Google's driverless cars use Bayesian methods.
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Google has a small fleet of robotic cars that since autumn have driven themselves for thousands of miles on the streets of northern California without once striking a pedestrian, running a stoplight or having to ask directions. The cars’ ability to analyze enormous quantities of data—from cameras, radar sensors, laser-range finders—lies in the 18th-century math theorem known as Bayes’ rule. The formula has survived decades of controversy and marginalization to emerge as the cornerstone of some of the most sophisticated robotics projects now under way around the world.
Discovered by English clergyman Thomas Bayes, the formula is a simple one-liner: Initial Beliefs + Recent Objective Data = A New and Improved Belief. A modern form comes from French mathematician Pierre-Simon Laplace, who, by recalculating the equation each time he got new data, could distinguish highly probable hypotheses from less valid ones. One of his applications involved explaining why slightly more boys than girls were born in Paris in the late 1700s. After collecting demographic data from around the world for 30 years, he concluded that the boy-girl ratio is universal to humankind and determined by biology.
Many theoretical statisticians over the years have assailed Bayesian methods as subjective. Yet decision makers insist that they bring clarity when information is scarce and outcomes uncertain. During the 1970s John Nicholson, the U.S. submarine fleet commander in the Mediterranean, used Bayesian computer analysis to figure out the most probable paths of Soviet nuclear subs. Today Bayesian math helps sort spam from e-mail, assess medical and homeland security risks and decode DNA, among other things.
Now Bayes is revolutionizing robotics, says Sebastian Thrun, director of Stanford University’s Artificial Intelligence Laboratory and Google’s driverless car project. By expressing all information in terms of probability distributions, Bayes can produce reliable estimates from scant and uncertain evidence.
Google’s driverless cars update information gleaned from maps with new road and traffic data from sensors mounted atop the vehicles. Google hopes that robotic cars will one day halve the number of road fatalities, cut energy consumption, fit more densely onto crowded roads and free commuters for more productive activities—like dreaming up even better ways to use a 250-year-old theorem.
This article was originally published with the title Why Bayes Rules.
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10 Comments
Add CommentOf course, Bayes' Rule only works when relevant data is taken seriously, and not rejected out of hand. It may be emotionally satisfying to say that cats (as an example) hate you when a specimen mouths your hand, but observation has shown that cats express affection by chewing your knuckles.
Reply | Report Abuse | Link to thisI assume that the full article actually gives the formula for Baye's Rule. It may be better, though, that it did not appear in the first paragraphs, since scientific evidence suggests that most people would misuse it.
Reply | Report Abuse | Link to thisHere is an explanation that gradually leads up to the formula:
http://yudkowsky.net/rational/bayes
That guy's explanation may or may not be good, but he's a nut. Never been formally educated, a fan of the Singularity and seems to believe in conspiracy theories. But hey, Bayesian Statistics is so simple his stuff is probably right on, right?
Reply | Report Abuse | Link to thisWhat is this "initial belief" thing? One cannot even HAVE an initial belief unless one has acquired a certain amount of data. Does a newborn have "beliefs"?
Reply | Report Abuse | Link to thisWhat this proposition is actually saying is simply, "Relevant data is required to make a prediction and the addition of more relevant data improves the odds of accurate prediction".
I am also very edgy about the use of the word "belief" in this context. It implies "an acceptance of something as true without sufficient evidence to support that conclusion".
This article confuses "Bayes rule" which is a
Reply | Report Abuse | Link to thistheorem in probability, and "Bayesian inference"
which is an approach to statistical inference.
It is only the latter that is controversial
because it takes a subjective view of probability.
"Bayes rule" is not controversial and is used all the time.
Bayes formula looks a lot like the scientific method and control theory used in electronics, both of which have also been proven useful. Any theoretical statistician who thinks that this sort of feedback system does not work needs to broaden their horizons a bit.
Reply | Report Abuse | Link to thisI am inclined to agree with scribblerlarry regarding the use of the word "belief" in this sort of scientific situation. Many people seem to have defined belief as being something you think as true without any supporting evidence. I think using words like "hypothesis" or "initial guess" would communicate what is going on better.
All feedback systems rely on improving their performance or accuracy on improved data. If any data which deviates from the original guess is ignored (like the person determined to "prove" that what he wants to believe is correct), then no improvement is possible. If the initial guess is revised by accurate data, then the accuracy of the initial guess is irrelevant. If you start by initially assuming that the best way to cool a house on a hot day is to turn on the furnace, it does not take long to figure out that turning on the AC works better.
Bayes works because the universe has laws of probability which we evolved to instinctively and intuitively rely on as our predictive logic.
Reply | Report Abuse | Link to thisWe can try now to describe it mathematically but we don't do it mathematically, we do it by mixing and matching a host of learned expectations that themselves contain subjective assessments of the probable.
Yet somehow we've used those laws of the probable to convince ourselves that's there's a certainty somewhere in their consistency. Inventing mathematics in the process based almost purely on that flawed assumption.
I want a self driving car. What does it take to get one? Think of the mobility enhancements for the infirm and elderly. Also they should be required for the sociopathic pinheads that can't be bothered to pay attention when driving.
Reply | Report Abuse | Link to thisFor those who would like to see Bayes Formula derived, you can go to an entry on my math blog at: brightstartutors.com/blog/2010/12/29/bayes-formula/
Reply | Report Abuse | Link to thisSebastian Thrun, a Stanford Computer Science Professor, was desciplined because of his involvement into multiple crimes in Stanford, including murder of an innocent student May Zhou from their own school in Stanford during their fight with authorities (see if he dare deny it to the public): Look-Inside-Dumbfounded [ http://tysurl.com/BsEnQ4 ]
Reply | Report Abuse | Link to thisHis bosses from Stanford Computer Science department, e.g. Professor Ed Feigenbaum, had constantly intruded my privacy and destroied all my private relations in this world which made me no one to trust for many years ( a very inhuman action ), and he had abused his power in scientific world to hype Thrun in name of Stanford Computer Sicence Department and AAAI.ORG in order to cover up Thrun in these crimes; Thrun's student, Mr. David Stavens, had lied to Stanford police on behalf of Thrun's side trying to prevent me from filing complain at authorities against their crimes.
--- A scandal unheard of in history of college education.