Question on Bias Variance Tradeoff
In the book and the lecture, it is said that generally a larger hypothesis set (more complex model) has lower bias and higher variance. This is intuitively explained by the pictures on page 64. Bias is shown as distance of ḡ (gbar) from the target function f. Variance is illustrated with a shaded region around the target function f.
My question is: It appears from the picture that it should be possible to increase the hypothesis space in a way so that it does not include the target function f. E.g. If we include hypotheses in the direction "further away" from the target function f, then we may have managed to still keep the bias high (or even increase the bias). From this line of reasoning, it appears that adding complexity to a model or a larger hypothesis space does not necessarily imply a decrease in bias (and/or an increase in variance). The decrease in bias occurs only when the hypothesis space grows in a way so that ḡ ends up being closer to f. But this need not happen always (theoretically at least). Is this conclusion correct? If yes, then should it be kept in mind when applying these concepts in practice? Also if this is correct, then could you give some examples that can illustrate how adding complexity can still increase the bias. 
Re: Question on Bias Variance Tradeoff
Quote:
It is indeed possible to construct cases where you enlarge the hypothesis set without decreasing the bias. In practice, since the target function is unknown, the chances are enlarging the hypothesis set will result in including some hypotheses that are closer to the target, so the bias goes down. Now, if you want the bias to actually increase as a result of increasing the complexity of the hypothesis set, that cannot be done by enlarging the set, but by going for an alternative hypothesis set that is more complex and also further away from the target. For a specific construction, take the more complex hypothesis set as the set of all hypotheses excluding those that happen to be close to the target function. This will a very complex hypothesis set but still with a large bias, by construction. Of course that's a deliberately unfriendly hypothesis set that does not happen in practice. 
Re: Question on Bias Variance Tradeoff
Thanks for the quick reply. I have some follow up questions in regards to variance.
How does increasing bias artificially in this way (by choosing a pathological hypothesis space) affect the variance? Variance appears to only depend on ḡ and is independent of f. Perhaps it could be considered to indirectly depend of f to the extent that each g tries to approximate f. Thus is variance affected by whether ḡ is close to f or not? Is it possible to increase complexity/hypothesissetsize without increasing the variance? It is not obvious that this is not possible although the intuitive explanation is that a larger hypothesis set will result in a larger variance. 
Re: Question on Bias Variance Tradeoff
Quote:

All times are GMT 7. The time now is 02:57 PM. 
Powered by vBulletin® Version 3.8.3
Copyright ©2000  2020, Jelsoft Enterprises Ltd.
The contents of this forum are to be used ONLY by readers of the Learning From Data book by Yaser S. AbuMostafa, Malik MagdonIsmail, and HsuanTien Lin, and participants in the Learning From Data MOOC by Yaser S. AbuMostafa. No part of these contents is to be communicated or made accessible to ANY other person or entity.