The Flaws of Fermi Estimates

Why don’t we use more Fermi estimates?[1] Many of us want to become more rational. We have lots of numbers we can think of and important variables to consider. There are a few reasons.

Fermi calculations get really messy. After a few variables introduced, they could quickly become difficult to imagine and outline a problem. Many people, especially those who were not used to writing academic papers, do not practice the skills of formalizing inputs and outputs. It can be tedious for those who do.

Fermi models typically do not include estimates of certainty. Certainty propagates. It creates bottlenecks. As a Fermi model grows, specific uncertain assumptions could underscore the result. Certainty estimates are typically not measured, and when they are they require formalization and significant calculation.

Fermi calculations are not fun to share. Most of them are pretty simple; they just involve multiplication and addition and 3–5 variables. However, in order to write them one must formalize them as few lines of math him or few long paragraphs which really should be math.

We propose the use of simple graphical models in order to represent estimates and Fermi models. We think these have the capacity to solve the issues mentioned above and make complex estimations more simple, more sharable, and more calculable. A formal and rigorous graphical model could not only improve on existing Fermi calculations, but it could also extend them to functions they have not yet been used for.

Multiplication

Let’s say we are trying to estimate the number of smiles per day in a park. A first attempt at this may be to guess the number of people in the park and to estimate the number of smiles on average per person in the park.

This is easy to calculate directly. 100 People x 10 smiles/(day * person) = 1000 smiles/day.

As a model, we can represent the variables as lines and the function as a box in between them. This fits nicely with similar diagramming standards. The function of multiplication acts as an object with inputs and outputs.

Independent variables, or user selected variables, are shown in black, and dependent variables are shown in blue.

We can condense this diagram by moving the number of smiles per day per person into the multiplication block.

Say we wanted to find the total smiles per year in the park. We can simply extend the model as follows.

Addition

Perhaps we think that kids and adults have different rates of smiling and would like to separate our model accordingly. We estimate the number of kids in the park, the number of adults in the park, and their corresponding smiling estimates. Then we add them with a similar block as we used for multiplication.

Uncertainty

If we have uncertainty estimates we can make them explicit. Estimates of certainty typically get left out of Fermi calculations, but become essential when making large models.

It is not clear what the best way is to annotate an uncertainty interval. In this case, the intervals described are meant as 90% Gaussian confidence intervals, but these could vary. They do not have to be Gaussian-like intervals, but could be complex probability distributions. These may require graphical representations and additional software. However, for many estimations, even simple models of uncertainty would be advantageous.

Estimate Combination

If two people give two estimates for a number, they could be combined to find the resulting probability distribution.

Uncertainty distributions are valuable for this. If two agents both state their uncertainty distributions, we can find a weighted average of their estimations with a calculated resulting uncertainty distribution.

Model Combination

We can combine models by combining their resulting estimates. So far we have shown two unique attempts at modeling the number of smiles in a park. They produced the same unit output, so they can be combined.

Both of them still have predictive power, and a combination could produce a more accurate estimate than either alone. The model with greater certainty, in this case the adult/child split model, will have more influence in the final calculation, but it will still be moderated by it. Combining many properly calibrated models will always give a more accurate result.

Abstraction

Large sections can be combined into black boxes.[2] Black boxes can be used to summarize large models into simple objects with specified inputs and outputs. This means that one can work on a very large total model in small pieces and have it be manageable.

Decision Making

Say we must decide between two options. One common way to do so is to estimate a value for each, and choose the one with a higher (or lower) value.

In this case we make a decision of which lemonade will sell better. We use a decision ‘block’, which could hold any arbitrary decision function. In this case, it simply outputs the value of the highest input value.

This can be useful if one can assume the use of the best option of alternatives. In a larger model, there may be many decisions determined by model. The outputs of these decisions could be used for later estimations or decisions.

Larger Models

These techniques can be combined to produce large and intricate models. As these increase in size they can become more valuable.

In the model above, a person is attempting to find the best use of their time to produce money. There are several options to sell lemonade, and there’s also the opportunity to work overtime. The estimator makes an estimate for each and uses the model to understand them in relation to each other.

This larger model demonstrates the option of configuration in these models. The profit percentage of lemonade sales was expected to be similar for different kinds of lemonade in different locations. It could have instead been multiplied individually for each one, but it was simpler to move it after the decision block between them.

In this case it may have been reasonable to use a table instead of a graphical model. However, a table would not necessarily demonstrate the unique constraints and considerations of each type of input. For instance, lemonade sales had a margin of profit, and overtime work had a different net income number. In tables many of the important calculations are often difficult to read at the same time as the data. We believe this form of modeling helps make the numbers understandable as well as the assumptions and certainties that go into those numbers.

Possible Automated Analysis

Once we arrive at the model above, we would have enough information to calculate the value of information (VOI) of additional certainty for each metric. For instance, a reduction of uncertainty of the variable ‘Regular Lemonade at Dolores Park’ to 0 could produce an expected few dollars per hour, assuming that resulting decisions would be made using the model.

The value of new options could also be calculated easily if one could come up with a probability distribution of their expected earnings per hour.

While these kinds of analysis are well established in academia, they are currently difficult to use. If estimations could be simply mapped, it may make them significantly more accessible.

Similar Work

This work can be seen as similar to Unified Modeling Language (UML) in that it attempts to graphically specify a complex system of knowledge. UML was an attempt to define a graphical language for software architecture. There were claims that programs that produced UML could be used to produce their corresponding programs. This hasn’t really happened. The UML spec went through several versions and became so specific and complex that few programmers now bother with it. However, it did encourage the use of whiteboard modeling for other programmers and experiences some popularity with larger projects.

Graphical computer software is challenging. Most attempts have failed, but a few companies have had success with it. LabView is a popular visual programming tool used by scientists and engineers. It uses a Dataflow programming paradigm, which would also be appropriate for Graphical Assumption Modeling.

The theory of this work is similar to that of Probabilistic Graphical Models. These are typically more formal models aimed at computer input and output rather than direct human interaction.

Future Work

This research is very young. The diagrams could use more experimentation and exploration. We have not included a method for subtraction or division, for example. Even if they were better established, it could take a long time for them to become accepted by other communities.

It’s obvious that if these models are useful, it would be valuable to have a computer program to make them. Ozzie Gooen has made a simple attempt called Fermihub. Fermihub is functional, free, and open source. However, it applies only a few simple analytic approximations and does not incorporate Monte Carlo simulations. For accurate or large models, Monte Carlo simulations will be necessary.

There could be more research done in this kind of estimation. While much of the math has already been solved, the art of efficiently creating large models and collaborating with others has a lot of work left. There is also some debate on the proper way to combine estimates, which is crucial for large models.

Note: I realize that the math in the models above, specifically in the combinations of estimates, is incorrect.  I'm currently investigating how to do it correctly.   

References

1. Fermi Estimates, LukeProg. 2013
2. See wikipedia for a high level understanding of black boxes. They are a fundamental unit for systems research, which in part has lead to many diagrams we see today.