Monte Carlo Comparison (N = 100,000 samples)
mu_p
0.10
True KL
--
E[k1] = -E[log r]
--
E[k2] = E[1/2 (log r)^2]
--
Bias (E[k2] - KL)
--
Relative bias
--
Var(k1)
--
Var(k2)
--
q = N(0,1), p = N(mu_p, 1). k1 is unbiased; k2 is biased but always non-negative. The bias is small because k2 agrees with KL to second order near r=1 (same f''(1)=1).