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Among other things, he was known for Fano's Inequality, which gives a limit for how well you can predict the value of a random variable based on knowledge of a

Among other things, he was known for Fano's Inequality, which gives a limit for how well you can predict the value of a random variable based on knowledge of a related variable. Apparently this is quite useful in signal processing for communications over a noisy channel, where the signal that you receive is not quite the same as what was sent. https://en.wikipedia.org/wiki/Fano%27s_inequality https://en.wikipedia.org/wiki/Fano%27s_inequality

hyperbovine 10y ago
Very useful in statistics too: if you can show that your estimator attains the Fano lower bound rate of convergence then it's optimal--nothing else can possibly beat it (in the minimax sense anyways).
- thearn4 10y ago
  Though I'm not a stats researcher, that sounds very similar to the Cramer Rao bound (a lower limit on the variance of any unbiased estimator, something akin to the Cauchy-Schwartz inequality only for point estimation).
  - gajjanag 10y ago
    That is good intuition, but note that one of the general drawbacks of Cramer Rao is that it applies for the unbiased case only (technically no, there is a version for the biased case, I have never seen it used though), and that too only well under certain regularity conditions. Indeed, there is a term for estimators where equality is met in CR, they are called "efficient" estimators. There is significant research effort these days on going beyond unbiasedness and asymptotics, which in many cases has been beaten to death and is "classical", though there are still plenty of questions. Here is a simple illustration of unbiasedness/going beyond it. Consider n iid (independent and identically distributed) coin flips of coin of P(heads) = p, and based on the observations, we want to estimate p, call the estimate p'. Optimum estimate in the sense of minimax - min (over estimators) max (over p in [0, 1]) E[(p-p')^2] is not the simple maximum likelihood which is the sample average. Sample average yields a score of (1/4)(1/n), while optimum is (1/4)(1/(sqrt(n)+1)^2), which is slightly lower. It is natural to wonder why people care, typically the motivation is in high dim statistics where there is dependence on the dimension as well. In such a case fine grained characterization of performance sometimes affects the scaling in terms of the target loss amount and dimension, i.e the asymptotics now has at least 2 parameters, sometimes more. My personal unhappiness with these problem formulations is their lack of robustness to transformations - suppose I now want to estimate p^3 instead of p, the optimum min-max will change in most cases, and definitely does here. Plugging in the sample average and then cubing is far more intuitive, and often works fine even though it is not optimal. There are more extreme examples of this, but that will take the thread too far afield. If one is interested in the topic of biased estimators, I suggest reading about James-Stein estimators and more generally shrinkage estimators.
    - thearn4 10y ago
      Good read, I've been meaning to make more time for reading up on the state of biased-but-low-variance estimation.
- gajjanag 10y ago
  There is a beautiful paper by Bin Yu on three common methods for getting minimax lower bounds on risk title "Assouad, Fano, and Le Cam" - https://link.springer.com/chapter/10.1007/978-1-4612-1880-7_29 https://link.springer.com/chapter/10.1007/978-1-4612-1880-7_....
adiM 10y ago
Surprisingly, the article on Fano's inequality on Scholarpedia is written by Fano! http://www.scholarpedia.org/article/Fano's_inequality http://www.scholarpedia.org/article/Fano's_inequality
- vanderZwan 10y ago
  > Robert Mario Fano (2008), Scholarpedia, 3(10):6648. So he was 90 years old when he wrote that. That is really cool, especially in the context of this section of the obituary: > In many respects, Fano was one of the world’s first open-source advocates. He frequently described computing as a public utility that, like water or electricity, should be accessible to all. His writings in the 1960s often discussed computing’s place in society, and predated today’s debates about the ethical implications of technology. > “One must consider the security of a system that may hold in its mass memory detailed information on individuals and organizations,” he wrote in a 1966 paper he co-authored with Corbató. “How will access to the utility be controlled? Who will regulate its use?”

hyperbovine 10y ago

Very useful in statistics too: if you can show that your estimator attains the Fano lower bound rate of convergence then it's optimal--nothing else can possibly beat it (in the minimax sense anyways).

thearn4 10y ago
Though I'm not a stats researcher, that sounds very similar to the Cramer Rao bound (a lower limit on the variance of any unbiased estimator, something akin to the Cauchy-Schwartz inequality only for point estimation).
- gajjanag 10y ago
  That is good intuition, but note that one of the general drawbacks of Cramer Rao is that it applies for the unbiased case only (technically no, there is a version for the biased case, I have never seen it used though), and that too only well under certain regularity conditions. Indeed, there is a term for estimators where equality is met in CR, they are called "efficient" estimators. There is significant research effort these days on going beyond unbiasedness and asymptotics, which in many cases has been beaten to death and is "classical", though there are still plenty of questions. Here is a simple illustration of unbiasedness/going beyond it. Consider n iid (independent and identically distributed) coin flips of coin of P(heads) = p, and based on the observations, we want to estimate p, call the estimate p'. Optimum estimate in the sense of minimax - min (over estimators) max (over p in [0, 1]) E[(p-p')^2] is not the simple maximum likelihood which is the sample average. Sample average yields a score of (1/4)(1/n), while optimum is (1/4)(1/(sqrt(n)+1)^2), which is slightly lower. It is natural to wonder why people care, typically the motivation is in high dim statistics where there is dependence on the dimension as well. In such a case fine grained characterization of performance sometimes affects the scaling in terms of the target loss amount and dimension, i.e the asymptotics now has at least 2 parameters, sometimes more. My personal unhappiness with these problem formulations is their lack of robustness to transformations - suppose I now want to estimate p^3 instead of p, the optimum min-max will change in most cases, and definitely does here. Plugging in the sample average and then cubing is far more intuitive, and often works fine even though it is not optimal. There are more extreme examples of this, but that will take the thread too far afield. If one is interested in the topic of biased estimators, I suggest reading about James-Stein estimators and more generally shrinkage estimators.
  - thearn4 10y ago
    Good read, I've been meaning to make more time for reading up on the state of biased-but-low-variance estimation.
gajjanag 10y ago
There is a beautiful paper by Bin Yu on three common methods for getting minimax lower bounds on risk title "Assouad, Fano, and Le Cam" - https://link.springer.com/chapter/10.1007/978-1-4612-1880-7_29 https://link.springer.com/chapter/10.1007/978-1-4612-1880-7_....

adiM 10y ago

Surprisingly, the article on Fano's inequality on Scholarpedia is written by Fano! http://www.scholarpedia.org/article/Fano's_inequality http://www.scholarpedia.org/article/Fano's_inequality

vanderZwan 10y ago
> Robert Mario Fano (2008), Scholarpedia, 3(10):6648. So he was 90 years old when he wrote that. That is really cool, especially in the context of this section of the obituary: > In many respects, Fano was one of the world’s first open-source advocates. He frequently described computing as a public utility that, like water or electricity, should be accessible to all. His writings in the 1960s often discussed computing’s place in society, and predated today’s debates about the ethical implications of technology. > “One must consider the security of a system that may hold in its mass memory detailed information on individuals and organizations,” he wrote in a 1966 paper he co-authored with Corbató. “How will access to the utility be controlled? Who will regulate its use?”