For-Loop with numpy array too slow I'm trying to speed up the execution of the following problem:

I have two functions which are used together.

Here are the functions explained:

The first function simulates N coin flips and returns an array of strings with the results:

def throw_a_coin(N):
    return np.random.choice(['H','T'], size=N)


The second function takes in number_of_samples, sample_size
and returns sample_probs.
The function simulates number_of_samples runs of sample_size coin flips and stores the result of num_heads / sample size in sample_probs for each run.

Here is the code:

def make_throws(number_of_samples, sample_size):
    sample_probs = np.zeros(number_of_samples)
    for i in range(number_of_samples):
        num_heads = sum(throw_a_coin(sample_size) == "H")
        sample_probs[i] = float(num_heads) / sample_size
    return sample_probs


Now I want to simulate multiple runs with different sample_sizes and 200 runs for each sample size. So sample_size = 1 I would do 200 runs of 1 coin flip, for sample_size = 10 I would do 200 runs of 10 coin flips and so on.

Here is the code that takes so long:

mean_of_sample_means = np.zeros(len(sample_sizes))
std_dev_of_sample_means = np.zeros(len(sample_sizes))

for i in range(len(sample_sizes)):
    prob = make_throws(200, sample_sizes[i])
    mean_of_sample_means[i] = np.mean(prob)
    std_dev_of_sample_means[i] = np.std(prob)   


I am pretty sure that this process can be improved by getting rid of the for loops and using array-operations instead. But I just cannot think of how to apply throw_a_coin or make_throws to an array.
My toy example I build from your snippets:

import numpy as np


def throw_a_coin(N):
    return np.random.choice(['H','T'], size=N)


def make_throws(number_of_samples, sample_size):
    sample_probs = np.zeros(number_of_samples)
    for i in range(number_of_samples):
        num_heads = sum(throw_a_coin(sample_size) == "H")
        sample_probs[i] = float(num_heads) / sample_size
    return sample_probs

sample_sizes = [np.random.randint(1, 1e3) for idx in range(500)]

mean_of_sample_means = np.zeros(len(sample_sizes))
std_dev_of_sample_means = np.zeros(len(sample_sizes))

for i in range(len(sample_sizes)):
    prob = make_throws(200, sample_sizes[i])
    mean_of_sample_means[i] = np.mean(prob)
    std_dev_of_sample_means[i] = np.std(prob)  


cProfile (ordered by cumtime) reveals the problem (runs in 82.359 seconds):

ncalls  tottime  percall  cumtime  percall filename:lineno(function)
 133/1    0.003    0.000   82.359   82.359 {built-in method builtins.exec}
     1    0.002    0.002   82.359   82.359 fast_flip.py:1(&lt;module&gt;)
   500    0.940    0.002   82.200    0.164 fast_flip.py:8(make_throws)
100000   79.111    0.001   79.111    0.001 {built-in method builtins.sum}
100000    0.058    0.000    2.148    0.000 fast_flip.py:4(throw_a_coin)
100000    1.455    0.000    2.090    0.000 {method 'choice' of 'mtrand.RandomState' objects}
100000    0.207    0.000    0.635    0.000 fromnumeric.py:2456(prod)
100000    0.026    0.000    0.429    0.000 _methods.py:34(_prod)
101500    0.409    0.000    0.409    0.000 {method 'reduce' of 'numpy.ufunc' objects}
     6    0.000    0.000    0.176    0.029 __init__.py:1(&lt;module&gt;)


There is a steep gap after buildins.sum, i.e. you spend most time there. We can use np.sum instead (pushing it down to 3.457 seconds):

ncalls  tottime  percall  cumtime  percall filename:lineno(function)
 133/1    0.003    0.000    3.457    3.457 {built-in method builtins.exec}
     1    0.002    0.002    3.457    3.457 fast_flip.py:1(&lt;module&gt;)
   500    0.905    0.002    3.307    0.007 fast_flip.py:8(make_throws)
100000    0.046    0.000    1.869    0.000 fast_flip.py:4(throw_a_coin)
100000    1.287    0.000    1.823    0.000 {method 'choice' of 'mtrand.RandomState' objects}
201500    0.702    0.000    0.702    0.000 {method 'reduce' of 'numpy.ufunc' objects}
100000    0.172    0.000    0.536    0.000 fromnumeric.py:2456(prod)
100000    0.136    0.000    0.532    0.000 fromnumeric.py:1778(sum)
100000    0.023    0.000    0.378    0.000 _methods.py:31(_sum)
100000    0.021    0.000    0.364    0.000 _methods.py:34(_prod)
     6    0.000    0.000    0.178    0.030 __init__.py:1(&lt;module&gt;)


further we can replace the strings "H" and "T" with a Boolean and stay in numpy for longer (down to 1.633 seconds):

ncalls  tottime  percall  cumtime  percall filename:lineno(function)
 133/1    0.003    0.000    1.633    1.633 {built-in method builtins.exec}
     1    0.001    0.001    1.633    1.633 fast_flip.py:1(&lt;module&gt;)
   500    0.101    0.000    1.485    0.003 fast_flip.py:11(make_throws)
100000    0.169    0.000    0.893    0.000 fast_flip.py:4(throw_a_coin)
100000    0.724    0.000    0.724    0.000 {method 'uniform' of 'mtrand.RandomState' objects}
100000    0.122    0.000    0.491    0.000 fromnumeric.py:1778(sum)
100000    0.024    0.000    0.354    0.000 _methods.py:31(_sum)
101500    0.334    0.000    0.334    0.000 {method 'reduce' of 'numpy.ufunc' objects}
     6    0.000    0.000    0.178    0.030 __init__.py:1(&lt;module&gt;)


next we can get rid of throw_a_coin and instead sample a numer_of_samples x sample_size array of uniformly distributed random numbers and threshold them. This also allows us to vectorize the for loop and stay in numpy even longer (0.786 seconds):

ncalls  tottime  percall  cumtime  percall filename:lineno(function)
 133/1    0.003    0.000    0.786    0.786 {built-in method builtins.exec}
     1    0.003    0.003    0.786    0.786 fast_flip.py:1(&lt;module&gt;)
   500    0.053    0.000    0.634    0.001 fast_flip.py:4(make_throws)
   500    0.526    0.001    0.526    0.001 {method 'uniform' of 'mtrand.RandomState' objects}
     6    0.000    0.000    0.179    0.030 __init__.py:1(&lt;module&gt;)


Here is the code:

import numpy as np


def make_throws(number_of_samples, sample_size):
    # True == Heads
    throws = np.random.uniform(size=(number_of_samples, sample_size)) &gt; 0.5
    sample_probs = np.sum(throws, axis=1) / sample_size
    return sample_probs

sample_sizes = [np.random.randint(1, 1e3) for idx in range(500)]

mean_of_sample_means = np.zeros(len(sample_sizes))
std_dev_of_sample_means = np.zeros(len(sample_sizes))

for i in range(len(sample_sizes)):
    prob = make_throws(200, sample_sizes[i])
    mean_of_sample_means[i] = np.mean(prob)
    std_dev_of_sample_means[i] = np.std(prob)


At this point we could start to tackle the for loop, but it would start to get very micro optimized. One might consider aggregating the results in prob and then using np.mean(prob, axis=1) and np.std(prob, axis=1) outside the loop, but that only nets like 20ms so it's more of a personal preference thing.