Bayes’ Theorem for Making Predictions?

A practical approach for forecasting and making wise choices.

Bayes’ Theorem is a futurist method for making predictions based on conditional probability. Conditional probability refers to the likelihood or chances that some outcome will occur given that another event has also occurred, usually in the past.

At its most basic level, it describes the relationship between the probability of an event occurring (the “prior probability”) and the probability of it occurring in the future given certain evidence (called the “posterior probability”).

At its most complex level, its uses statistics and mathematical formulas that describe the relationship between the probability of an event occurring (the “prior probability”) and the probability of it occurring given certain evidence (the “posterior probability”).

Why is Bayes’ theorem so important? One of the most important parts is that it allows us to keep track of the probability that a particular hypothesis holds, given evidence. Here we

In a way, Bayes’ theorem is the codification, and formalization of the reasoning behind the scientific method: if you have an extraordinary hypothesis, you will need extraordinary evidence (if the prior probability is low, you must have a lot of data in support of the hypothesis to make the posterior probability large).

Today, Bayes’ is increasingly being used in machine learning, there is an entire branch called Bayesian learning, where we use Bayes’ rule (or approximate it if the posterior does not have a nice form that can easily be stored and worked with).

Bayes’ Theorem is easy to explain.

If you flip two coins, what’s the probability they both come up heads?

Answer — There are four possible outcomes;

Coin #1: Heads, Coin #2: Heads

Coin #1: Heads, Coin #2: Tails

Coin #1: Tails, Coin #2: Heads

Coin #1: Tails, Coin #2: Tails

Only one of these outcomes can be described as “both coming up heads.” Therefore, there’s a one in four chance that we’ll get two heads. This is a pretty basic construct of simple probability.

Where the “conditional” part comes in is where we start throwing additional information into the mix.

So now, let’s ask the question: “what’s the probability that we’ll get two heads given that coin 1 came up heads?” Going through this:

Here math and statistics can tell us the probability of something happening. Cncerning flipping coins can get pretty dry, even boring.

Let’s make this more interesting…

You’re talking to your friend. Your friend refuses to use gendered pronouns (e.g. he, she), so when your friend starts talking about this new awesome person they’ve met, you don’t know if this new awesome person is male or female. For whatever reason, this is information you really want to know.

Then your friend says this new awesome person really likes Sex and the City. Now you can make some some quick assumptions about the world (and your friend’s friend):

1) The world is roughly 50% male, 50% female.

P(M)=12�(�)=12

P(F)=12�(�)=12

2) 60% of women like Sex and the City. In other words, given that a person is a woman, there is a three in five chance said person likes Sex and the City

P(S|F)=35�(�|�)=35

3) 10% of men like Sex and the City. In other words, given that a person is a man, there is a one in ten chance said person likes Sex and the City.

P(S|M)=110�(�|�)=110

This leads to the subsequent conclusion that 35% of people in the world like Sex and the City — 10% of 50% of the population plus 60% of another 50% of the population is equal to 35% of the total population.

What we want to find is P(F|S)�(�|�) — the probability that a person is a woman given that said person likes Sex and the City.

P(F|S)=P(S|F)P(F)P(S)�(�|�)=�(�|�)�(�)�(�)

Swapping in the numbers, we find that there’s an 85.7% chance that this new awesome person is female.

If this Sex and the City example is making your eyes glaze over, let me state it in even simpler terms.

Bayes’ Theorem is a way of updating what we think about the world based on what we know about the world. And what we find is that when tested we soon learn that the world’s is different from what we might assume it to be.

For example, let’s say that we’ve got a drug test. 99% of people who use the drug will show up positive. 99% of people who don’t use the drug will show up negative. One person in five hundred uses the drug. What’s the probability that a person selected at random from the population who tests positively for having done the drug will have actually done the drug?

16.5%

Bayes’ Theorem matters because the math shows that the intuitive understanding of the world is often wrong.

Here is another example of Bayes’ Theorem. Let’s say you have a roommate who’s a bit of a slacker. Now this lazy person is trying to convince you that money can’t buy happiness, citing a Harvard study showing that only 10% of happy people are rich.

After giving it some thought, it occurs to you that this statistic isn’t very compelling. What you really want to know is what percent of rich people are happy. This would give a better idea of whether becoming rich might make you happy.

Bayes’ Theorem tells you how to calculate this other, reversed statistic using two additional pieces of information:

1. The percent of people overall who are happy

2. The percent of people overall who are rich

The key idea of Bayes’ theorem is reversing the statistic using the overall rates. It says that the fraction of rich people who are happy is the fraction of happy people who are rich, times the overall fraction who are happy, divided by the overall fraction who are rich.

So if…

1. 40% of people are happy; and

2. 5% of people are rich

And if the Harvard study is correct, then the fraction of rich people who are happy is:

10%×40%5%=80%10%×40%5%=80%

So a pretty strong majority of rich people are happy.

It’s not hard to see why this arithmetic works out if we just plug in some specific numbers. Let’s say the population of the whole world is 1000, just to keep it easy. Then Fact 1 tells us there are 400 happy people, and the Harvard study tells us that 40 of these people are rich. So there are 40 people who are both rich and happy. According to Fact 2, there are 50 rich people altogether, so the fraction of them who are happy is 40/50, or 80%.

When I am doing work as a predictive futurist, I often use Bayes’ Theorem. Mathematics is not my strength so I am likely to collaborate with someone who is good at basic math.

I often research for criticism concerning and theory related to pribabilites. For instance there are always some criticisms if ideas drawn from Freakanomics. Surpisingly, there aren’t really any criticisms of Bayes’ Theorem among forcasters and futurists. This is because the only goal of Bayes’ theorem, is to identify what is the likelihood that something is going to happen given certain information.

That’s it!

Now, of course, as you accumulate more data your statements about what the probability is of something happening will need to be update.

This is one of the reasons that I have a personal rule about offering my opinion on anything unless I must do so.

There is nothing controversial about that. However, different schools of probabilists/statisticians often interpret the probabilities and date in different ways. For example, a frequentist might think make prediction based on long-term probabilities and here Bayes’ Theorem is likely to be less accurate than other approaches.

Let’s look at the distinction a bit more closely.

• Frequentist statistics (for long term predictions) never uses or calculates the probability of a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a specific number of people, places, or things. Frequentist methods do not demand construction of prior events or data and depend on the probabilities of observed and unobserved data.

• Bayesian statistics uses probabilities of data, as well as information on past occurences that took place within certain perameters, and concerning previous probabilities and how they are related.

So to review. Bayes’ theorem is a fundamental concept in probability theory and statistics that relates conditional probabilities of events. It can be used to update prior beliefs about a hypothesis or event based on new evidence or observations.

When can we not use Bayes’ theorem and when should it be used instead?Bayes’ theorem can be applied to a wide range of problems, such as classification, regression, and decision-making. However, there are some cases where it may not be appropriate to use Bayes’ theorem, or where alternative approaches may be more suitable.

Here are some situations where Bayes’ theorem may not be appropriate:

1. Violation of Independence Assumption: Bayes’ theorem assumes that the events or variables being considered are independent of each other. If the events are not independent, the application of Bayes’ theorem can lead to incorrect results.

2. Missing Data: Bayes’ theorem requires complete information about the variables being considered. If there is missing data or incomplete information, it may not be possible to apply Bayes’ theorem.

3. Complexity: Bayes’ theorem can become computationally complex for large datasets or models with many variables. In such cases, approximation techniques or other methods may be more appropriate.

Here are some situations where Bayes’ theorem can be used effectively:

1. Prior Knowledge: Bayes’ theorem is useful in situations where prior knowledge or beliefs about the variables being considered are available. It can be used to update these prior beliefs based on new evidence or observations.

2. Decision-Making: Bayes’ theorem can be used in decision-making problems where the goal is to choose an action that maximizes a certain objective function. By incorporating prior knowledge and new evidence, Bayes’ theorem can help make more informed decisions.

3. Classification and Regression: Bayes’ theorem can be used in classification and regression problems, where the goal is to predict the value of a variable based on other variables. Bayesian methods can provide a principled way to incorporate uncertainty and variability into the model.

The Takeaway

Much of life is random. We are buffeted about by unexpected events, Black Swans, and emotional sucker punches.

Ultimately, even for a beginner seeking to have more order in their life, as well as the advanced forecaster, Bayes’ theorem is a powerful tool for dealing with randomness, as it models uncertain events and updating beliefs based on new evidence. However, it may not always be appropriate or feasible to use, depending on the specific problem and the available data. In general, Bayes’ theorem is most effective when prior knowledge or beliefs are available and when the events or variables being considered are independent.

Here is an article you might enjoy

Here is one by A.S. Deller

This is an excerpt from my Module on Futurism, Predictions, and Forecasting, in my Course “The Self Improvement Lifestyle”.

Author: Lewis Harrison is a professional forecaster. He is the creator of the Ask Lewis Mentoring Method as well as HAGT — Harrison’s Applied Game Theory. He and is the Executive Director of the International Association of Healing Professionals an educational organization that offers programs around the world in Intentional Living. He is also Independent Scholar and a Results-Oriented Success Coach, with a passion for knowledge, personal development, self-improvement, creativity, innovation, and problem-solving. You can read all of his Medium stories at [email protected].

For a decade, Lewis was the host of a humor-based Q & A talk show on NPR (National Public Radio) affiliated WIOX FM in NY.

Here is the humorous promo for that radio show…

“I am always exploring trends, areas of interest, and solutions to build new stories upon. Please share this article with others. It is appreciated.

If you have any ideas you would like me to write about, just email me at [email protected] or check out all of my books, blogs, and videos through my portal www.asklewis.com

I want to acknowledge Robby Goetschalckx, and Terry Moore. Their postings in Quora.com inspired me to research this concept more than I would have otherwise.