Confusion about the equations in the NCE subsection (equation 4,5,6,7)

[FengWang]

Hi, it is a great work, with simple idea and effective results. But I am confused about the mathematical modeling of NCE in 3.2.

From equation (4), you define the p(i|v) which should be normalized along the random variable i instead of v, but equation (5) gives the normalization along random variable v, which is not constrained by that the summation of p(i|v) along i is equal to 1. Question also emerges in equation (6) and (7), in (6) you define p_n(i) however the f_i is fixed, you choose different v.

So according to my understanding, you should model p(v|i) instead of p(i|v), the normalization is correct and eq(6) should be p_n(v) so in eq(7), the negative samples (v in memory bank) are sampled from P_n, which corresponds your symbols.

I wonder if my formulation is correct, or i have misunderstood your modeling, I am appreciate if you give me a reply.

Thanks.