Last week there was a tragic news of death by an Uber’s self-driving car. According to this news and the fatality report by IIHS, some estimated the probability of the crash happened if Uber’s autonomous vehicles (AV) are as safe as non-AV using negative exponential distribution. The answer is around 3\%, which can also happen by bad luck. Specifically, from the IIHS data, it was obtained there was 1 fatal crash for every 93 million miles travelled by non-AV cars (i.e. 34,439 fatal crashes in 3,220,667 million miles in the US). The author also extrapolated from a report by the time the crash happened (i.e. last week), Uber’s AV would have collected 3 million miles.

Using the same data, my question is slightly different, “how likely are Uber’s AV safer than non-AV on average?” To answer the question, we can use the Poisson distribution,

P(k|λ)=λkeλk!

where k is the number of occurrence and λ is the expected number of occurrence. In 3 million miles travelled, the expected number of fatal crashes for non-AV is λnAV3/930.0323. The Uber AV would be safer if λAV<λnAV. Given the information that there is k=1 fatal crash in 3 million miles for Uber AV, we can infer the expected number of occurrence with Bayesian inference,

P(λAV|k)=P(k|λAV)P(λAV)0P(k|λAV)P(λAV) dλAV.

The term P(k|λAV) is the Poisson distribution given in the equation 1. The prior distribution can take different forms to capture our prior belief on how safe the AV is. As a general form, we can take the prior distribution to be

P(λAV)λpAV.

Putting the equation 3 to the equation 2 with k=1 gives us

P(λAV|k=1)=λ1+pAVeλAVΓ(p+2)

where Γ(z) is the gamma function.

Let’s take 3 forms of prior distributions: (1) uniform, p=0, (2) log-uniform, p=1, and (3) the Jeffreys prior for Poisson distribution, p=0.5. The log-uniform and Jeffreys prior put a lot of belief of small λAV, which assumes the AV tends to be safe. Here is the plot of all prior distributions mentioned.

By substituting the values of p to the posterior distribution equation 4, we can plot the posterior distribution of λAV as shown in the figure below.

To calculate the likelihood it is safe, we can integrate the area under the curve for λAV<λnAV with λnAV0.0323 from equation 4, which gives us

L(λAV<λnAV)=λnAV0P(λAV|k=1) dλAV=1Γ(p+2,λnAV)Γ(p+2)

where Γ(z,x) is the incomplete gamma function.

For uniform (p=0), log-uniform (p=1), and Jeffreys (p=0.5) priors, the likelihood of Uber AV being safer than non-AV respectively are 0.00051, 0.032, and 0.0043. From these calculation, we can see even if we have strong prior that the Uber AV is safer (i.e. log-uniform prior), there still a small chance 3.2% of the Uber AV is now safer than non-AV. Personally I would prefer Jeffreys prior as it is invariant under re-parameterization, so I belief that only miniscule chance, 0.43%, that Uber AV is safer, which means the non-AV is almost certainly safer than Uber AV, for now. I believe (and hope) Uber will improve to reduce the expected number of fatal crashes in the future.