Increase performance of Bayesian likelihood equation I need to calculate a Bayesian likelihood in its negative logarithmic form:

$$-\ln L = N\ln M -\sum\limits_{i=1}^{N} \ln\left\{P_i \sum\limits_{j=1}^M \frac{\exp\left[ -\frac{1}{2}\left(\sum_{k=1}^D \frac{(f_{ik}-g_{jk})^2}{\sigma_{ik}^2 + \rho_{jk}^2}\right) \right]}{\prod_{k=1}^D \sqrt{\sigma_{ik}^2 + \rho_{jk}^2}} \right\}$$

where these $$P_i\;\;;\;\; f_{ik} \;\;;\;\; \sigma_{ik}^2\;\;;\;\; g_{jk} 
\;\;;\;\; \rho_{jk}^2$$ are all known floats. This is my code:

import numpy as np

# Define random data with correct shapes.
N, M = np.random.randint(10, 1000), np.random.randint(10, 1000)
D = np.random.randint(2, 5)
f_N, g_M = np.random.rand(N, D, 2), np.random.rand(M, D, 2)
P_i = np.random.uniform(0., 1., N)

# N summatory.    
# Store here each 'i' element from the M summatory.
sum_Mi = []
for f_i in f_N:

    # M summatory.
    sum_M_j = 0.
    for g_j in g_M:

        # Obtain ijk element inside the M summatory
        D_sum, sigma_ijk = 0., 1.
        for k, f_ik in enumerate(f_i):
            # Where: f_ik[0] = \f_{ik}       ; g_j[k][0] = \g_{jk}
            # and:   f_ik[1] = \sigma_{ik}^2 ; g_j[k][1] = \rho_{jk}^2
            D_sum += np.square(f_ik[0] - g_j[k][0]) / (f_ik[1] + g_j[k][1])
            sigma_ijk = sigma_ijk * (f_ik[1] + g_j[k][1])

        sum_M_j += np.exp(-0.5 * D_sum) / np.sqrt(sigma_ijk)

    sum_Mi.append(sum_M_j)

sum_Mi_Pi = P_i * sum_Mi
# Remove 0. elements before applying the logarithm below.
sum_N = sum_Mi_Pi[sum_Mi_Pi != 0.]

# Final negative logarithmic likelihood
lik = len(f_N) * np.log(len(g_M)) - sum(np.log(sum_N))
print(lik)


I'm after an optimization done at the code level, ie: not using multi-threading.

I've tried numpy broadcasting on the inner loop

# Obtain ijk element inside the M summatory
D_sum = sum(np.square((f_i[:, 0] - g_j[:, 0])) / (f_i[:, 1] + g_j[:, 1]))
sigma_ijk = np.prod(f_i[:, 1] + g_j[:, 1])


but that actually takes a longer time to complete than using the for loop!

I also tried using the logsumexp function from scipy combined with the broadcasted arrays:

# Store here each 'i' element from the M summatory.
sum_N = 0.
# N summatory.
for f_i in f_N:

    # M summatory.
    A, B = [], []
    for g_j in g_M:
        # Obtain ijk element inside the M summatory
        A.append(sum(
            np.square((f_i[:, 0] - g_j[:, 0])) / (f_i[:, 1] + g_j[:, 1])))
        B.append(np.prod(f_i[:, 1] + g_j[:, 1]))

    sum_N += logsumexp(-.5 * np.array(A), b=1. / np.sqrt(B))

# Final negative logarithmic likelihood
lik = len(f_N) * np.log(len(g_M)) - sum(np.log(P_i[P_i != 0.])) - sum_N
print(lik)


This resulted in a marginal improvement over just using broadcasting, but still slower than the original code.
This is much faster.  I added the seed to better compare results across runs.

import numpy as np

# Define random data with correct shapes.
np.random.seed(0)
N, M = np.random.randint(10, 1000), np.random.randint(10, 1000)
D = np.random.randint(2, 5)
f_N, g_M = np.random.rand(N, D, 2), np.random.rand(M, D, 2)
P_i = np.random.uniform(0., 1., N)

# replace loops with broadcasted array operation, followed by sums    
fgsum = f_N[:,None,:,1] + g_M[None,:,:,1]
fgdif = f_N[:,None,:,0] - g_M[None,:,:,0]
print(fgsum.shape)   # e.g. (694, 569, 3)

Dsum = (fgdif**2 / fgsum).sum(axis=-1)
sigma_ijk = np.prod(fgsum, axis=-1)
sum_M_j = np.exp(-0.5 * Dsum) / np.sqrt(sigma_ijk)
sum_Mi = np.sum(sum_M_j, axis=-1)

# as before
sum_Mi_Pi = P_i * sum_Mi
# Remove 0. elements before applying the logarithm below.
sum_N = sum_Mi_Pi[sum_Mi_Pi != 0.]

# Final negative logarithmic likelihood
lik = len(f_N) * np.log(len(g_M)) - sum(np.log(sum_N))
print(lik)