Some blind men, an elephant and estimation theory

Originally published on Ribbrish

 

I am going to start with a story here, perhaps you have heard it earlier, perhaps you haven’t. If you have, please bear with me. The story is about four blind men who wanted to know what an elephant was. They lived in a remote village and elephants were rather uncommon in that part of the country so they had never come across one. So one fine day these men were informed that an entourage had stopped by their village and travelling with this entourage was an elephant. A real bull elephant. Excited, our heroes rushed to the spot to finally know what an elephant was.

Confident, as they were of their sense of touch, each of them grabbed hold of a different part of the elephant in order to know it better. Suddenly, the man who had grabbed the legs of the beast exclaimed, “I get it, an elephant is a pillar of flesh”. The one holding the trunk shouted “No it is not, it is more like the snake we felt last year.” The third one, who was gripping the tail interjected, both of you are wrong, though the second one is almost right, but it is more like a rope than a snake.” The man holding the ear was completely baffled. To him the elephant was like a fan, a big huge automatic fan that could be a source of relief on hot day, if only he could make it move faster. “Liars” he shouted “it is a fan, an automatic fan. Can you imagine the market potential of this thing if we figure out how to make it move faster. Maybe you already know it, that is why you want to keep me out of the group.”

“It is a frigging snake you half-wit inbred bozo, it was here when you were having fun with your female relatives.” Shouted the second man at the top of his voice. Not wanting to be left out, the other two men also joined in. What followed can be seen in the comments section of any you-tube video or on the twitter timeline of anyone remotely famous.

At the end of this tale, the reader is left confused, weather to laugh at these men or pity them. They like most of us are unable and even incapable of perceiving the reality, and yet fully that their sense of judgement is far superior to everyone around them. Incapable of knowing the whole truth and yet ready to tear down anyone who disagrees with, let alone challenges their version of the absolute truth.

However, this process of trying to figure out an elephant just by the sense of touch (though I agree that the elephant should have much patience for this sort of thing) is not uncommon and has been given the name “estimation theory” by signal processing people. The estimation problem may simply be stated as the problem of approximating one or more unknown parameters from noisy measurements of a function of those parameters. In simpler words, it means that we try to figure out something we cannot measure directly from something that depends on it and can be measured directly. This is somewhat similar to the idea of estimating the degree of depravity of a guy by the amount of dowry that he asks for. One can always say that one who is greedy in one aspect of life would follow suit in other walks as well, so his demands from his future in laws can serve as a good estimate of his moral character. However, one cannot judge a man’s character by a dowry case registered against him or his family. That thing is a double edged sword.

However, coming back to the point estimation involves estimating a quantity, not directly measurable by the use of erroneous measurements of quantities depending upon itself. It follows that an error in these functions will simply result in an error in our estimate. This error is random and as it may occur due to a variety of reasons. In general, the magnitude of the error tends to reduce as we increase the number of independent observations being used to obtain the estimate. It is observed that the estimate improves as we increase the number of independent samples that contain some information about the parameter of interest. Specifically, the variance of the estimate is found to be inversely proportional to the number of independent unbiased observations about the phenomenon that also contain information about the phenomenon.

So, in general we can say that the more we read about some topic, the more fitting our onions regarding it will be. However, this comes with a catch. The above statement mentions independent unbiased observations containing information about the phenomenon. Both of these are quite rare to achieve practically. Independent means that, they should have nothing in common except for the fact that they contain nothing in common except for the information about that topic. Which is often not the case, much of the information that we receive from mainstream/ social media (including this post) comes from people with similar backgrounds and world views and one might not like to read something that is too different from his or her own perception of reality or too preachy (If you have heard Rajdeep Sardesai, Yogendra Yadav speaking you know what I mean.) The second thing is that the observations should be unbiased which is hardly the case with anything reported by a human. It seems to be a part of human nature to add personal views to facts for any event (Ref: And they hanged Yakub). Therefore, the collection of information from multiple sources may as well result in what we call a biased estimate of the truth. With the biases added consciously or unconsciously. However, those are the only observations that we have and we have to subtract those biases and minimize the variance of the estimator on our own.

However, there is one important point where estimation theory differs from human behaviour. Even in the presence of independent unbiased samples, it can only be said that the estimation error is limited to this much variance. No estimator claims to know the exact value of the underlying parameter, it only claims that the error made by it is minimized. Every estimator acknowledges the existence of an orthogonal error lying outside the dimensions (fields of perception) of that estimator. Humans however are too arrogant for such things.

Perhaps it is this limit of perception, that the seer in Isavasaya Upnishad refers to as the Golden Vessel when he says

हिरण्मयेन पात्रेण सत्यस्यापिहितं मुखम्। तत्त्वं पूषन्नपावृणु सत्य धर्माय दृष्टये॥

Meaning:

Truth lies concealed by a golden vessel.

O Sun! Open the entrance of that cover

So as the truth concealed be visible to me,

a devotee,who by nature is truthful.

Note : This finishes my rather long and boring series on rants based on signal processing. Thanks for bearing. Special Thanks to Kartik who first heard this nonsense.

Peace

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