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From Tesla's statement: "contrasted against worldwide accident data, customers using Autopilot are statistically safer than those not using it at all"

I think this is a foolish statement: fatal accidents in the US are 1.3 per 100M miles driven. Tesla reports autopilot has driven 100M miles. That's not enough data collected to draw this kind of conclusion.

There's a parallel here with software testing. I've seen bugs that happen 25% of the time, for example, and take 10 minutes to run a test. Our test guys have great intentions, but if they test 5 times and see no bug, they think the bug is gone. There is no instinctive understanding that they have insufficient data to draw a conclusion.




I think this is a foolish statement: fatal accidents in the US are 1.3 per 100M miles driven. Tesla reports autopilot has driven 100M miles. That's not enough data collected to draw this kind of conclusion.

Also, not all miles are equal. I'd hope that people are enabling Autopilot primarily while driving on straightish roads with good driving conditions -- in other words, at times when the rate of fatalities with a human driver is far less than 1.3 per 100M miles.


I thought most fatalities are on the highway during good driving conditions, where the road is so smooth and boring that humans lose attention, fall asleep or staring at their phones. Speed is much higher when driving on those roads and there is a bigger chance a car will rollover/bigger impact.


On a per mile basis, they most definitely are not. Here's one set of US DOT numbers[0] from 2007 that show the rate on interstates is 0.7/100M miles while the rate is highest on collector roads at 1.99/100M miles followed closely by local roads at 1.94/100M miles. Interstates have less than half the fatalities on a per mile basis as local roads.

[0] http://safety.fhwa.dot.gov/speedmgt/data_facts/docs/fataltbl...


Surely per mile makes no sense, as you travel faster on a motorway?

The only measure that makes sense in this context is time, bringing distance into it is statistical trickery.


Absolutely not the case. If you're driving from A to B, you have to cover similar distance regardless of type of road. (Yeah not conpletely true.) It turns out that highways are both quicker and safer, but the two are not necessarily correlated.


That's not how it works.

If you expose yourself on a live fire range for 1 second and travel 100m/s, it's far less dangerous than exposing yourself on a live fire range for 100 seconds travelling at 1m/s.

It's the same thing with cars, the longer you are physically on the road timewise, the longer you are exposing yourself to the chance of an accident, regardless of distance travelled.

Not sure why you think it's "absolute", you haven't listed a single reason or argument, just a tautology.


The analogy is poor: on a live fire range the bullets move much more quickly than you. The speed of the bullets relative to you is largely determined by the bullets, and your speed is irrelevant - so how fast you move is naturally irrelevant (assuming random fire, of course). The chance of collision will be determined largely by the number of bullet-meters made. If you will: it's as if the bullets are exploring the space in a probabilistic/ballistic fashion - the more bullets and the faster they go, the more they can explore.

Conversely, when you hit something on the highway, it's likely to be moving much more slowly than you (e.g. the chicken crossing the road), or at a speed similar to you (the drunk guy driving on the left side). So your speed matters - you're more likely to reach that obstacle in any given period of time if you can explore more space in that time.


Correcting for time (I am making up an average speed): Highway 49.0/1Mhours @70mph Collector 99.5/1Mhours @50mph Local 77.6/1Mhours @40mph


That's not how roads work, the average speed of a road isn't simply the max speed of the road. Especially not in towns, compared to motorways, which will skew it heavily towards motorways.


Thanks for pointing this out, I was trying to figure out why my brain didn't like per mile for things moving at vastly different speeds.


When traveling, most people do so for a given distance, from hither to yon, not for a given period of time.


Again, not true, you fundamentally misunderstand the majority of car trips.

If my job is 2 hours away because there is no motorway, I will get a different job. If there is a motorway, making the commute 40mins, I will take that job.

People care how long the journey is, not how far the journey is.

People live by commute time, not commute distance. Same for service areas, if the roads all have low speed limits you would need more drivers, they wouldn't just drive for longer.


Personally when facing decision between highways, local roads and other means of transportation I almost always have a specific destination in mind. Accidents per mile is the statistic I want.

Accidents per hour of driving would be more useful for public policymakers – given fixed average commute 1 hour per day, what's the tradeoffs of building highways (worker mobility, commerce, pollution, landscape, accidents).


Out of curiosity, if the motorway is shut down due to an accident or construction, would you drive for 40 minutes and then pull over to the side of the road and work from there?


At least in Germany, driving on the Autobahn is about 3x to 4x safer than driving on other road types: https://en.wikipedia.org/wiki/Autobahn#Safety, not sure how it compares to the US, but "fast and boring roads" seem to be safer.


Getting a drivers license in Germany requires a lot more training (and money) than in the US. The people who do get licenses in Germany are generally better prepared. As a result, they get into less accidents.

https://en.wikipedia.org/wiki/Driving_licence_in_Germany


Germany is a unsuitable country for comparison, as they have many highways without speed limits. I would argue that this invalidates the boring part.


I think the missing speed limit isn't that big of a factor, as most traffic is still flowing at around the same not-so-high speed, maybe somewhere between 130 and 160 km/h (except for trucks on the right lane). It's the relative speed differences that matters, and there the differences on other roads where traffic is stopping and accelerating all the time, with other vehicles, pedestrians and cyclists crossing the road which makes urban roads so dangerous.


Seems you have never been to Germany ;-) No offense. Most highways have a speed limit only some have none.

And most of the highways that don't have a speed limit are quite narrow so while it is possible, no sane driver would do so.

If you look at highway A5 as an example. This highway has 8 lanes, which makes driving fast quite dull since you can not really feel the speed relative to the environment.


The figure I've seen is 50% of Autobahn is without speed limit, 25% with permanent limits (due to noise control, permanently high traffic volume or intersections) and 25% with temporary limits (depending on buildings works, weather, congestion). Of course the perception is probably worse because more people use the congested parts with speed limits, and due to the limits they spend more time on those parts...


It's the same in all European countries.


Germany is the only EU country having roads without speed limits - https://en.wikipedia.org/wiki/Speed_limit#Roads_without_spee... .

Outside of EU, the only other European location that would qualify would be Isle of Man (as referenced in that article).


And one road in the Northern Territory, Australia: https://en.wikipedia.org/wiki/Speed_limits_in_Australia


I don't think the Isle of Man has any dual carriageways, it certainly doesn't have any motorways.


I was referring to the highways being more safe than other road types.


In the UK the motorways (straight, multi-lane, higher speed limits, barriers) are the safest roads, in terms of death or serious injury per vehicle mile. The most dangerous roads are typically single carriage A roads that are twisty, contain blind junctions and so on.

For those familiar with the UK road network see: http://roadsafetyfoundation.org/media/32639/rrm_britain_2015...


I would agree that autopilot miles are not equal to overall miles. I was not able to find data on which are most risky. My guess is that most accidents happen on surface streets but most fatalities on highways.


I think most accidents are within a couple miles of home. It's so familiar people stop paying attention.


Is that most accidents, or most per mile driven? Because for many, quite a bit of driving is within a few miles from home, skewing the data.


Couple this with the fact that most people are still concentrating on the road while autopilot is in action. So in effect you still have a human driver, they are just not fully engaged with the cars controls.

I know when I tried my friend's auto-pilot that I was concentrating 10x more on the road than I would be normally. Obviously if I got more comfortable with the tech then I would be concentrating less but I can't imagine that I would ever stop paying attention unless I knew the system was fool-proof.


>most people are still concentrating on the road while autopilot is in action //

This must be harder than concentrating when you're actively engaged?


RAND conducted a study that criticized the statistical analysis that most autonomous vehicle companies are doing to try and prove their safety:

"Given that current traffic fatalities and injuries are rare events compared with vehicle miles traveled, we show that fully autonomous vehicles would have to be driven hundreds of millions of miles and sometimes hundreds of billions of miles to demonstrate their safety in terms of fatalities and injuries. Under even aggressive testing assumptions, existing fleets would take tens and sometimes hundreds of years to drive these miles — an impossible proposition if the aim is to demonstrate performance prior to releasing them for consumer use. Our findings demonstrate that developers of this technology and third-party testers cannot simply drive their way to safety. Instead, they will need to develop innovative methods of demonstrating safety and reliability."

http://www.rand.org/pubs/research_reports/RR1478.html


Google does a lot of driving in simulation, using data captured by real vehicles. They log all the raw sensor data on the real vehicles, so they can do a full playback. They do far more miles in simulation than they do in the real world. That's how they debug, and how they regression-test.

For near-misses, disconnects, poor decisions, and accidents, they replay the event in the simulator using the live data, and work on the software until that won't happen again. Analyzing problems which didn't rise to the level of an accident provides far more situations to analyze. They're not limited to just accidents.

See Chris Urmson's talk at SXSW, which has lots of playbacks of situations Google cars have encountered.[1]

[1] https://www.youtube.com/watch?v=Uj-rK8V-rik


RAND's point is true but irrelevant. As Tesla likes to point out, when you roll out to an entire fleet (rather than a few dozen test cars), you rack up hundreds of millions of miles almost overnight. 'Hundreds of millions' may sound like a lot, but Americans drive trillions of miles per year.


One of the points addressed in this paper is the degree of testing that already needs to be done in order to proclaim that these systems are safer than humans with statistical rigour. Tesla says that their systems are _statistically safer_ than human drivers, but there is simply not enough data to make this conclusion. I respectfully suggest that you read the paper.


I did read the RAND paper, when it came out months ago, and I double-checked the traffic accident per mile citation and the power calculation as well. Their point is irrelevant because their fleet size estimate is ludicrously small, and their statistics is a little dodgy as well: it should be a one-tailed test (since the important question is only if the Tesla is worse than a human driver), and if one wanted to debate the statistics, this is somewhere that Bayesian decision theory minimizing expected lives lost would be much more appropriate, and that approach would roll out self-driving cars well before a two-sided binomial test yielded p<0.05.


From the paper: "Therefore, at least for fatalities and injuries, test-driving alone cannot provide sufficient evidence for demonstrating autonomous vehicle safety."

Note that the number of crashes per 100 million miles is a lot bigger than the number of injuries. One would hope a statement about safety would look at all of the data.


I still can't believe that they are using statistically safer in an official response. It now have been widely discussed everywhere that 1 occurrence is not enough to draw any reliable statistical conclusion!

PS: Beside should one more fatalities occur soon, by they're own standard, Model S would become statistically more deadful. To much risk of backfire, this PR is madness...


That's not really true. Using a beta distribution model (i.e. treating each mile driven as a single event), we can say that with 99% probability the true accident rate is at least 1/1B miles and no more than 7/100M miles.

Before the accident occurs, these numbers were 1/100B miles and 5.3/100M miles.

If a second fatality occurred, those numbers would increase to 3.4/1B miles and 9.3/100M miles.

However, I'm pretty sure that their claims of safety are based on more than just this calculation: "...a wealth of internal data demonstrating safer, more predictable vehicle control performance..."

100M miles of driving is easily enough time to observe things like fewer near misses, greater distance between the Tesla and other cars, etc. In math terms, what Tesla is most likely doing is working off some model relating driving behavior with accident rates and observing that 1 single accident does not invalidate this model.


"How many miles would autonomous vehicles have to be driven to demonstrate that their failure rate is statistically significantly lower than the human driver failure rate? ... Suppose a fully autonomous vehicle fleet had a true fatality rate that was A=20% lower than the human driver fatality rate of 1.09 per 100 million miles, or 0.872 per 100 million miles. We apply Equation 7 to determine the number of miles that must be driven to demonstrate with 95% confidence that this difference is statistically significant ... It would take approximately 5 billion miles to demonstrate that difference. With a fleet of 100 autonomous vehicles test-driven 24 hours a day, 365 days a year at an average speed of 25 miles per hour, this would take about 225 years."

This is from http://www.rand.org/pubs/research_reports/RR1478.html

I'd love to hear any problems with RAND's analysis, because it'd be great for this to not be the case.


Their calculation is frequentist, but a quick Bayesian calculation gives similar numbers.


Hmmn. If you assume that increased adoption still leaves us with a similar cohort of drivers traveling on a similar collection of roads with similar driving habits, then this is statistically legitimate.

But each of those assumptions is problematic at best. It's a variant of the "Soldiers in Vietnam" problem. In 1961, the U.S. had 3,200 soldiers in Vietnam, and only 16 of them died. If straight extrapolation worked, you could infer that in 1968, when troop levels peaked at 536,000, we'd be looking at 2,680 deaths.

Actually, 16,899 American soldiers died in 1968. The mortality rate climbed 6x, because the war became quite different. American troops were exposed to all sorts of lethal situations that weren't part of the mix in 1961. Data is here http://www.archives.gov/research/military/vietnam-war/casual... http://www.americanwarlibrary.com/vietnam/vwatl.htm

In Tesla's case, it's likely that a lot of the test miles so far have been incurred on California freeways or their equivalent, where lanes are wide, markings are good, rain is rare and snow is unheard of. Start moving more of the traffic load to narrower roads where nastier weather is common, and you're adding stress factors. Demographics change, too. The high price tag of Tesla vehicles today means that most drivers are likely to be in their prime earning/wealth years (let's say 35 to 70.) If the technology becomes more ubiquitous, we'll need to see how it performs in the hands of people of teenagers and the elderly, too. Once again, the risks rise.


This could be true. If so, their secondary metrics (e.g. # of near misses, avg distance to other cars, etc) should rise as Tesla use expands out of CA.

Again, please note that all I'm saying is that the secondary metrics are the primary predictor here since they are statistically significsnf. The primary metric is just an order of magnitude check since we don't have enough data (note the width of my posterior) to do more than reject the secondary metrics if they are crazily off.


What are you basing that on? Looks like you've misplaced the decimals there someplace, because you're concluding that the likely rate is around 1/10 of the observed rate. Depending on your priors, you can get some difference of course, but that sounds absurd - can you more precisely explain the method you used to arrive at that conclusion?


Take a uniform prior for gamma = fatality probability | drive one mile. Use Bayes rule. Here is a tutorial:

https://www.chrisstucchio.com/blog/2013/bayesian_analysis_co...

Finding that the lower end of a credible interval close to zero is an order of magnitude below the top end is pretty common. It just means you need more data, that the beta hasn't approached a gaussian yet.


Some people, when their posterior doesn't match the facts, update their prior.


Are you asserting that the posterior I described doesn't match the facts? If so, can you explain?


More detail; you're assuming a binomial likelihood function? What exactly is "uniform prior for gamma" supposed to mean?


Please go read the tutorial I linked to. It explains exactly how this analysis works. Here's the code:

    In [1]: from scipy.stats import beta
    In [2]: from numpy import percentile
    In [3]: data = beta(1, 100e6+1).rvs(1e7)
    In [4]: percentile(data, 0.5), percentile(data, 50), percentile(data, 99.5)
    Out[4]: (5.0460240722165642e-11, 6.9295652915902601e-09, 5.3005747758871231e-08)


You're definitely messing something up somewhere in that analysis. Where's that numpy/betadistribution snippet coming from?

Back to basics: If you assume that each mile is uncorrelated, (on the large not a crazy assumption), then the overall number of accidents over N miles can be modeled by a binomial distribution with N trials and some crash probability p. You're trying to use bayesian analysis to estimate that p.

The core concept there is that the probability of the observation given a certain p is directly proportional to the probability of the p given a certain observation.

Given p, it's trivial to compute the chance of the observation; just plug it into the pmf. For exactly one crash (and a small p), you can even quite accurately approximate it by Np/exp(Np) due to the trivial binomial coefficient for k=1 and the taylor expansion of the base-near-1 power.

So regardless of the numerical analysis that lead you to believe otherwise: you can just plug in numbers can get a numeric approximation of the (scaled) distribution of the probability mass for p. When I do that, both with an exponential prior (i.e. p in 0.1-0.01 is as likely as p in 0.01-0.001) or a uniform prior (i.e. p in 0.1-0.2 is as likely as p in 0.2-0.3) I get the unsurprising result that the maximum likelihood estimation for p is 1/N, and that the median p is a little lower; at around 1 accident in 1.5e8 miles

In practice that means that common sense is correct: if you observe 1 accident in N miles, a reasonable approximation of the best guess of the true accident rate per mile is 1/N.

Now, you can do this analytically, and the conjugate prior of the binomial is indeed a beta distribution. But that forces you to pick parameters for the prior from that beta distribution, and it's easy to do that wrongly.

A reasonable prior might be Beta[1,3] which assumes p is likely closer to 0 than 1. Analytically [then](http://www.johndcook.com/blog/conjugate_prior_diagram/), the posterior distribution is Beta[1+1,3+1e8-1]; with a mean p of 2e-8 (but note that the details of the chosen parameters definitely impact the outcome). This is quite close to the numerically derived p, though probably less accurate since it is constrained to a unreasonable prior.

So I'm not sure exactly what your python script is computing, but commonsense, numerical analysis, and Bayesian analysis all arrive at roughly 1/N in my case - you probably made some mistake in your reasoning somewhere (if you elaborate in detail how you arrive at this numpy snippet, maybe we'll find the error).


Whoops, you are right, my posterior should have been Beta[2,100e6+1] (I chose a uniform prior). With that I get a median probability of 1.7e-8, and a mean probability of 2e-8. Good catch!

Note that this is not particularly sensitive to whether you choose beta[1,1] or Beta[1,3] as the prior.


Yeah, that makes virtually no difference. I'm not familiar with picking these priors, so I could imagine picking (say) Beta[0.5,1] or Beta[5,30] as prior too, and that does make some difference (as in the prior alpha has some impact).

Is there some principled reason to pick one over the other, or is it just "whatever looks reasonable"?

I also can't explain why plain numerical simulation doesn't come up with almost identical p values - I get 1/1.5e8 to 1/1.7e8, but that's 6.25e-9. I mean, it's the same order of magnitude, but it's nevertheless quite different.

Oh well, unless you've got some insight there, I guess I'm going to give up on that analysis - that kind of nitty gritty stuff is a timesink ;-).


That they are using it in official statements should tell you exactly how intelligent Tesla is being with self-driving technology.

This is also the company that thinks level 4 self-driving can be done without LIDAR. They're idiots, frankly, completely ignoring the difficulties of computer vision (both in computation and optics) that will not be overcome anytime soon. Even their fatality seems to have been rooted in their camera's inability to distinguish the color of the truck from the color of the sky.

Tesla hasn't even taught a car to navigate an intersection on its own, much less all the things Google's car is capable of these days.

I think what all this shows is the importance of regulators getting off their asses and establishing standards for self-driving vehicles. Perhaps the fatality was the event that will get them in gear. Poor guy.


Probably because "Our cars are super safe, if in perfect weather conditions, otherwise you are on your own", wouldn't sell very well outside of California.


> I still can't believe that they are using statistically safer in an official response. It now have been widely discussed everywhere that 1 occurrence is not enough to draw any reliable statistical conclusion!

They probably mean that 1 occurrence is not enough to invalidate a statistical model built on wealth of other data.


Except that the other data is certainly suspect. I mean - what other source of realistic data can they possibly have that is realistic enough and large enough to be more significant that 100000000 miles of actual usage? At this level, it's the odd, unpredictable events that are going to get you, and whatever models they have cannot include all of those.


I was wondering about the 1.3/100m number and found this:

http://www-fars.nhtsa.dot.gov/Main/index.aspx

It has data from 1994 to 2014, including this plus several other statistics. In particular, it looks like they are tracking on the order of trillions of miles driven per year, so you're right, making any kind of statement after only 100 million is more shameful marketing than anything else.

Edit: this is US-only, while Tesla is claiming vs. worldwide. Clearly more miles driven worldwide, and probably higher deaths per 100m miles.


Not only is the statistics a big issue, but the apples-to-oranges comparison is what really irks me. The NHTSA average is over all car models on the road. Even in 2014, a non-negligible fraction of those are cars without even decent airbags and crumple zones.

The proper comparison would be with non-Autopilot Tesla miles. But that comparison makes Autopilot look bad.


It also seems like they should control for the demographic subset of riders that own Teslas. It seems possible that affluent car owners have factors at play that affect the likelihood of a fatal accident vis-a-vis the global average.


Just compare Teslas fatalities per mile with and without autopilot.

Remove the mileage driven from the cars that have never used autopilot, to control for drivers too cautious to trust their lives to beta software, and you should have your demographic.


In fact, there is a study which suggests that affluent car owners drive more recklessly than not so wealthy ones. Here's an article about it: http://usa.streetsblog.org/2013/07/16/study-wealthier-motori... I remember reading a more detailed article about the study somewhere but I can't find it right now.


Also autopilot can only be used in certain scenarios (freeway driving). Are hours driven by autopilot in this ideal environment being compared with millions of hours spent driving in non-autopilot-eligible roads/conditions?


Autopilot _should_ only be used on divided highways. However in practice it can be activated on most roads as long as there are lane markings. It's up to the driver to assess the driving conditions and decide whether it's safe to activate Autopilot.


If you torture the data long enough it will confess


It's not a fooling statement, it's a savvy political statement by a company's PR department that cares about self-interest.

You raise whether there is statistical significance about which is safer, autopilot or a typical driver. However it's worth noting that in a lawsuit the burden of proof is on the plaintiff.


Isn't it "balance of probabilities" for civil cases?


It's doubly foolish because the rates for autopilot are for human plus autopilot, not autopilot alone.

So at best they matched what a human alone can do. Meaning the autopilot has done nothing at all.


Not only that using worldwide data is being very disingenuous. Basically zero Teslas are driven in the places with the highest vehicle fatality rates.

https://en.wikipedia.org/wiki/List_of_countries_by_traffic-r...

>Seventy-four per cent of road traffic deaths occur in middle-income countries, which account for 70% of the world’s population, but only 53% of the world’s registered vehicles, burdens for 74% of world's road deaths. In low-income countries it is even worse. Only one percent of the world's registered cars produce 16% of world's road traffic deaths.


from the article: "...with zero confirmed fatalities and a wealth of internal data demonstrating safer, more predictable vehicle control performance" and "That contrasted against worldwide accident data, customers using Autopilot are statistically safer than those not using it at all." (emphasis mine)

i also stumbled over this statement, but i think they are using all telemetry data (e.g. what would the autopilot have done vs what the human actually did) in all accidents, not just fatalities per miles driven.


You can use the same argument when Tesla has 10000M miles driven. Fatal accidents in the US are 130 per 10000M miles driven.


Not if you understand statistics.


Maybe a nitpick, but if your bug happens 25% of the time, then you haven't yet found the root cause. Until you do so, no amount of data will help you draw a conclusion.


If I say "I have found and understood the bug", we still have to re-test to verify that the bug is not still present. Assuming that we are looking for a bug which, based on a race condition, occurs 25% of the time...




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