reprint becomes worthwhile?

Let $p$ be the probability of success:

It turns out that we do not need to be 70% or 80% confident in success. Given this particular balance between potential profit and potential loss, the decision to order an additional print run already has a positive expected value at a probability of success of roughly 36%.

The reverse is also true: a high probability of success does not automatically make a decision good. Imagine a project where success earns us 100,000 rubles while failure costs us one million. Even with an 80% probability of success:

Probability answers the question of how confident we are that a certain outcome will occur. Expected value adds another question: how much do we gain or lose if we are right or wrong? Together, these two things turn an assessment of uncertainty into a management decision.

A good manager does not need to correctly guess in advance whether a book will succeed. They need a reasonable starting expectation, an understanding of which new data should actually change it, and a willingness to revise the decision as that data appears.

Bayes’ theorem does not remove uncertainty. It offers a more disciplined way to work with it: start with a base rate, update your confidence only when meaningful signals appear, and separate the probability of an outcome from the value of the decision itself.

In the end, a manager’s job is not to be right on the first try every time, but to become progressively less wrong as new information arrives.