# Problems with zero probability events in Bayesian Networks

In the book Probabilistic Graphical Models: Principles And Techniques by Daphne Koller, the author at one place (Box 3c), states the challenges in picking probabilities for a Bayesian network model. One of the challenges stated, is in assigning zero probabilities to events. Am finding difficulty interpreting the author's statement. Am quoting the exact text of the section below.

Zero probabilities: A common mistake is to assign a probability of zero to an event that is extremely unlikely, but not impossible. The problem is that one can never condition away a zero probability, no matter how much evidence we get.

What is meant by "condition away"? Please clarify this above point.

The text following the passage you quote provides one concrete example. Box 3.D discusses a medical diagnosis system (diseases in lymph nodes) in which experts ascribed zero probability to certain events occurring instead of merely a very tiny probability. As a result, "10 percent of the cases were diagnosed incorrectly."

Fleshing out that example, imagine that in the 1990's, when this system was constructed, doctors almost never saw swollen lymph nodes resulting from vaccinations (I have no idea whether that's true at all, but just pretend). Let's say that in 1992, only .0001% cases of swollen lymph nodes were the result of vaccinations (again, fictionalizing). But the experts who created the Bayes net diagnostic system wrote off that .0001% as insignificant and just set the probability of swollen lymph nodes resulting from vaccination to exactly zero.

Now imagine that same Bayes net is still deployed in 2021 in a country where vaccinations are widespread and where the Covid vaccine is causing many women (but few men--this, I think, is true) to get swollen lymph nodes in the days after their vaccination. In 2021, most human physicians would have been well aware of several facts that would have dramatically increased a doctor's willingness to entertain recent vaccination as a possible cause of swollen nodes, especially when presented with a female patient:

(1) the pandemic resulted in a vastly larger segment of the general population receiving a vaccine in 2021 than was the case in 1992,

(2) the Covid vaccine was much more likely than other vaccines to cause swollen lymph nodes (again, let's pretend)

(3) women were more likely than men to get this particular side effect

Given those three facts, a human physician seeking to diagnose the cause of a female patient's swollen nodes in 2021 would have instantly thought to inquire whether that patient was recently vaccinated because when conditioning on those three facts alone, the chances of vaccination being a cause would have dramatically increased. If her answer were "yes," the probability of that vaccination being the cause would again increase dramatically, perhaps to over 99%. Each of the four facts known to the physician would increase the probability, and taken together those four facts could render a human physician extremely confident in a diagnosis--even if, in the absence of such facts, the probability would still have been a meager .0001%. The point is that conditioning on a combination of factors can alter the probability of a medical diagnosis by orders of magnitude.

If, however, the system initially ascribed 0% probability to such an event, such magnification becomes impossible. Regardless of how many additional pieces of evidence the system might condition on, a probability of zero will forever remain a probability of zero (because multiplying or dividing a zero by some other number can have no effect). By contrast, if one starts with a tiny probability of .0001% of a particular diagnosis (vaccination being the cause of the swollen nodes, as opposed to some other cause), one leaves the door open to eventually reaching 99% confidence in one's diagnosis. Whereas non-zero probabilities can be dramatically increased and decreased--"conditioned away"--a zero probability event remains immovable. As a result, an expert system might fail to even consider the most probable explanation for a patient's symptoms. This apparently happened to the system described in Box 3.D.

Presumably what they mean here is that for any observable data $$\mathbf{x}$$ and any hypothesis $$\mathcal{H}$$ you have:

$$\mathbb{P}(\mathcal{H}) = 0 \quad \quad \implies \quad \quad \mathbb{P}(\mathcal{H}|\mathbf{x}) = 0.$$

That is, if you assume that an event has prior probability zero then its posterior probability will still be zero.

• unfortunately I can accept only one answer. But this was equally helpful in clarifying my doubt as the one given by Bill Vander Lugt. Commented Sep 20, 2022 at 3:15