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,

where $$k$$ is the number of occurrence and $$\lambda$$ is the expected number of occurrence. In 3 million miles travelled, the expected number of fatal crashes for non-AV is $$\lambda_{nAV} \approx 3/93 \approx 0.0323$$. The Uber AV would be safer if $$\lambda_{AV} < \lambda_{nAV}$$. 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,

The term $$P(k|\lambda_{AV})$$ is the Poisson distribution given in the equation $$\ref{eq:poisson-distribution}$$. 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

Putting the equation $$\ref{eq:prior-lambda}$$ to the equation $$\ref{eq:posterior-distribution}$$ with $$k = 1$$ gives us

where $$\Gamma(z)$$ 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 $$\lambda_{AV}$$, 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 $$\ref{eq:posterior-distribution2}$$, we can plot the posterior distribution of $$\lambda_{AV}$$ as shown in the figure below.

To calculate the likelihood it is safe, we can integrate the area under the curve for $$\lambda_{AV} < \lambda_{nAV}$$ with $$\lambda_{nAV}\approx 0.0323$$ from equation $$\ref{eq:posterior-distribution2}$$, which gives us

where $$\Gamma(z,x)$$ 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.