## Averages are misleading I learned about Jensen's Inequality very early in my career, but it wasn't until recently that I started to grasp how powerful it is.  Philosophy and religious scholars have spent many years debating things such as experience versus reality, freedom and choice, perception, and emotions, and so on. Daniel Kahneman and Amos Tversky pioneered Prospect Theory and started the research of cognitive biases. Neuroscientists today are researching how the reality we perceive is not necessarily the reality in front of us. Many self-help books over the years have talked about pursuing healthy habits and avoiding bad habits.  Finally, there is a lot of good work done on mental models and feedback loops that show how our environment, behavior, and emotions are tightly linked. What has reinvigorated my interest in Jensen's Inequality is that it is the glue that ties all these varied concepts together in a nice framework for life. Simply put Jensen's inequality tells us that averages are misleading. Not very insightful, I know, but by putting it into an equation it gives us a framework to understand our biases and how our perception of reality can be misleading. ![[Pasted image 20220721131738.png]] *Jensen's Inequality tells us that the average of the orange line is greater than the average of the blue line.* There was an experiment done with rats that gives us a good start in explaining the power of this seemingly obvious mathematical fact. In the experiment rats could push a lever ten times, and on every 10th push, they would get a food pellet as a reward. The rats quickly figured out this function and pushed the lever as much as they could to get the treats. When the reward function was turned off the rats quickly figured out that it wasn't working anymore and stopped pushing the lever. The fascinating part of this experiment is that when the researchers introduced _[[Uncertainty]]_ into the reward function, a pellet on _average_ for every ten lever pushes, the rats continued to push the lever far longer after the reward function had been turned off. The introduction of uncertainty, or a curved function, distorted the rat's view of reality. They had been trained to expect a convex function, so they continued in their efforts for far longer than when they were on a fixed reward schedule; the _hope_ of a pellet kept them going for much longer. They failed to recognize when the orange line changed to the blue line. Why does uncertainty have such a powerful effect on our behavior? ## Complexity Life is [[Complexity|complex]] and we are wired to seek out answers that are available, easy to retrieve, vivid, and fluent. This came from the early need to survive. If the bushes were rustling, there might be a lion that wants to eat you so you better run. It didn't matter if there was a lion there or not, living to fight another day is all that mattered. Luckily, we don't have to worry about lions in bushes anymore, but [[Complexity#Reductive Effort|what we gained in security came at the cost of increased complexity]]. Intelligence strengthens our ability to reduce complexity into easily understandable and fluent pieces, however in doing so we lose important details that the complexity is communicating. There is a perpetual conversation happening between the world and us, it is overwhelming, and the urge is to simplify, but in doing so we introduce bias and skew our perception of the truth. We rely on abstractions, or average expectations of reality because we feel that we must decide, and to do so in a highly complex and uncertain environment it is preferable to base those decisions on a model of reality versus no model. ## Change and Choice Free will or the power to choose is one of the greatest powers we possess. However, many times we have no idea what to do with our freedom and the choices we end up making tend to restrict our freedom. There was only one bad choice in the Garden of Eden, and Adam and Eve wasted no time making that one wrong choice. They took their status quo for granted and believed that increasing their knowledge was attractive. What they failed to see when they ate the fruit and did not die was that they started a long process that would lead to their eventual death. What we choose to believe, or the abstractions of reality we anchor to influences what we do and the outcomes we experience. By choosing to believe in a certain model or abstraction of the world we try to reduce complexity and uncertainty, but we cannot eliminate either. >_A model might show you some risks, but not the risks of using it. Moreover, models are built on a finite set of parameters, while reality affords us infinite sources of risks._ >-Nassim Taleb via [FS Blog](https://fs.blog/map-and-territory/) When our outcomes do not match our expectations, we have a very hard time adjusting, just like the rats we are blind to the change that has occurred. This is part of the reason why it is so easy to stay in bad habits that make us feel good now and it is hard to form good habits that are hard now but will make us feel good in the long run. This is bigger than just long-term thinking versus short-term thinking, it is a willful blindness to the truth in front of us. ## How can Jensen help? >“One’s philosophy is not best expressed in words; it is expressed in the choices one makes. In the long run, we shape our lives and we shape ourselves. The process never ends until we die. And the choices we make are ultimately our own responsibility.” >-Eleanor Roosevelt We have learned that we make abstractions of reality or mental models in the interest of speed and cognitive ease, but that these models are poor representations of the true nature of the world. We are not blazing any new territory here, but what have learned is that we have a better tool to help navigate the uncertainty and complexity of the world. This tool helps us understand which direction our measurements and predictions will be biased. It also helps us to remember to slow down, engage our slower, more deliberative pre-frontal cortex, and recognize when the functions we base our decisions on have changed or when our models are poor abstractions of reality. For example, selling covered calls for income is a popular portfolio management strategy where Jensen's Inequality can be applied. Selling any option is a negatively convex function which means that your long-term expected outcome of engaging in that activity is less than any single outcome you may realize by engaging in that activity. You may sell the call for $2, and it expires worthless, free money! Do it again, bang, $2 more dollars. But the next time the stock rallies and your stock is called away. Or worse the stock drops $10 and you only [[Hedging|hedged]] $2 of that loss. I'm not saying you should never sell options, in fact MBS are bonds with embedded short options and I believe they can be terrific investments, I am just saying Jensen gives us a better framework for making these sorts of decisions. It also shows us how habits compound over time. Smoking is a negatively convex function, but smoking one cigarette doesn't do a lot of harm. If you assume that a single observation tells you all you need to know about the function that you are engaging in then your assumption will be wrong with potentially fatal long-term consequences. **Case Studies** - [[CHG Issue 169 The Morality of Black-Scholes]] - [[CHG Issue 117 The Game of Life]] - [[CHG Issue 101 Russian Roulette]] - [[CHG Issue 77 Sampling]] Explore Further: [[Complexity]] | [[Decision Making]] | [Jensen's Inequality as an Intuition Tool](https://moontowermeta.com/jensens-inequality-as-an-intuition-tool/) Tags: #evergreen Your support for Cedars Hill Group is greatly appreciated <form action="https://www.paypal.com/donate" method="post" target="_top"> <input type="hidden" name="hosted_button_id" value="74PGN8ZXHQVHS" /> <input type="image" src="https://www.paypalobjects.com/en_US/i/btn/btn_donate_LG.gif" border="0" name="submit" title="PayPal - The safer, easier way to pay online!" alt="Donate with PayPal button" /> <img alt="" border="0" src="https://www.paypal.com/en_US/i/scr/pixel.gif" width="1" height="1" /> </form>