chat_sample_multidim_integral

# To apply importance sampling correctly, we need to account for the fact that the
# importance distribution q(x) is not exactly the same as the integrand's distribution.

# We will use lambda = 1 for the exponential distribution q(x).
lambda_param = 1.0

# Compute the importance sampling weights
# Since the joint pdf of the exponential distribution for 10 independent variables is
# q(x_1, ..., x_10) = lambda^10 * exp(-lambda * (x_1 + ... + x_10)), the weights would be
# the function we want to integrate divided by this q(x).
# However, because we're sampling from the exponential distribution itself, we can ignore the q(x) in the denominator.

# Now, compute the f(x)/q(x) for each sample
weights = np.exp(-1/(2**20 * np.prod(samples, axis=1))) * \
          np.prod(samples ** (np.arange(1, 11)/11 - 1), axis=1) * \
          (lambda_param ** 10)

# Estimate the integral as the mean of the weights
integral_estimate = np.mean(weights)

integral_estimate