Source: Steel Seal of Thought

Author: People Are Excited Together

At the end of investment analysis is Bayesian probability

# 1. Multiple probabilities of the same thing

You have studied a company's financial report. You think the data is very good, the industry space is also very large, there is industrial policy support, and the investment logic is very smooth. You are considering buying;

However, during the research, you met an employee who left the company and learned that the company's management is chaotic and that the leaders are not enterprising. In this highly competitive industry, you think that the company's competitive position is actually slowly declining, so you are hesitating;

Then you visited the dealer and found that the company has strong control over the channel, and the dealer's feedback also shows that consumers are very sticky, and sales have continued to be strong recently.

You discussed it with your peers again, and the information you got was even more confusing...

The above situation is normal in investment. Different analysis results are obtained from different angles, which correspond to different probability of winning.

However, in terms of operation, there are only two options: “buy or not buy”. If you buy, the result is only one of the two: “meeting profit expectations” and “not meeting profit expectations”. Why is there a different probability of one thing?

It depends on how you understand “probability.”

Some people think that there is no probability; the result of an investment is either profit or loss; whether it is 0 or 100%; others think that if an investment has a probability but cannot be calculated, it means that there is no probability.

There are two explanations for probability. The “classical explanation” sees probability as an objective independent numerical value, such as:

It is known that there are 9 red balls and 1 white ball in the pocket, so the probability that you can figure out one is a red ball with your eyes closed is 90%.

If at this point, you take a look at the ball in your hand, throw it away, and keep touching another one with your eyes closed, because I don't know what ball you just touched, so I can only think that the probability of you touching a red ball is still 90%, but since you know that you lost a red ball, the probability that the next one will still be a red ball becomes 88.89%.

There are two probabilities of the same thing.

This is another explanation for probability — Bayesian probability, which is a belief-based, subjective, variable value that changes as you learn new information.

From the perspective of Bayesian algorithms, probability can not only be calculated, but can also change with information, and changes in stock prices depend on marginal changes in information, so changes in probability can also trigger changes in stock prices, that is, they can be used in investment decisions.

Let's take a look at an actual investment question: There is a large company that has backdoor listing, and companies A, B, and C have three alternative goals. After some research, you think they're all similar, so you chose A.

Later, you found someone who knew the story behind the scenes and told him that you bought A, but he didn't want to give you an answer directly; he only told you that Company B was impossible.

Excuse me, was this information useful to you? In other words, there are only two companies left, A and C, do you want to replace A with C?

Many students may have seen that this is a variant of the “three-door problem.”

# 2. The three-gate problem and Bayesian algorithm

Considering that there are still many readers who don't know about the “Three Questions,” I'll briefly repeat it:

This is a quiz TV show. There are three closed doors on stage. Two of the doors are sheep behind the door, and a car is behind the door. You can choose any of them; if it's a car, it's up to you.

So, you randomly selected one (let's say A).

According to the rules, the host (who knows which door has a car behind it) opens one of the doors (let's say B), shows you a sheep behind this door, and gives you a chance to change the door (that is, change from A to C).

Do you choose “exchange” or “no exchange”?

The answer to this question is intuitively judging that the probability of “changing” and “not exchanging” is the same, but in reality, you should change; the probability of getting a car after changing is higher.

The standard explanation for the three-door problem is this: because there are two sheep and one car, the probability of choosing a sheep in the beginning is 2/3, and the probability of choosing a car is 1/3. After the host opens a door, if you switch, the one you chose before will definitely become a car; if you chose a car before, it will definitely become a sheep, and the probability is completely interchangeable. It became “2/3 chance to choose a car, 1/3 chance to choose a goat”.

If the text is still not easy to understand, the pictures will be more clear:

If you still can't figure it out, you can simulate it with a deck of playing cards.

The answer to the three-question question is the answer to the restructuring question. If you move your position from A to C now, the probability of betting has increased from 33% to 67%.

It's amazing. Just one piece of relevant news, even if it has nothing to do with companies A or C, can change your current probability.

Let's change the above conditions again. The person familiar with the matter also said, of course, anything can happen before the restructuring is over, and B is not completely out of the game, but the possibility is relatively low.

According to our previous analysis method, if you move a position from A to C, it won't rise to 66%, but since the probability of B is less than 33%, the result of exchanging is still better than not exchanging.

Let's expand the above example from “insider trading” to a normal investment decision scenario.

If you don't research a stock, the probability of achieving the expected return after buying it is 50%.

As your research deepens — whether it's fundamental analysis or technical analysis or advanced points, or even if you just research other companies, every time you learn new information, it's equivalent to an omniscient host closing a “door” for you, and the probability of achieving the expected return after buying begins to change, from 50% up or down.

If you use computer language to describe the process of an investment expert's research decision making, it must be as described above. This is called a “Bayesian algorithm.”

The Bayesian algorithm is the foundation of artificial intelligence. When you ask ChatGPT a question, every word that pops up is a word corresponding to the maximum probability value calculated by the Bayesian algorithm. When you tell it that you just said something wrong and added a new piece of information, it immediately incorporated this new information into the previous results and produced a new series of highly probable written results — this time it was the answer you were looking for.

Seeing this, many people, even if they understand it, don't know why it became like this; it's so counterintuitive. This is also the greatest characteristic of probability—it can be calculated, but it's hard for you to feel it.

So, if you want to understand probability, the best way is to “calculate” — find an example from life and do the math yourself using the Bayesian formula.

Bayesian calculation has a numerical formula (the one Shelton wrote on the blackboard). In order not to scare everyone away, I used a graphical interface to display it to ensure that no middle school or higher number formulas appear.

# 3. Graphical interface for Bayesian computation

A customer walks into the store, looks at the shelves, and asks you about the condition of a certain product. Please ask: How likely is this customer to end up buying an order?

For a sales veteran, this problem is equivalent to a fundamentalist reading financial reports and a technical expert looking at the picture. Through the customer's every move, they can determine the customer's transaction probability, decide how much time to spend selling to the customer, select corresponding sales priorities, and decide how much discount to win over the customer.

Before answering, you need to know a “priori probability” — sales conversion rate, or “all customers who come in”. This is a historical empirical value that any salesperson should know, assuming this store is 20%.

The following picture divides all the people entering the store into two parts. The left side is the 20% portion of the transaction, and the right side is the 80% of the non-transaction.

Next, we need to consider a new message — “I asked you carefully about the condition of a certain product” based on prior probability.

At this point, we need to know the two “conditional probabilities” of this new information: the inquiry rate of customers who have made transactions and the inquiry rate of unsuccessful customers — this is also historical empirical value, that is, the proportion of customers who have carefully inquired in the past, experienced sales, and that the heart should have a rough estimate of these two probabilities.

Let's first look at the transaction customer inquiry rate, that is, “carefully inquired transactional customers/all transaction customers”. Assuming 50%, that is, open the left half five or five, and then receive the above inquiry, accounting for 20% * 50% = 10% of the total transaction customer;

Let's also look at the inquiry rate of unsold customers, that is, “unsold customers who inquire carefully/all unsold customers”. Assuming 30%, the right half is three or seven. The total proportion of unsold customers inquired above is 80% * 30% = 24%.

Each of the four corners in the picture above represents the four situations. What we are experiencing today is the upper half — consulting the customer, so first remove the lower half of the situation and just look at the top half.

What we want to analyze now is — customers who have carefully consulted and made transactions account for the proportion of all customers who have made transactions. Obviously, the upper left corner accounts for the upper half:

As a result, among consulting clients, the probability of a final transaction is: 10%/(10% +24%) = 29.4%.

Therefore, when a customer walks into a shopping mall, when he opens an inquiry, his probability of a transaction increases from 20% to 29.4%. Experienced salespeople should pay attention to this sales lead.

Using this method, we can also continue to estimate that for a customer who doesn't ask, the probability of a transaction will drop from 20% to 15.2%.

A veteran salesman collects information and makes probabilistic judgments every step of the way, so the next step is for an experienced salesperson not to introduce the product in a dry way, but to further inquire about the customer's needs. Different demands correspond to different transaction probabilities.

OK, we have the same problem as before. Even if the probability increases from 20% to 29%, I still don't know what to do?

# 4. Probability until operationally meaningful

Customers know whether they will buy something before they arrive. Assuming this person must buy it today, the actual transaction probability is 100%.

However, the salesperson doesn't know this; he only knows that in the end, customers only have two possibilities: buy (100%) and not buy (0%).

29% is just the result of the first step. He can also continue to search for new information and change the probability through the “Bayesian algorithm” to get closer to the actual target probability — whether it is 0% or 100%.

This is the meaning of Bayesian probability compared to classical probability; we must find a signal of operationally meaningful probability.

As a result, the salesman noticed that the customer asked about another completely unrelated product — bad. Experience told him that the probability of a transaction under such circumstances would decrease, because many unwilling customers like to ask questions east and west.

But how much will it actually drop? We started the second “Bayesian calculation” and then introduced two conditional probabilities. Among the customers who traded, 30% asked about other completely unrelated products, and 40% of the unsold customers.

The following is a diagram of the second Bayesian calculation. What needs to be explained is that the prior probability is no longer 20% before, but about 29% after the previous calculation:

This result shows that when the customer asks about another completely unrelated product, his chance of making a sale drops again from 29% to 8.7%/(8.7% +28.4%) = 23%

OK, after asking, the customer directly started talking about the price. Very well. According to the “third Bayes formula” of price discussion behavior, the probability of a transaction eventually skyrocketed to 70%...

70%! It was you who waited, and the salesperson didn't hide it. They directly took out the big killer — the discount, successfully won over the customer, and finally fixed the transaction probability at 100%.

In this process, although at the beginning you only have a prior probability that is very different from the actual result, as you get more information, this probability will get closer and closer to the actual situation — 0 or 100%. Once you reach a certain value, you can respond.

Many people must be asking, how can I know the probabilities of these conditions? The answer is just two words — try first.

These are all gradually summed up from numerous sales practices in the past, and are constantly being updated. For example, today's middle-aged man, assuming an 85% transaction probability and not buying in the end, this experience will change the prior probabilities of sales personnel and the probability of a series of subsequent conditions.

The so-called “experience” means that you have mastered a priori probability and a large number of conditional probabilities in a certain field of expertise.

At this point, we can use the “Bayesian algorithm” to answer the initial investment opportunity analysis questions.

# 5. Probability in investing

Everyone has their own best research method. The probability that stocks selected using this method will meet the expected return within a certain period of time (such as 1 year) can be used as a “prior probability.”

This probability is not too high. For example, it is generally impossible to exceed 60% (unless it is a particularly long-term method, or a method with very few targets that meet the requirements). Otherwise, you only need this indicator, select 20 stocks, and you can get excessive profits year after year.

If you've had a good record using this method before, then you can assume 55%.

Next, you can substitute conditional probability: What is the probability of poor management among all companies that can/cannot achieve your expected earnings.

In fact, the probability of these two conditions is not much different — this conditional probability difference is called the “degree of differentiation,” because your consideration time is one year. In such a short time, the management factors are almost negligible. Moreover, the probability of leaving a job evaluating the company's “chaotic management” is actually very high. Otherwise, the reason for leaving a job must not be “poor ability, right?”

Let's assume that in all companies that can or cannot meet your expected benefits, the probability that the leavers think the management is good is 20%/25%, respectively.

The result after the second Bayesian calculation was 53%.

Due to management factors that are not sufficiently differentiated in one-year investments, the probability has only declined slightly and is still above 50%.

Investors are particularly prone to “one-vote veto” the target due to personal likes and dislikes. In reality, the distinction is not that great. Without Bayesian probability, it is impossible to talk about rational investment.

As for the next conditions, the impact of “good sales” on the one-year investment results is mostly differentiated. It is 50% and 30% for targets that meet or do not meet expectations, respectively.

The greater the degree of differentiation, the greater the impact of this condition. After adding the “good sales” requirement, the probability that the return on investment will meet expectations rises to 65%.

Next, as new information is discovered, you can use Bayesian algorithms to update the probability of “meeting the expected return.”

Investment experts will set a probability of buying, such as 70%. Once the new conditions raise the probability to 70%, you can buy, then statistically analyze the probability based on the new information, and continue to rise to a certain level, such as 80%, then continue to increase positions. If it falls to a certain probability, such as less than 55%, the investment will end.

# 6. Three types of Bayesian investors

Summarize the method above:

First, judging investment opportunities = judging the probability of achieving the expected return

Second, with the advent of new information, this probability is also constantly changing

Third, as the probability changes, corresponding operations must also be carried out

People often leave comments in the background saying, “Just now, I found a company that is comparable to Apple's Buffett. How about you take a look?

Unfortunately, Apple's success was not calculated by Buffett at the beginning; it met expectations year after year and “the leftovers are king.” There are 99 companies behind an Apple that do not meet Buffett's requirements to continue holding, because of the new information, the probability of post-test testing declined.

Investing is a long-distance race, and Bayesian probability is your guide.

From the perspective of Bayesian probability, there are three types of so-called masters:

The first type is a Bayesian master with excellent computing power.

The most typical example is a quantitative program. Where a machine cannot do it is that the machine uses a fixed algorithm to search for investment opportunities that meet the requirements in all targets, and people use their feelings and experience to think about investment opportunities that generally meet the requirements in a limited number of targets, and sometimes consider what kind of investment methods to use.

So a real investment expert, you might ask him what “Bayesian computing” is. He is dumbfounded, and that's because he has completely internalized Bayesian calculation.

For example, Buffett once said, “Multiply the probability of loss by the amount of possible loss, then use the probability of profit multiplied by the amount of possible profit, and finally use the profit result to subtract losses. This is the method we have always tried to do.” ——This is the calculation of the expected rate of return.

The second category is experts who are good at digging for differentiated information.

From the previous example, it can be seen that most of the new information is very limited in terms of differentiation. The information you think is useful may also be useful for bad stocks, which is not enough to greatly improve the final probability.

Therefore, the most common experts have mastered some “conditional probabilities” that few people have mastered, such as focusing on a certain industry and gaining insight into some special rules and phenomena in this industry, so as to explore investment opportunities with a high winning rate earlier than others.

There are also short-term experts with a “fresh move+quick trading+decisive stop loss”. They don't need a high win rate; they only need good graphics (a priori probability) + timing (slightly higher win rate).

Even more powerful are experts at observing changes in market style. The degree of differentiation of the same type of information was also different in different periods. For example, in 2017-2020, the differentiation of the ROE index was very good, but it became invalid after 2021, while the dividend rate index had no differentiation before 2021, but after 21 years, the differentiation increased greatly.

Such experts are good at understanding changes in the differentiation of common indicators over different periods, as well as the macro factors behind them, increasing the most effective factors in a timely manner, and changing their stock selection style to suit different markets.

Third-class experts have a higher “prior probability”.

Most people have similar “prior probabilities” during the stock selection stage. They rely on finding differentiated new information later, while the legendary third-class experts have higher “prior probabilities” during the stock selection stage. Afterwards, they only need to use “elimination indicators” to filter out targets that do not meet the requirements.

The most typical ones are some big players with an information advantage in the core resource circle, and those with big capital that can actively guide topics and market sentiment. They only need “a priori probability” to stand invincible.

There is also a type of talented and patient person among these experts. They have a very high set of “prior probability signals,” but there are very few cases that meet the requirements. In most cases, they wait patiently, and as soon as the signal appears, they immediately increase leverage.

****

Many people will tell you that investing is likely to do the right thing, such as buying a white horse.

However, this classical idea of probability often results in “all four banks lose” in investment, because human behavior changes probability. Everyone says that white horses are good, and white horses will be raised to a price with no odds. Low-probability events that no one can avoid often present opportunities with extremely high odds.

Those transcendental, stable, and knowable high-probability events conceived by classical probability are bound not to occur. Probability in real investment varies from person to person, and often causes sudden changes in probability due to epiphany.

However, classical probability is so intuitive that investors always have the illusion that they are “probabilistically doing the right thing.”

Editor/jayden