What is the role of 方 in 貽笑大方？ Consulting a dictionary on the meaning of 貽 makes the meaning of the expression rather clear. I would say “give (others cause for) great laughter”, “become laughingstock”. Yet I have trouble understanding 方。 The only function I can guess is that of a classifier. But there are two problems.

Firstly, it occurs in somewhat literary Chinese, which, I am given to understand, does not use classifiers. 


  清·李汝珍《鏡花緣》第17回：“才女才說學士大夫論及反切尚切瞠目無語，何況我們不過略知皮毛，豈敢亂談，貽笑大方！


On the other hand the use of 我 perhaps shows it is not really classical, but a hybrid literary Chinese?

A lesser problem is that none of my Pleco dictionaries （PLC, TL, CC, UNI) list 方 as a classifier for 笑。 Google search of 一方笑 doesn’t remotely support it as a classifier.
the 國語辭典 listed: 貽笑大方 is derived from 見笑大方

http://dict.revised.moe.edu.tw/cgi-bin/cbdic/gsweb.cgi?o=dcbdic&amp;searchid=Z00000093396

then 見笑大方 is derived from 莊子﹒秋水

http://ctext.org/zhuangzi/floods-of-autumn/zh


  吾長見笑於大方之家


it's a comment made by 河伯 (uncle river), when he reached the sea.

again, the 國語辭典 explained "大方之家" as "懂得大道理的人" (~ experts)

http://dict.revised.moe.edu.tw/cgi-bin/cbdic/gsweb.cgi?o=dcbdic&amp;searchid=Z00000040658

and "大方" as "有名的大家"

http://dict.revised.moe.edu.tw/cgi-bin/cbdic/gsweb.cgi?o=dcbdic&amp;searchid=Z00000040654

"方" is not a classifier in this usage. as i "google", some remarks of "方" in literatures of yore were "道" (tao, the proper way).