Fastest computation of N likelihoods on normal distributions In the context of a Gibbs sampler, I profiled my code and my major bottleneck is the following: 

I need to compute the likelihood of N points assuming they have been drawn from N normal distributions (with different means but same variance).

Here are two ways to compute it: 

import numpy as np
from scipy.stats import multivariate_normal
from scipy.stats import norm

# Toy data
y = np.random.uniform(low=-1, high=1, size=100) # data points
loc = np.zeros(len(y)) # means

# Two alternatives
%timeit multivariate_normal.logpdf(y, mean=loc, cov=1)
%timeit sum(norm.logpdf(y, loc=loc, scale=1))



The first: use the recently implemented multivariate_normal of scipy. Build the equivalent N-dimensional gaussian and compute the (log)probability of a N-dimensional y. 

1000 loops, best of 3: 1.33 ms per loop
The second: use the traditional norm function of scipy. Compute the individual (log)probability of every point y and then sum the results.

10000 loops, best of 3: 130 µs per loop


Since this is part of a Gibbs sampler, I need to repeat this computation around 10.000 times, and therefore I need it to be as fast as possible.

How can I improve it? 

(either from python or calling Cython, R or whatever)
You should use a line profiler tool to examine what the slowest parts of the code are.  It sounds like you did that for your own code, but you could keep going and profile the source code that NumPy and SciPy use when calculating your quantity of interest.  The [Line profiler](https://pypi.python.org/pypi/line_profiler/) module is my favorite.  

import numpy as np
from scipy.stats import multivariate_normal
from scipy.stats import norm
%lprun -f norm.logpdf norm.logpdf(x=np.random.random(1000000), \
                                  loc=np.random.random(1000000), \
                                  scale = np.random.random())




Timer unit: 1e-06 s

Total time: 0.14831 s
File: /opt/local/Library/Frameworks/Python.framework/Versions/2.7/lib/python2.7/site-packages/scipy/stats/_distn_infrastructure.py
Function: logpdf at line 1578

Line #      Hits         Time  Per Hit   % Time  Line Contents
==============================================================
  1578                                               def logpdf(self, x, *args, **kwds):
  1579                                                   """
  1580                                                   Log of the probability density function at x of the given RV.
  1581                                           
  1582                                                   This uses a more numerically accurate calculation if available.
  1583                                           
  1584                                                   Parameters
  1585                                                   ----------
  1586                                                   x : array_like
  1587                                                       quantiles
  1588                                                   arg1, arg2, arg3,... : array_like
  1589                                                       The shape parameter(s) for the distribution (see docstring of the
  1590                                                       instance object for more information)
  1591                                                   loc : array_like, optional
  1592                                                       location parameter (default=0)
  1593                                                   scale : array_like, optional
  1594                                                       scale parameter (default=1)
  1595                                           
  1596                                                   Returns
  1597                                                   -------
  1598                                                   logpdf : array_like
  1599                                                       Log of the probability density function evaluated at x
  1600                                           
  1601                                                   """
  1602         1           14     14.0      0.0          args, loc, scale = self._parse_args(*args, **kwds)
  1603         1           23     23.0      0.0          x, loc, scale = map(asarray, (x, loc, scale))
  1604         1            2      2.0      0.0          args = tuple(map(asarray, args))
  1605         1        13706  13706.0      9.2          x = asarray((x-loc)*1.0/scale)
  1606         1           33     33.0      0.0          cond0 = self._argcheck(*args) &amp; (scale &gt; 0)
  1607         1         5331   5331.0      3.6          cond1 = (scale &gt; 0) &amp; (x &gt;= self.a) &amp; (x &lt;= self.b)
  1608         1         5625   5625.0      3.8          cond = cond0 &amp; cond1
  1609         1           84     84.0      0.1          output = empty(shape(cond), 'd')
  1610         1         6029   6029.0      4.1          output.fill(NINF)
  1611         1        11459  11459.0      7.7          putmask(output, (1-cond0)+np.isnan(x), self.badvalue)
  1612         1         1093   1093.0      0.7          if any(cond):
  1613         1        58499  58499.0     39.4              goodargs = argsreduce(cond, *((x,)+args+(scale,)))
  1614         1            6      6.0      0.0              scale, goodargs = goodargs[-1], goodargs[:-1]
  1615         1        46401  46401.0     31.3              place(output, cond, self._logpdf(*goodargs) - log(scale))
  1616         1            4      4.0      0.0          if output.ndim == 0:
  1617                                                       return output[()]
  1618         1            1      1.0      0.0          return output


It looks like a not-insignificant amount of time is being spent checking and removing invalid arguments from the function input.  If you can be sure you will never need to use that feature, just write your own function to calculate the logpdf.

Plus, if you are going to be multiplying probabilities (i.e. adding log probabilities), you could use algebra to simplify and factor out common terms from the summand for the normal distribution's pdf.  That will lower the number of function calls to np.log etc.  I did this in a hurry, so I probably made a math mistake, but:

def my_logpdf_sum(x, loc, scale):
    root2 = np.sqrt(2)
    root2pi = np.sqrt(2*np.pi)
    prefactor = - x.size * np.log(scale * root2pi)
    summand = -np.square((x - loc)/(root2 * scale))                         
    return  prefactor + summand.sum()



# toy data
y = np.random.uniform(low=-1, high=1, size=1000) # data points
loc = np.zeros(y.shape)
​
# timing
%timeit multivariate_normal.logpdf(y, mean=loc, cov=1)
%timeit np.sum(norm.logpdf(y, loc=loc, scale=1))
%timeit my_logpdf_sum(y, loc, 1)
1 loops, best of 3: 156 ms per loop
10000 loops, best of 3: 125 µs per loop
The slowest run took 4.55 times longer than the fastest. This could mean that an intermediate result is being cached 
100000 loops, best of 3: 16.3 µs per loop