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Bias + Variance Tradeoff
Bias + Variance Tradeoff
Question
Suppose you are trying to estimate the (unknown) probability density function of a random variable X which takes on values only within the interval [0,1], by drawing a sample of 100 points, making a histogram using these points using b equally sized bins over the interval [0,1], and finally normalizing so that total area under the histogram is 1.
(i) Argue that the choice of “b” represents a bias-variance tradeoff.
(ii) Will a solution using a small value of b tend to have high bias or high variance?
Answer
In the base case of b=1, there will be a large bias as the result is always a uniform pdf (all elements fall in [0,1]) but no variance as the result stays the same irrespective of the drawn sample. On the other extreme, as b tends to infinity (continuous distribution), there is no bias as the expected frequency of each value is exactly correct but very high variance as small variations in the sample distribution will yield different values in each bin.
Queen side counterattack against the Kings Indian
Queen side counterattack against the Kings Indian
To the left is one of my favorite games I've played in chess against one of my friends, specifically against the King's Indian defense. I've put some time into studying one counterattack against the King's Indian, based on pushing up the Queens Pawns. The main philosophy is to set up your bishops and knights and castle followed by doing the Queens side attack. I've played against some AIs and it has truly resulted in some beautiful games. I'll go into more detail below.
The Kings Indian is typically these moves 1. d4 NF6 2. c4 g6
There are variations such as bringing out the e pawn before the c pawn to e4 for white and bringing out the d pawn before the b pawn to d6 for black.
I've looked up some videos online that suggest that the in response to the Kings Indian, white can bring out the b pawn, both knights (connect the knights) and bishops, then the e pawn, and then castle... After that, they should begin a Queen's side attack. Below is how this can look
After this, white is supposed to push up the a-c pawns in a way that attacks black while also maintains center control.
One thing I decided to try doing against some friends was do start this Queen's side attack before actually setting anything up to see if I can catch the opponent off guard.
There are a few things that are problematic here. For instance, the dark squared bishop on the black side can take white's rook and/or knight if you're not careful with the d4 pawn across the main black diagnoal.
However, this is chess right!? So you should try and experiment with some wacky positions!
In this game, I pushed up the b4 pawn early and my friend responded by mostly ignoring it and focusing on the center.
Eventually, I interlocked the center pawns, pushed up my a pawn even more, let him attack on the Kings side, jumped my g knight around, let it get exchanged, allowing for my d pawn to make it to the e file. This allowed for a nice pawn fork against his Queen and rook allowing me to get his rook followed by me forking his Queen, forcing him to surrender...
The next few games he totally pummeled my early Queenside counter attack... but this one game I gotta beautiful win.
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