Approaching Consulting from a Bayesian Lens

Bayes theorem quantifies and systematizes the idea of changing beliefs. So what can we learn from it when it comes to doing our day-to-day jobs?

Bayes theorem quantifies and systematizes the idea of changing beliefs

What fascinates me about Bayes' theorem is its ability to succinctly quantify and generalize how we learn, understand, and accept the world in just one equation.

Richard Price made the following analogy to illustrate the essence of Bayes' theorem when he first introduced the theorem to the world, after discovering it upon Thomas Bayes's death.

Imagine a primitive caveman emerging from his cave to witness the sunrise for the first time in his existence. Gazing upon the blazing celestial orb, casting luminous lays and emanating scorching heat, he is flabbergasted. The stark contrast to the stygian cave he once lived overwhelms his senses. Yet, as the sun slowly sinks below the horizon, the man quickly sinks into deep thought. "Will this radiant spectacle be repeated tomorrow, in the ensuing week, and every forthcoming day?" he ponders. Although there is no way for him to validate his conjecture and ascertain the sun’s return, with each successive sunrise, the man becomes a little bit more confident than the day before. Over time, having borne witness to a myriad of sunrises, the man’s belief becomes inexorably stronger, and he starts to be convinced that this is the nature of the world.

And that is basically what Bayes' theorem captures - the man's osmosis of the world as he watches the sunrise each day, or more broadly, how will our hypothesis/prior knowledge change given the emergence of new evidence.

\(P(H\vert E)=\frac{P(E\vert H)\times P(H)}{P(E\vert H)\times P(H) + P(E\vert \neg H)\times P(\neg H)}=\frac{P(E\vert H)\times P(H)}{P(E)}\)

The mathematical representation of Bayes' theorem is actually quite straightforward if we strip away the math symbol. What we want to accomplish with the theorem is to find out the posterior, the probability that your hypothesis holds true given the appearance of new evidence. Then to compute the posterior (P(H|E)), we would need three additional inputs:

1. Prior - P(H) - This is the hypothesis about the probability of an event occurring before you are given any additional evidence

2. When new evidence emerges (P(E)), the probability of it happening can be broken down into two pieces:

   1. Sensitivity of the evidence - P(E|H) - how sensitive the evidence is to the accuracy of your hypothesis

   2. The false positive rate - P(E|¬H) - how likely it is that the evidence is due to some confounding factor and not specifically to your hypothesis

(3Blue1Brown, a YouTube channel that specializes in teaching math in simple and digestible terms, has a video that better explains Bayes’ theorem visually.)

While the fundamentals of Bayes’ theorem have been widely applied to solve STEM-related problems, I have not seen it get the exposure that it deserves in the realm of strategy consulting.

Hence, in this article, I will explore how the theorem can revolutionize the way we approach business problem-solving, and will delve into the key lessons that the theorem imparts and unveil its practical implications within the consulting industry. Hopefully, by the end of this article, you will have gained a fresh view of problem-solving, armed with a powerful mental model that can guide you through complex decision-making processes.

What we can learn from Bayes' theorem in business problem-solving

If you are familiar with management consulting or have ever done any casing, then you probably know that a hypothesis-driven mindset is highly valued in most consulting/strategy-related vocations. The ability to formulate your hypothesis as well as test its validity by analyzing data/evidence from various sources are skills which professional consultants spend their lives honing.

And guess what...it just so happens that these two capabilities are also the cornerstone of Bayes' Theorem. So, how does its underlying principle provide insight into solving business problems? Here are a few of my thoughts.

Having a prior matters, but it is even more important to keep updating it

The prior is often the hardest part of Bayes' theorem to figure out. First, not everything has a prior, and sometimes calculating it is no better than flipping a coin. Second, and most importantly, the prior is completely subjective; different people have different priors based on their previous understanding and experience with the problem, which can end up with different answers.

In fact, in the world of statistics, the prior is categorized as so-called "subjective probability" because of its use as a depiction of a state of belief or knowledge rather than randomness. And there is a real push in academia to get the prior out of the statistics, which later leads to the formation of a new school of thought called the frequentist.

While the subjective nature of the prior may drive statisticians nuts, it is inextricably linked to the business world where reality is far more complex than a simple equation. Solving business problems is often more art than science, so having robust and relevant priors actually matters a lot in the sense that it helps us to cut through the noise more quickly and efficiently. For experienced professionals who have a strong prior, it will take them substantially fewer iterations or pieces of evidence to arrive at a solution than for novices who may formulate their hypothesis based only on intuition. This is why we see experts with specific domain knowledge/experience becoming coveted targets for companies because of their highly valuable priors, which effectively limit the search space and reduce the number of evidence needed to reach a cogent conclusion.

That being said, priors only matter to a certain extent. In fact, from a Bayesian perspective, the discrepancy in priors between an expert and a neophyte doesn't really matter. Remember, Bayes' theorem doesn't tell us how to set our prior beliefs. We really shouldn't be debating our prior beliefs, but rather looking at how the new evidence changes them. Since we often deal with only a piece of the problem at the beginning, we won’t know the entirety perfectly. All we can do is to update our understanding and continue to test our prior as we delve deeper into the problem. Of course, we would never be completely sure of the answer, but the hope is that with each new piece of evidence, we would get closer and closer to the optimal answer, regardless of what priors one has to begin with.

Look at both sides of the coin - be careful when evaluating evidence

From Bayes' theorem, we know that the denominator (P(E)) can be broken down into two components - the sensitivity (P(E|H)) and the false positive rate (P(E|¬H)) of the evidence. There is a clear distinction between the two parameters in that we are looking at two separate spaces here - one with only cases where the hypothesis is true (P(H)) and the other examining the opposite (P(¬H)). This is why the ratio of events in these two spaces plays a huge role in determining the posterior. Let's look at an illustrative example to see how all of these play out.

Suppose that we are the owner of a large grocery store chain, and our objective is to integrate online delivery services into our existing establishments. The idea is that the delivery service can potentially augment sales by attracting a broader customer base, but we are also curious about the profitability of such an investment. Currently, when we open a store in any given location, there is a 30% chance of yielding profitability. Looking at the current market landscape, 70% of the profitable stores support delivery, whereas a mere 25% of the unprofitable stores have invested in this capability. With a 70% chance of being profitable, adding delivery to the business sounds like a no-brainer, right? Well, it is more nuanced than that. Let's use Bayes' theorem to look at the true probability that the store will end up profitable given the added service.

\(P(\text{profit}\vert \text{delivery})=\frac{P(\text{delivery}\vert \text{profit})\times P(\text{profit})}{P(\text{delivery}\vert \text{profit})\times P(\text{profit}) + P(\text{delivery}\vert \neg \text{ profit})\times P(\neg \text{ profit})}\)

\(=\frac{0.7 \times 0.3}{0.7\times 0.3 + 0.25\times 0.7}\approx 0.55\)

While adding a delivery service may increase the odds of profitability, it doesn't guarantee success. The ratio of profitable stores to unprofitable stores is still relatively low, and there are still unprofitable stores even with the added service. Therefore, the probability of a store becoming profitable with the delivery service is not as high as initially expected. Nevertheless, the addition of the delivery service increases the odds in favor of a profitable store, updating the chance of stores reaching profitability to 55%, which is substantially higher than our 30% prior.

Bayes’ theorem underscores the complementary nature of the two facets of evidence, akin to the two sides of a coin, working together to offer a holistic picture of the underlying truth. However, when it comes to solving problems in real life, we have this tendency to fixate solely on the sensitivity side of the coin, as in what percentage of instances does the evidence support a hypothesis given that the hypothesis holds true. Such a blind spot may not be obvious, or might even be counterintuitive at first. Yet, if we can embrace the theorem as a mental model, we can better systematically evaluate the incoming evidence and avoid jumping to conclusions based on potentially skewed information.

Final Words

This is a little bit about the foundational principles of Bayes' theorem and its relevance in the context of work of day-to-day business engagement. In my view, statistics offers a rigorous and unbiased lens through which we can decode the intricacies of our world, though often sheathed in complex mathematical language. This post, in which I explain the theorem with a business twist, is my attempt to translate statistics into relatable lessons.

More importantly, what truly excites me is the exercise of forging connections between knowledge drawn from disparate domains. It allows me to perceive the familiar world we inhabit from a fresh vantage point and teaches me to be curious, flexible, and open-minded for today, tomorrow, and every day that follows.