May be it’s because of my recent circumstances, but Traffic Flow Theory (informally the mathematics of traffic flow) has always interested me.  A monograph on this subject can be accessed here and the second page in Chapter 4 has some useful notations relating to this matter. Although mathematicians always enjoyed a prophetic ability of prediction- whether it be predicting an eclipse or the next perfect number- they cannot still harness their powers in plethora of avenues such as seismic events’ precognition.

Thus I ponder how much can we lasso the focus to scrawl an equation on blackboard that will enable predicting a car accident before it happens. Surprisingly, Google seemed to have a good match for the searches and it took me precisely to a link which has discussed this issue: The numbers behind NUMB3RS: solving crime with mathematics

Now admittedly, I am not a big fan of the show, simply because I feel that it has tried too hard to popularize the sober art of mathematics for lay audience. Not everything must be edutainment and while Scott Free productions has some excellent visuals,  it has very much watered down the mathematics insulting audience intelligence. But my intention was not a critique of the show and it seems there is an episode which aired titled Manhunt that dealt with Bayesian Inference.

As the afore-cited free link of Google Books shows that apparently 9/11 attack on the Pentagon was predicted on May 2001 via a software called Site Profiler utilizing Bayesian network (also known as belief network).  The term Bayesian is named after Reverend Thomas Bayes, a Presbyterian minister, who composed two works in his lifetime, the first one of which was on theology Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures (1731).

Bayes’ Theorem which has wide applications from Artificial Intelligence to drug testing, figuring out guilty party in taxicab accident (see here) to aiding entomologist identify a “rare subspecies of beetle” (frequentist example) can be best appreciated with the following example (Wikipedia):

If someone told you he had a nice conversation in the train, the probability it was a woman he spoke with is 50%. If he told you the person he spoke to was going to visit a quilt exhibition, it is far more likely than 50% it is a woman. Call W the event he spoke to a woman, and Q the event “a visitor of the quilt exhibition”. Then: P(W) = 0.50, but with the knowledge of Q the updated value is P(W | Q) that may be calculated with Bayes’ formula as: P(W|Q)=\frac{P(Q|W)P(W)}{P(Q)}=\frac{P(Q|W)P(W)}{P(Q|W)P(W)+P(Q|M)P(M)}

in which M (man) is the complement of W. As P(M) = P(W) = 0.5 and P(Q | W) > > P(Q | M), the updated value will be quite close to 1.

Returning to the original topic, can Bayesian inference from a mathematical model aid in assisting a car accident before it happens? Research does not predict accidents at will; they only reconstructs model of past accidents.  Journal of Transportation and Statistics has a paper by Marjan Simoncic of Slovenia titled A Bayesian Network Model of Two-Car Accidents along with Gary A. Davis’s Bayesian reconstruction of traffic accidents, the abstract of which has been reproduced below from Oxford Journal:

Traffic accident reconstruction has been defined as the effort to determine, from whatever evidence is available, how an accident happened. Traffic accident reconstruction can be treated as a problem in uncertain reasoning about a particular event, and developments in modeling uncertain reasoning for artificial intelligence can be applied to this problem. Physical principles can usually be used to develop a structural model of the accident and this model, together with an expert assessment of prior uncertainty regarding the accident’s initial conditions, can be represented as a Bayesian network. Posterior probabilities for the accident’s initial conditions, given evidence collected at the accident scene, can then be computed by updating the Bayesian network. Using a possible worlds semantics, truth conditions for counterfactual claims about the accident can be defined and used to rigorously implement a ‘but for’ test of whether or not a speed limit violation could be considered a cause of an accident. The logic of this approach is illustrated for a simplified version of a vehicle/pedestrian accident, and then the approach is applied to four actual accidents.

Clearly the question posed in the heading doesn’t seem that much in the realm of sci-fi as given by first impression. For an additional resource see the paper
PREDICTION UNDER BAYESIAN APPROACH OF CAR ACCIDENTS IN URBAN INTERSECTIONS

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