線性鏈條件隨機(jī)場(chǎng)
DEFINITION
%3D%5Cfrac%7B1%7D%7BZ(%5Cmathbf%7Bx%7D)%7D%20%5Cprod_%7Bt%3D1%7D%5E%7BT%7D%20%5Cexp%20%5Cleft%5C%7B%5Csum_%7Bk%3D1%7D%5E%7BK%7D%20%5Ctheta_%7Bk%7D%20f_%7Bk%7D%5Cleft(y_%7Bt%7D%2C%20y_%7Bt-1%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5Cright)%5Cright%5C%7D)
%3D%5Csum_%7B%5Cmathbf%7By%7D%7D%20%5Cprod_%7Bt%3D1%7D%5E%7BT%7D%20%5Cexp%20%5Cleft%5C%7B%5Csum_%7Bk%3D1%7D%5E%7BK%7D%20%5Ctheta_%7Bk%7D%20f_%7Bk%7D%5Cleft(y_%7Bt%7D%2C%20y_%7Bt-1%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5Cright)%5Cright%5C%7D)
ESTIMATION
損失函數(shù)為
%3D%5Csum_%7Bi%3D1%7D%5E%7BN%7D%20%5Clog%20p%5Cleft(%5Cmathbf%7By%7D%5E%7B(i)%7D%20%7C%20%5Cmathbf%7Bx%7D%5E%7B(i)%7D%20%3B%20%5Ctheta%5Cright))
即%3D%5Csum_%7Bi%3D1%7D%5E%7BN%7D%20%5Csum_%7Bt%3D1%7D%5E%7BT%7D%20%5Csum_%7Bk%3D1%7D%5E%7BK%7D%20%5Ctheta_%7Bk%7D%20f_%7Bk%7D%5Cleft(y_%7Bt%7D%5E%7B(i)%7D%2C%20y_%7Bt-1%7D%5E%7B(i)%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5E%7B(i)%7D%5Cright)-%5Csum_%7Bi%3D1%7D%5E%7BN%7D%20%5Clog%20Z%5Cleft(%5Cmathbf%7Bx%7D%5E%7B(i)%7D%5Cright))
對(duì)
求導(dǎo)結(jié)果為
%7D%2C%20y_%7Bt-1%7D%5E%7B(i)%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5E%7B(i)%7D%5Cright)%20%5C%5C%20%26-%5Csum_%7Bi%3D1%7D%5E%7BN%7D%20%5Csum_%7Bt%3D1%7D%5E%7BT%7D%20%5Csum_%7By%2C%20y%5E%7B%5Cprime%7D%7D%20f_%7Bk%7D%5Cleft(y%2C%20y%5E%7B%5Cprime%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5E%7B(i)%7D%5Cright)%20p%5Cleft(y%2C%20y%5E%7B%5Cprime%7D%20%7C%20%5Cmathbf%7Bx%7D%5E%7B(i)%7D%5Cright)%5Cend%7Baligned%7D)
關(guān)注一下%7D%5Cright))
是怎么得來(lái)的：
%7D%5Cright)%7D%7B%5Cpartial%20%5Ctheta_%7Bk%7D%7D%20%26%3D%20%5Cfrac%20%7B1%7D%7BZ%5Cleft(%5Cmathbf%7Bx%7D%5E%7B(i)%7D%5Cright)%7D%20%5Csum_%7B%5Cmathbf%7By%7D%7D%20%5Cfrac%7B%5Cpartial%20%5Cleft%5C%7B%20%5Cprod_%7Bt%3D1%7D%5E%7BT%7D%20%5Cexp%20%5Cleft%5C%7B%5Csum_%7Bk%3D1%7D%5E%7BK%7D%20%5Ctheta_%7Bk%7D%20f_%7Bk%7D%5Cleft(y_%7Bt%7D%2C%20y_%7Bt-1%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5Cright)%5Cright%5C%7D%20%5Cright%5C%7D%20%7D%20%7B%5Cpartial%20%5Ctheta_k%7D%20%5C%5C%20%26%3D%5Cfrac%20%7B1%7D%7BZ%5Cleft(%5Cmathbf%7Bx%7D%5E%7B(i)%7D%5Cright)%7D%20%5Csum_%7B%5Cmathbf%7By%7D%7D%20%5Cleft%5C%7B%20%5Cprod_%7Bt%3D1%7D%5E%7BT%7D%20%5Cexp%20%5Cleft%5C%7B%5Csum_%7Bk%3D1%7D%5E%7BK%7D%20%5Ctheta_%7Bk%7D%20f_%7Bk%7D%5Cleft(y_%7Bt%7D%2C%20y_%7Bt-1%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5Cright)%5Cright%5C%7D%20*%20%5Csum_%7Bt%3D1%7D%5E%7BT%7D%20f_%7Bk%7D%5Cleft(y_%7Bt%7D%2C%20y_%7Bt-1%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5Cright)%20%5Cright%5C%7D%20%5C%5C%20%26%3D%20%5Csum_%7B%5Cmathbf%7By%7D%7D%20%5Cleft%5C%7B%20p%5Cleft(%20%5Cmathbf%7By%7D%20%7C%20%5Cmathbf%7Bx%7D%5E%7B(i)%7D%5Cright)%20*%20%5Csum_%7Bt%3D1%7D%5E%7BT%7D%20f_%7Bk%7D%5Cleft(y_t%2C%20y_%7Bt-1%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5E%7B(i)%7D%5Cright)%20%5Cright%5C%7D%20%5C%5C%20%26%3D%20%5Csum_%7Bt%3D1%7D%5E%7BT%7D%20%5Csum_%7By%2C%20y%5E%7B%5Cprime%7D%7D%20f_%7Bk%7D%5Cleft(y%2C%20y%5E%7B%5Cprime%7D%2C%20%5Cmathbf%7Bx%7D_%7Bt%7D%5E%7B(i)%7D%5Cright)%20p%5Cleft(y%2C%20y%5E%7B%5Cprime%7D%20%7C%20%5Cmathbf%7Bx%7D%5E%7B(i)%7D%5Cright)%20%5Cend%7Baligned%7D)
解釋：
從第一行到第二行可以將累乘符號(hào)轉(zhuǎn)換為exp中的累加即得。
從第三行到第四行是將中括號(hào)內(nèi)部的求和符號(hào)放在外面去。然后進(jìn)行邊緣概率的計(jì)算。
理解清楚這里就可以利用prml里面的sum-product algorithm來(lái)進(jìn)行優(yōu)化求導(dǎo)。
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