The Belief of a Search System

… and doing the best despite uncertainties

Sunday, Jan 16, 2022

Some thoughts/aids/ideas that helped me understand some of the concepts like fundamental to so many things in Information Retrieval, as well as other areas.

So you’re a Search System. Everything you know is comprised in a Library of Knowledge full of scrolls and docs

The World.png

and your job is to find and give Seekers what they need based on what they tell you.

The Truth.png

You have a knack for finding things, and for doing what you’re told, so you imagine that for any need of knowledge that they might have, If your docs-world has the truth, in these docs, you’ll find and guide them to the Truth and nothing else.

The Truth Retrieved.png

In an ideal, rainbows-and-unicorns world.

Unfortunately for you, there is an interface the user’s need has to pass through before reaches you: which is how well they can express their thoughts.

All you can get is a query, an rough idea, of what their need is. With that, you have to guess your best to divide your docs world into what you think is correct and not.

And given how far off the query was from the need, your guess might be … well, a bit patchy.

The Guess.png

The Guess Retrieved.png

You decide to find at least know just how far off you really are, rather than leaving everything upto chance, so you can maybe find tricks to do better.

The Elusive Truth

But just how far your guess is from the real deal? You notice three main areas

The Extra Part.png

The Missed Part.png

And then of course,

The Correct Part.png

The Dilemma

There seem to be two ways to correct the error then. You don’t have control on the real part, just on the guess. So, you can,

But this can’t be done without a Seeker’s help.

So the next seeker that comes up to you and states his need, you give the docs and ask,

You: “Hey, so any thoughts about the results?”

Seeker: “Yeah they’re nice, though there are some wrong docs in here too.”

You: “I can think of a way to imporve them maybe. Could you try to be more specific, and maybe remove some words?”

The  Guess No Extra (no text).png

Seeker: “Yeah now all of these seem fine, there’s no wrong ones, and looks like there are quite a few missing”

The Guess No Extra.png

You: “Well, what if you added more words and were more general?”

The Guess Catching everything (no text).png

“Yeah now there’s everything I wanted, what are all these other things?”

The Guess Catching Everything.png


It seems there are two very different ways of doing well, each of which is good in one way, but bad in another.

And, you realize, unless the initial offset between query and need is cured, or you’re very lucky, you’re never going to do well on both.

The Types of Goodnesses

So it seems you need to track both these cases of your searches.

Two points strike you that might be good pointers:

Quantity: How well you were able to catch everything. It shows you correct things removed mistakenly.

That would basically be your catch vs everything correct.

That is, your correct guess ÷ truth


The other one is Quality: How well did you avoid being wrong. It shows wrong things added mistakenly.

Which would be your catch vs everything you got

That is, your correct guess ÷ guess


Not bad. The only trouble now is that you’re having some difficulty measuring all those areas.

“Finding things” is everywhere

To fix the squiggly shape area problem, you dumb it down removing unnecessary details, that makes it easier.

To Boxes.png

Hang on. This looks familiar.

It look just like something your friend, who is an Object Detection System, does.

Object Detection.png

You’re quite excited and show it to her. She’s very interested, but then points out that it’s actually not surprising, and in fact your measures are real and actually have names:

New Names.png

What’s more, there are even more names. The extra part, for example is called a False Postive. The missed part on the other hand is called a False Negative. And the correctly guessed part is called a True Positive.

FN TP FP.png

Woah, that’s neat.

You start to ask her how does she measure them, but them remember that she really just has all the box widths and pixel values which are pretty convenient.

But your corpus and guesses do not have any apparent shape …

… which actually might be a good thing. Maybe you don’t even have to stop at this box shape.

The Proportions

You keep trying things out, and then something hits you.

Maybe there’s no real reason for the guess and real blobs to be in the middle. (And let’s just increase the sizes so they’re easier to work with)

The Transformation.png

Ah, now everything is proportional to the big Docs Box. That should make measuring things a bit easier.

So Precision and Recall would now look like,


Working with proportions seems pretty convenient.

Say if there are 10% true docs, then we don’t have to know the actual number of docs.

Prior 10 percent.png

Also, you had heard somewhere that percents can be thought of as measuring the chance, or “probability”.

So if you just got 1 doc from the entire store, it would have a 10% chance of being correct.

To put in an another way, P(was true) = 10%

You decide to go even further. To see what happens when you guess.

Say the Seeker gives you a query, and 70% of correct docs satify it, along with 5% of the incorrect ones.

Percents After Guess.png

This means you know you got 70% of correct ones.

It sounds very familiar … wait isn’t that just Recall?

Also, 70% of correct ones is also just another way to say the chance that a doc was guessed, given it was true (limiting ourself to just true docs), is 70%.

That is, P(doc_was_got | doc_was_true) = 70%

Which means, all of these are equivalent:

Recall Calculation.png

So we have had our Recall with just proportions or rather, probabilities.

Can we do precision too in a similar way?

The Bayes Case

Precision Box diag.png

Precision looks like it is very similar. The correct docs (green part) are still in the focus, but here you compare it with respect to the guess (yellow) part rather than the true part.

So looks like here Precision would also be a probability, but a flipped one; it would say: “Given you got a doc, what is the chance it was true?”,

Which is P(doc_was_true | doc_was got)

But for that you need the areas of the green and yellow. Not a problem, you can find them with the percents you know.

Precision Components Areas.png

This means Precision is,

Precision Calculation.png

Oh, but wait.

We already know that the

In the same way,

Also, that denominator is just the entire guess, which is P(doc_was_got)

And so, you find Precision to be,

Precision Probabilities.png

Woo! Can’t wait to show this interpretation your friend, bet she hasn’t seen this one before.

You slowly begin to realize that whatever are any of those docs, even if you don’t know their exact count, just a rough ratio, or a chance how many they might be, you still have a way to know how good we’ll be.

Of course, everything still much upto chance. But something hits different.

It’s like you now know exactly just how much that chance is :)


Concepts: Information Retrieval (IR) by Chris Manning, Bayes Theorem by 3Blue1Brown

Style: Lauren Ipsum, Wait But Why, Vihart, Advent of Code, HPMoR

Based On: Precision and Recall as relevance measures in Search, and the Probabilistic approach to IR