Tag Archives: philosophy

A problem well stated is a problem half-solved

Lately, I’ve been reflecting on how we can prevent “things from getting worse” in a variety of settings — at work, within our families, and even more broadly in society. I’ve come to believe that what truly matters is our ability to course correct: to adjust and improve situations once we understand that they are deteriorating.

The difficulty, of course, lies precisely in that word — understand. Recognizing that things are getting worse is often far from easy. In many cases, it is hard even to articulate clearly what we mean when we say that something is “not going in the right direction.” And in most environments, it can be genuinely difficult to speak up and say, “Something is not right here.”

This is what brings to mind the quote often attributed to Charles Kettering, former head of research at General Motors: “A problem well stated is a problem half solved.” There is considerable wisdom in that statement, particularly when dealing with complex issues and when trying to mobilize groups of people toward meaningful course correction.

Why do I say this? Fundamentally because, in my experience, when problems or opportunities are not well stated, a host of negative dynamics tend to emerge.

  • People begin to adapt to problems instead of solving them — a powerful driver of many organizational and social failures.
  • A lack of clarity makes it difficult for competent individuals to take the lead.
  • Those who have adapted successfully to a flawed situation often resist change, even when the overall impact is negative.
  • Morale and energy decline as conditions worsen and collaboration becomes harder.

Within the limits of this post, I want to share some thoughts on what can help prevent this outcome. In particular, I want to highlight one family of tools that humans have developed to state problems with exceptional clarity: quantitative models. To be clear, the core point is not about mathematics per se, but about fostering clarity of language and transparency in order to enable course correction. Quantitative models are simply a particularly powerful way to achieve that.

Quantitative models: well-stating problems the hard way

Quantitative models are not always available — after all, they require measurable quantities — but when they are, they are remarkably effective. They force assumptions to be explicit, make key trade-offs visible, and provide a shared and precise language that greatly facilitates collaboration. It is no coincidence that the extraordinary progress of physics, chemistry, and engineering from the 16th century onward coincided with the widespread adoption of mathematical modeling.

In some cases, such as physics, we are even able to state incredibly complex problems with great precision without fully understanding them. Quantum mechanics is a striking example: we can formulate models that answer factual questions with astonishing accuracy, even when their interpretation in plain language remains deeply contested.

Another major benefit of mathematical models is that they uncover relationships that would otherwise remain hidden. A simple example illustrates this.

Suppose you run a sales effort in which you provide services at a loss, hoping that a fraction of prospects will eventually convert to paid customers. You would like to understand how to balance this investment in order to maximize profit. Assume that you are sophisticated enough to have an estimate of the probability of conversion for different prospect profiles.

A natural question arises: up to what probability of conversion should we be willing to provide services at a loss?

At a high level, the answer is intuitive: the gains from converted prospects must offset the losses from those who do not convert. In quantitative terms, this can be written as:

Conv\% \cdot (ConvValue-Investment) \ge (1-Conv\%) \cdot Investment

Solving this inequality at breakeven yields the minimum conversion probability required for profitability:

Conv\% = \frac{Investment}{ConvValue}

This expression immediately clarifies several things. Profitability depends only on the investment per prospect and the value of a converted customer. And if the investment exceeds the conversion value, there is simply no viable business.

A less obvious insight concerns sensitivity. The conversion threshold depends on investment and conversion value in exactly opposite relative terms: increasing investment by 10% raises the required conversion rate by 10%, while increasing customer value by 10% lowers the threshold by 10%. This kind of elasticity-based reasoning is extremely hard to see without writing the problem down explicitly.

Of course, this model is simplified. In practice, conversion rates often depend on investment — for example, offering a richer free trial may increase the likelihood of conversion. At first glance, this seems to make the problem much harder: the conversion rate depends on investment, but investment decisions depend on the conversion rate.

Yet writing this down actually simplifies the situation. If conversion probability is a function of Conv\%(Investment), profitability requires

Conv\%(Investment) \ge \frac{Investment}{ConvValue}​

Rather than a fixed threshold, we now have a relationship defining a region of profitability. Far from being an obstacle, this opens the door to optimization: by segmenting prospects by expected value, we can refine investment levels and improve outcomes.

This is a general principle in quantitative modeling: relationships between variables may complicate the mathematics, but they expand the space of possible strategies.

From thresholds to overall profits

So far, the discussion has focused on whether to pursue a given prospect segment. But what about overall profitability once we act?

If conversion rates are not easily influenced by investment, total profits can be written as:

Profits = MarketSize \cdot ( P(Converting)\cdot(ConvValue - Investment) - P(Not\ Converting)\cdot Investment )

Suppose the baseline conversion rate is 30% and the long-term value of a converted customer is three times the investment. Plugging in the numbers yields a loss: on average, each prospect served generates a loss equal to 10% of the investment.

At this point, several strategic levers are available: improve the product without raising costs, improve it while raising costs but increasing conversion, reduce free-trial costs, or develop targeting models to focus on prospects more likely to convert.

How should a team — say, a group of founders with very different backgrounds — decide which lever to prioritize? Disagreement is inevitable, and mistakes will be made. This is precisely why course correction matters, and why developing a precise language around the problem is so important.

Consider targeting. Suppose we segment the market into two equal-sized groups: Segment A with a 40% conversion rate, and Segment B with a 20% conversion rate. Targeting only Segment A yields positive profits — a substantial improvement driven by a very rough segmentation (equivalent to a two-bin scorecard with a Gini of roughly 23%). See below:

SegmentAProfits=\frac{1}{2}\cdot MarketSize\cdot(40\%\cdot(3\cdot Investment-Investment)-60\%\cdot Investment)=\frac{1}{2}\cdot MarketSize\cdot(20\%\cdot Investment)

With further work, we could address questions such as: how valuable is improving targeting further? How does that compare with reducing free-trial costs or increasing customer lifetime value? Quantitative models allow us to ask — and answer — these questions systematically.

Clarity, knowledge, and course correction

One might object that quantitative models are difficult for many people to understand, and therefore limit broad participation in decision-making. This is a fair concern. But clarity is never free. Whether expressed mathematically or otherwise, precision requires effort.

Course correction depends on acquiring and applying new knowledge, and conversations about knowledge are rarely easy. We cannot hope to improve conversion through product enhancements without learning what users value most — and learning often requires time, attention, and risk. As Feynman put it, we must “pay attention” at the very least.

Recognizing knowledge, applying it, and revising beliefs accordingly is hard, even for experts. A well-known anecdote from Einstein’s career illustrates this. After developing general relativity, Einstein initially concluded — incorrectly — that gravitational waves did not exist. His paper was rejected due to a mistake, which he initially resisted. Yet within a year, through discussion and correction, he recognized the error and published a revision.

Even giants stumble. Progress depends not on being right the first time, but often on being willing — and able — to correct course.

Recommended book on how powerful quantiative models can be:

Galileo and the scalability of progress

As I write this blog I am surrounded by a social/political dialogue that does not seem to proceed, or be guided, by the standards and signposts I was educated to follow to aim for progress. It is all quite confusing and this has brought back memories of a great hero of mine: Galileo Galilei.

The man in the image below:

Galileo lived ~500 years ago and he is largely credited as one of the driving forces of the scientific revolution. That is the revolution that took us from 1-2bps of growth in output and standards of living for hundreds of thousands of years, to the skyrocketing acceleration of innovation, and quality of life, of the last few hundreds years. We all hope this pace will continue for as long as possible.

When I grew up I did get the point that Galileo was a genius, and that he was prosecuted for his heretic ideas about the Earth moving around the Sun, but I did not understand, as much as I do now, how fundamental his life and example had been in driving change and progress in a world that was intellectually paralyzed for millennia.

When Galileo started his scientific career, to be educated meant to be able to understand the body of Aristotle philosophy, the Bible, and to have a good account of Geometry as it was developed by the Ancient Greeks.

Instead of focusing on “How do we know, explain, or understand such and such…” the prevailing framework was to argue on the basis of “by what authority do we claim?”. The latter and prevailing intellectual tradition led to what is called “Scholasticism”. Scholasticism basically meant that everyone’s effort was to come up with some sort of top down explanation about facts that had to be aligned or, even better, justified by some passage of what Aristotle wrote, or what was in the Bible.

Facts and reason did not matter as much, it was all about narratives and how good a story was and how well it aligned with “first principles”, or whatever fashion was prevailing in a given epoch.

For example Aristotle, who was nevertheless a great philosopher, envisioned a world where Earth was at the centre, and somehow at the bottom of the universe, which, in his view, explained why everything fell down. He also devised a world in which different substances behaved differently, but everything liked to move in circles, the argument was that circles are perfect.

Aristotle never tried to really test his theories as we would do today, after Galileo’s lesson. All he did was providing plausible arguments and sensory examples. In this sense he was an empiricist, but not a scientist. I will clarify the distinction later, but here’s an example.

Aristotle observed that large bodies seemed to fall harder to the ground, and that very light and thin ones didn’t. He then provided an explanation that sounds “common sense” in his philosophy. Heavier objects have more “matter” and long to reach Earth, the mother of matter, as quickly as possible, others are happy to take a bit more time as they are also part of a substance that longs for the heavens (like air).

Believe it or not this was then to be the prevailing understanding of how things move for about two thousand year, in Europe at least – crazy how well a story can sell right?

Nevertheless Aristotle was wrong on all accounts (when it comes to Physics).

Here’s how this has to do with Galileo, and science. Galileo was of a different temper since he was young. He thought that all the philosophical speculations were a lot of fun, but that ideas needed to be accountable before reality. Galileo didn’t just believe in “experiments” and gathering data, what we called empiricism, but he developed an approach that was tougher and gave birth to western science in the process. Galileo thought ideas and conjectures needed to be put on “trial”. In italian he called that “cimenti” which, for the fans of Game of Thrones, best translates to “trial by combat”.

So when Galileo decided to investigate the motion of objects, he immediatly sought to test his view that objects moved following specific geometrical and mathematical laws, with the view of Aristotle, and famously dropped objects from the leaning tower of Pisa and behold!:

After two millennia someone actually bothered to find out that, after all, objects of a similar fabric, but of very different weights, fall at the same speed! (his “cimenti” actually were on inclined planes but… the power of a story). And in a fierce trial by combat Aristotle theory lies on the ground, dead as a stone. This is the key difference, Galileo didn’t tell stories about how his view “made sense”, he took is idea and put it out there, fly or get killed.

Why did it take two thousand years? – That is a long story, but we can all testify that the behavior that was followed for millenia is still around us: believe in authority, don’t ask quetions, optics are all that matters, tell a good story, its common sense etc.etc..

Galileo was, as Aristotle, also a great debater and writer and in his famous books written as dialogues he indeed asks the question to Simplicio, he goes:

“But, tell me Simplicio, have you ever made the experiment to see whether in fact a lead ball and a wooden one let fall from the same height reach the ground at the same time?”

This is science: truth is not determined by who said what, common sense, or examples, but by what nature/reality reveals when cornered. We discover the truth, we do not and should not create it!

This spirit is what led to the cures that help us when we are sick, and the tehcnologies that improved our lives beyond the wildest imagination of anyone living in Galileo’s time. This is what made progress scalable: a method to drive progress, that can be distributed everywhere. Everyone can pick things up from where Galileo left them today, and do the experiment and move things forward, there is no authority, no story or view to follow, no boundary beyond honesty.

This matters everywhere. In politics, in science, and indeed in business.

But Galileo was also a revolutionary genius specifically within Physics and I do want to close with what we today call the “principle of relativity”, and how Galileo deeply unlocked the power of imagination in science. This seems a contradiction as you might now think that Galileo was weary of speculations, but the truth is the opposite, once he provided an effective method for progress, he then unleashed imagination in a productive way.

Most pople that did not to go far in enjoying the study of science, and physics, believe that what is hard about science is something like the math, the jargon, or the baggage of things to memorize. In truth the concepts are hard, the formalism is often very easy and mechanical.

Listen to what Galileo said about motion carefully. Galileo observed the following:

“No one can actually tell whether he or she is moving, velocity is not real, there isn’t physical fact about velocity, as velocity is only but relative” (this was an important point for him to support that Earth was moving around the Sun).

He then goes on to the famous example of one sitting below deck in the cabin of a smooth sailing ship, where you won’t be able to tell whether the ship is moving, or not, in any way.

He was right but this concept is tough, how can velocity be an illusion? If the change in velocity (acceleration) is real, and we can tell if a ship is changing its speed, how can velocity not be physically real? How can the change in something “unreal”, like velocity, be real, like acceleration?

I won’t answer those questions now, but I hope the thirst for knowledge will push some of you toward picking up a physics and classical mechanics book to find out more, but my point is the following:

You don’t need maths and all the techincal baggage to understand more about Galileo and his science.

And my broader point is what is most at my heart in writing this blogpost:

You don’t need any title to go out there and defend what has served us well, and lifted us after hundreds of millenia of no progress, which is the methodical pursuit of truth, useful truth, I would say. Facts matter, reality is out there. Opinions, narratives, stories, optics, bad formulas and all that alone do not move us forward, they never did and never will.

We should all embrace the “cimenti” , the trial by combat of our scientific tradition, first and foremost on our own ideas and conjectures and never forget we are aiming for progress not fabrications.

To get a quick view on Galileo’s life and key ideas I recommend the following book:

Causal inference: suggested readings and thoughts

I have studied statistics and probability for over 20 years and I have been constantly engaged in all things data, AI and analytics throughout my career. In my experience, and I am sure in the experience of most of you with a similar background, there are core “simple” questions in a business setting that are still very hard to answer in most context where uncertainty plays a big role:

Looking at the data, can you tell me what caused this to happen? (lower sales, higher returns etc.etc.)

Or

If we do X can we expect Y?

…and other variation on similar questions

The two books I have read recently focus on such questions and on the story of how statisticians and computer scientists have led a revolution that, in the past 30-50 years, succeeded in providing us with tools that allow us to answer those questions precisely (I add “precisely” since, obviously, those questions are always answered, but through unnecessary pain, and often with the wrong answers indeed).

Those tools and that story is still largely unknown outside of Academia and the AI world.

Let’s come to the books:

1 – Causal Inference by Paul R.Rosenbaum

2 – The Book of Why by Judea Pearl and Dana Mackenzie

Both books are an incredibly good read.

Causal Inference takes the statistical inference approach and it tells the story of how causes are derived from statistical experiments. Most of you will know the mantra: “Correlation does not imply Causation”, yet this book outlines in fairly simple terms how correlations can be leveraged and are indeed leveraged to infer on causation.

Typical example here is how the “does smoke cause cancer?” question was answered, and it was answered, obviously without randomised trials.

The Book of Why is a harder read and goes deeper into philosophical questions. This is natural given the authors are trying to share with us how the language of causation has been developed mathematically in the last 30 years, and the core objective here is to develop the tools that would allow machines to answer causal queries.

I want to get more into the details of some of the key concepts covered in the books and also to give you a sense of how useful the readings could be.

Starting with Rosenbaum’s I would point out that this books is even overall a great book to get a sense of how statistical inference, and the theory of decisions under uncertainty, is developing.

This book is a gem, no less.

It starts very simple with the true story of Washington and how he died after having been bled by his doctor (common practice back then) and asks: Would have Washington died, or recovered, had he not been bled?

He then moves to explain randomised trials, causal effects, matching, instruments, propensity scores and more.

Key here is that the only tool for statistical inference that was well developed and accepted up to the 70s was the randomised trial, that is, for example in medicine, giving a treatment to a random sample of individuals, a placebo to the others and check the difference in outcomes to make inferences.

This procedure itself is not even causal, in principle it is still flawed with respect to answering a causal query (logically), but it works as follows:

  • I see outcome O associated with treatment T
  • What are the chances that I would see O regardless of treatment T?
  • If chances are low, that is evidence of treatment T causing outcome O (unlikely association after all is interpreted to imply causation)

Rosenbaum goes to explain why above works in the causal inference framework, as it is interpreted as a counterfactual statement supported by a particular type of experiment, but then moves to explain that even observational studies (where there is no placebo, for example) can provide answers that are as robust as randomised trials/experiments.

Other key points are really on the struggle that the statistical community had to go through and goes through today when working through observational studies. It is of note the historical account of the debate when the relationship between smoking and lung cancer was investigated with the unending “What ifs”… what if smokers are less careful? what if smokers are also drinking alcohol and alcohol causes cancer? etc.etc.

An illuminating read, which also sheds light to how the debate on climate changes is addressed by the scientific community.

Moving on to The Book of Why

I love one of the statements that is found earlier in the book:

“You cannot answer a question that you cannot ask, you cannot ask a question you have no words for”

You can tell this book goes deeper into philosophy and artificial intelligence as it really aims to share with us the development of a language that can deal with posing and answering causal queries:

“Is the rooster crowing causing the sun to rise?”.

Roosters always sing before sunrise, so the association is there, but can we express easily in precise terms the obvious concept that roosters are not causing the sun to rise? Can we even express that question in a way a computer could answer?

The authors go into the development of those tools and the story of what hindered this development, which is the “reductionist” school in statistics. Taking quotes from Karl Pearson and his follower Niles:

  • “To contrast causation and correlation is unwarranted as causation is simply perfect correlation”
  • “The ultimate scientific statement of description can always be thrown back upon…a contingency table”

The reductionist school was very attractive as it made the whole endeavour much simpler, and mathematically precise with the tools available at the time. There was a sense of “purity” to that approach, as it was self consistent although limited, but indeed attempted to imply that causation, which was a hard problem, was unnecessary (best way to solve a hard problem right?). Ironically, as the authors also point out, this approach of assuming that data is all there is and that association are enough to draw an effective picture of the world, it is something that novices in machine learning still believe today (will probably talk more about this in a later blogpost).

Pearson himself ended up having to clarify that some correlations are not all there is, and often misleading. He later compiled what his school called “spurious correlations” which are different from correlations that arise “organically” although what that meant (which is causal correlations) was never addressed..

The authors also introduce the ladder of causation, see below:

Which is referenced throughout the book and it is indeed a meaty concept to grasp as one goes through the 400 pages of intellectual mastery.

What Pearl and Mackenzie achieve, that Rosenbaum does not even aim to discuss, is to invite the reader to reflect upon what understanding is, and how fundamental to our intelligence causality is.

They also then share the tools that allow for answering very practical questions, e.g.:

Our sales went up 30% this month, great, how much of it is due to the recent marketing campaign and how much is it driven by other factors?

The data scientists among you know that tools to address that question are out there, namely structural equation modelling and do calculus, but this indeed is closely related to structural causal models that Pearl promotes, and, in ultimate instance, the framework of introducing causal hypothesis is unavoidable.

Conclusions:

I recommend the books to any knowledge seeker, and anyone that is involved decision making (or selling benefits that are hard to measure).

I would start with Rosenbaum’s book as it is less than 200 pages and, if time is scarce I would prioritise reading Pearl and Mackenzie’s book up to chapter 6 first (190 pages)