1:193260
2:51794
3:19112
4:9250
5:6486

pdf(x) = 1 / scale * exp ( - x / scale)

log_pdf(x) = - log(scale) - x / scale

def neg_log_likelihood(scale):
    total = 0.0
    for x, count in counter.iteritems():
       total += (math.log(scale) + x / scale) * count
    return total

import scipy.stats
import scipy.optimize
import math
import collections

def fit1(counter):
    def neg_log_likelihood(scale):
        total = 0.0
        for x, count in counter.iteritems():
           total += (math.log(scale) + x / scale) * count
        return total

    optimize_result = scipy.optimize.minimize(neg_log_likelihood, [1.0])
    if not optimize_result.success:
        raise Exception(optimize_result.message)
    return optimize_result.x[0]

def fit2(counter):
    data = []
    # Create an array where each key is repeated as many times
    # as the value of the counter.
    for x, count in counter.iteritems():
        data += [x] * count
    fit_result = scipy.stats.expon.fit(data, floc = 0)
    return fit_result[-1]

def test():
    c = collections.Counter()
    c[1] = 193260
    c[2] = 51794
    c[3] = 19112
    c[4] = 9250
    c[5] = 6486

    print "fit1 'scale' is %f " % fit1(c)
    print "fit2 'scale' is %f " % fit2(c)

test()

fit1 'scale' is 1.513437
fit2 'scale' is 1.513438