[Python] Assign columns in pandas multiindex

I have this dataframe:

mu_post
z c t
index a b
0 0.0 0.0 0.042824 0.051212
0.5 0.5 0.048293 0.058130
1.0 1.0 0.047267 0.074043
1 0.0 0.0 0.058205 0.054106
0.5 0.5 0.064153 0.063573
1.0 1.0 0.056918 0.059572
2 0.0 0.0 0.059032 0.052211
0.5 0.5 0.070616 0.066792
1.0 1.0 0.056892 0.045061


produced by:

import pandas as pd
df = pd.DataFrame({('mu_post', 'c'): {(0, 0.0, 0.0): 0.042824223871028126, (0, 0.5, 0.5): 0.04829260822563669, (0, 1.0, 1.0): 0.047267365970316805, (1, 0.0, 0.0): 0.05820509767743391, (1, 0.5, 0.5): 0.06415323721481726, (1, 1.0, 1.0): 0.0569177959009184, (2, 0.0, 0.0): 0.05903204294019807, (2, 0.5, 0.5): 0.07061613725719014, (2, 1.0, 1.0): 0.056892088025082874}, ('mu_post', 't'): {(0, 0.0, 0.0): 0.051212446939110846, (0, 0.5, 0.5): 0.058129980845875964, (0, 1.0, 1.0): 0.07404310411549644, (1, 0.0, 0.0): 0.05410577324029455, (1, 0.5, 0.5): 0.06357338131851693, (1, 1.0, 1.0): 0.0595723832219094, (2, 0.0, 0.0): 0.05221119083827467, (2, 0.5, 0.5): 0.06679207329135116, (2, 1.0, 1.0): 0.04506069626935631}})


I want to add odds.

def odds(p):
return p / (1-p)


I can assign like this:

df.assign(
odds_c=lambda x: odds(x[('mu_post', 'c')]),
odds_t=lambda x: odds(x[('mu_post', 't')]),
)

mu_post odds_c odds_t
c t
0 0.0 0.0 0.042824 0.051212 0.044740 0.053977
0.5 0.5 0.048293 0.058130 0.050743 0.061718
1.0 1.0 0.047267 0.074043 0.049612 0.079964
1 0.0 0.0 0.058205 0.054106 0.061802 0.057201
0.5 0.5 0.064153 0.063573 0.068551 0.067889
1.0 1.0 0.056918 0.059572 0.060353 0.063346
2 0.0 0.0 0.059032 0.052211 0.062735 0.055087
0.5 0.5 0.070616 0.066792 0.075982 0.071573
1.0 1.0 0.056892 0.045061 0.060324 0.047187


But what I really want is for the columns MultiIndex to be [(mu_post, c), (mu_post, t), (odds, c), (odds, t)]

If possible I'd like to use the pipe/apply/assign style of chaining methods together.

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