How To Actually Avoid The Transposed Conditional

The transposed conditional is another one of those subtle but very consequential misunderstandings in the field of statistics. I recently covered the topic of how to really understand statistical significance. In the discussion that ensued, someone mentioned that my essay did not cover the issue of the transposed conditional. So, here we are.

In this essay, I’ll start by explaining the notion of probability statements. Then, I’ll go on to explain how the issue of the transposed conditional surfaces from flawed probability statements. And finally, I will go over potential remedies and strategies for avoiding this issue in practice. Let us begin.

This essay is supported by Generatebg

What is a Probability Statement?

Probability as a mathematical concept is partly intuitive to human beings and is partly not intuitive at all. We engage in implicit probabilistic thinking all the time. When we make a split-second decision to run to catch a train, we are optimizing to increase our probability of making it.

On the other hand, we constantly underestimate the probability of improbable events. This is such a deceiving issue that I’ve written an entire essay covering the topic: How To Perfectly Predict Impossible Events?

A probability statement is a verbal or written statement that expresses the mathematical probability of occurrence of an event.

That shouldn’t be too hard, right? Well, it is! You see, although absolute probabilities are a thing in the mathematical world, most real-world phenomena (especially statistical phenomena) need to be expressed as conditional probabilities.

A conditional probability statement is a verbal or written statement that expresses the mathematical probability of occurrence of an event or observation given the certainty of underlying information or hypothesis.

In probability notation, the probability of a observing evidence “E” given the certainty of hypothesis “H” is expressed as: P(E|H) — the vertical line here can be read as “given” or “conditional on”.

For the remainder of our discussion in this essay, we will be focusing on conditional probability statements purely because of their practical worth.

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How does the Transposed Conditional Occur?

The very first thing that we need to clarify is that in statistics, we often hear statements of the form:

“The probability of observing this result purely by chance is very low.”

Such a statement implies that it is talking about the absolute probability of the event of observing. But what it actually implies here is a conditional probability under the condition that the underlying hypothesis is true.

Statistical inferences are seldom based on absolute probabilities. So, our default assumption should be that statistical inferences are based on conditional probabilities.

The issue of transposed conditional arises when statistical statements switch evidence and hypothesis in practical contexts. Let me clarify this using a simplified example.

Simple Example of the Transposed Conditional

Let us say that we are dealing with the hypothesis that cows have four legs and the animal you are currently observing is a cow. Following this, you come across the following set of statements:

1. The probability that an animal has four legs if it is a cow is one.

2. The probability that an animal is a cow if it has four legs is one.

The first sentence seems to make logical sense given our hypothesis. But there is something odd about the second statement. It seems to suggest that any animal with four legs is a cow. That cannot be true, right?

In any case, it is quite obvious that both these statements do not mean the same thing. Yet, in the statistical world, statement 2 is often used as a replacement for statement 1. This is the issue of the transposed conditional.

In technical terms, let “H” be the hypothesis that cows have four legs and you are looking at a cow. Let “E” be the evidence of observing an animal with four legs. The two statements from above can be mathematically expressed as:

1. P(E|H) = 1

2. P(H|E) = 1

In the Bayesian world, P(E|H) cannot equal P(H|E). In this context, we are interested in P(E|H) and not in P(H|E). You might be feeling sceptical about the “issue” that this example illustrates.

“Who in their right mind would switch these two statements?”, you might ask.

But think again! This is an extremely simplified example to illustrate this very issue. In real life, things are not so obvious. To get a feel for this, let us explore a more advanced example.

Advanced Example of the Transposed Conditional

Say that while investigating a crime scene, you find a blood stain of type “V” that matches with a sample from the prime suspect “Agent x”. Let “E” denote the evidence that the blood stain of type “V” was found at the crime scene and “H” denote the hypothesis that Agent x was responsible for the crime.

Now, let us say that you read the following two statements from the department of statistical forensics:

1. The probability that the stain has come from Agent x if it is of type “V” is 1000 in 1.

2. The probability that the stain would be of type “V” if it had come from Agent x is 1000 in 1.

On the surface, both seem equally convincing. But one of them is wrong. If we refer to the mathematical probability notion, the two statements could be written as follows:

1. P(H|E) = 1000/1

2. P(E|H) = 1000/1

Again, in the Bayesian world, P(E|H) is not equal to P(H|E). Furthermore, P(E|H) is what we are interested in. Therefore, the second statement is correct and the first statement is wrong (in this context).

There could be other contexts where the two terms switch roles as to which one is correct, but the main takeaway is that both these terms cannot have the same probability value assigned to them. Any decision that is taken from such flawed logic would be misleading and potentially dangerous.

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How To Actually Avoid the Transposed Conditional?

The issue of the transposed conditional reveals itself when we look at the facts and statements in writing. However, when these statements are made verbally, they go easily undetected. Consequently, they present a potent threat for misinformation and manipulation.

In order to avoid the issue of the transposed conditional, ask yourself the following questions:

1. Is the statistical statement you are reading offering a probability for the truth of a hypothesis? — If yes, be sceptical.

2. Does the statistical statement you are reading offer an explicit conditional qualifier such as “if” or “given”? — If no, be sceptical.

3. Is the statistical statement you are reading suggesting a conclusion without explicitly considering at least one alternative hypothesis beforehand? — if yes, be sceptical.

A reasonably well-formulated statistical statement for a non-technical audience would be along the following lines:

“The evidence strongly supports the hypothesis that the blood stain came from Agent x.”

Although there are no explicit qualifiers in this statement, it explicitly states that the evidence supports the hypothesis. It does not make any assertions about the probability of the hypothesis.

To conclude, whether you are reading a statistical statement or you are the one formulating one, be aware of the potential issue of the transposed conditional and work on actively avoiding it!

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References and credit: I.W. Evett.

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Further reading that might interest you: The Bell Curve Performance Review System Is Actually Flawed and How To Really Understand The Mathematics Of Language?

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