Why does max() sometimes return nan and sometimes ignores it?

Question

This question is motivated by an answer I gave a while ago.

Let's say I have a dataframe like this

import numpy as np
import pandas as pd

df = pd.DataFrame({'a': [1, 2, np.nan], 'b': [3, np.nan, 10], 'c':[np.nan, 5, 34]})

     a     b     c
0  1.0   3.0   NaN
1  2.0   NaN   5.0
2  NaN  10.0  34.0

and I want to replace the NaN by the maximum of the row, I can do

df.apply(lambda row: row.fillna(row.max()), axis=1)

which gives me the desired output

      a     b     c
0   1.0   3.0   3.0
1   2.0   5.0   5.0
2  34.0  10.0  34.0

When I, however, use

df.apply(lambda row: row.fillna(max(row)), axis=1)

for some reason it is replaced correctly only in two of three cases:

     a     b     c
0  1.0   3.0   3.0
1  2.0   5.0   5.0
2  NaN  10.0  34.0

Indeed, if I check by hand

max(df.iloc[0, :])
max(df.iloc[1, :])
max(df.iloc[2, :])

Then it prints

3.0
5.0
nan

When doing

df.iloc[0, :].max()
df.iloc[1, :].max()
df.iloc[2, :].max()

it prints the expected

3.0
5.0
34.0

My question is why max() fails in 1 of three cases but not in all 3. Why are the NaN sometimes ignored and sometimes not?

Answer 1

The reason is that max works by taking the first value as the "max seen so far", and then checking each other value to see if it is bigger than the max seen so far. But nan is defined so that comparisons with it always return False --- that is, nan > 1 is false but 1 > nan is also false.

So if you start with nan as the first value in the array, every subsequent comparison will be check whether some_other_value > nan. This will always be false, so nan will retain its position as "max seen so far". On the other hand, if nan is not the first value, then when it is reached, the comparison nan > max_so_far will again be false. But in this case that means the current "max seen so far" (which is not nan) will remain the max seen so far, so the nan will always be discarded.