# How to deal with missing data for Bernoulli Naive Bayes?

I am dealing with a dataset of categorical data that looks like this:

    content_1   content_2   content_4   content_5   content_6
0         NaN         0.0         0.0         0.0         NaN
1         NaN         0.0         0.0         0.0         NaN
2         NaN         NaN         NaN         NaN         NaN
3         0.0         NaN         0.0         NaN         0.0


These represent user downloads from an intranet, where a user is shown the opportunity to download a particular piece of content. 1 indicates a user seeing content and downloading it, 0 indicates a user seeing content and not downloading it, and NaN means the user did not see/was not shown that piece of content.

I am trying to use the scikit-learn Bernoulli Naive Bayes model to predict the probability of a user downloading content_1, given if they have seen downloaded / not downloaded content_2-7.

I have removed all data where content_1 is equal to NaN as I'm obviously only interested in data points where a decision was actively made by the user. This gives data as:

    content_1   content_2   content_3   content_4   content_5   content_6
0         1.0         NaN         1.0         NaN         NaN         1.0
1         0.0         NaN         NaN         0.0         1.0         0.0
2         1.0         0.0         NaN         NaN         NaN         1.0


In the above framework, NaN, is a missing value. For data points where a Nan is present, I want the algorithm to ignore that category, and use only those categories present in the calculation.

I know from these questions: 1, that there are essentially 3 options when dealing with missing values:

1. ignore the data point if any categories contain a NaN (I.e. remove the row)
2. Impute some other placeholder value (e.g. -1 etc.) or
3. Impute some average value corresponding to the overall dataset distribution.

However, these are not the best option for the following reason:

1. Every single row contains at least 1 NaN. This means, under this arrangement I would discard the entire dataset. Obviously a no go.
2. I do not want the missing value to add to the probability calculation, which will happen if I replace Nan with say -1. I'm also using a Bernoulli Naive Bayes, so as I understand, this requires singly 0 or 1 values.
3. As this is categorical data, it does not make sense for me to do this, in this way (it was either seen or not, and if not, it is not needed).

The answer here indicated that the best way to do this, is, when calculating probabilities, to ignore that category if it is a missing value (essentially you are saying: only compute a probability based on the specific categories I have provided with non missing values).

I do not know how to encode this when using the scikit-learn Naive Bayes model, whether to do this as a missing value.

Here's what I have so far:

df=pd.read_clipboard()
from sklearn import datasets
from sklearn.naive_bayes import BernoulliNB
# Create train input / output data
y_train = df['content_1'].values
X_train = df.drop('content_1', axis=1).values
# Loud Bernoulli Naive Bayes model
clf = BernoulliNB()
clf.fit(X_train, y_train)


Obviously, this returns an error because of the present NaNs. So how can I adjust the scikit-learn Bernoulli model to automatically ignore the columns with NaNs, and instead take only those with 0 or 1?

I am aware this may not be possible with the stock model, and reviewing the documentation seems to suggest this. As such, this may require significant coding, so I'll say this: I am not asking for someone to go and code this (nor do I expect it); I'm looking to be pointed in the right direction, for instance if someone has faced this problem / how they approach it / relevant blog or tutorial posts (my searches have turned up nothing).

Your search results are on point: without dropping or imputing data, there's no built-in way to do what you want with BernoulliNB.

There is, however, a way out: train separate Bayesian models on filtered samples from your data, and then combine their predictions by stacking them.

## Filtering

Filtering here means:

• Isolating samples from your original df, each having only a subset of df.columns. That way, you'd have a DataFrame only for content_2, one for content_2, content_3, in a sort of a factorial combination of columns.
• Making sure each sample is made only of rows that have no NaNs for any of the columns in the subset.

This part is somewhat straightforward in your case, yet a bit lengthy: you'd have $$n!$$ (n factorial) combinations of columns, each of which would result in a separate sample. For example, you could have a sample named df_c2 containing only content_2 rows valued 0 or 1, df_c2_c3 with only content_2 and content_3 columns filled, and so on.

These samples would make NaN values non-existent to every model you'd train. Implementing this in a smart way can be cumbersome, so I advise starting with the simplest of scenarios - e.g. two samples, two models; you'll improve gradually and reach a solid solution in code.

## Stacking Bayesian Models

This is called Bayesian Model Averaging (BMA), and as a concept it's thoroughly addressed in this paper. There, weight attributed to a Bayesian model's predictions is its posterior probability.

The content can be overwhelming to absorb in one go, be at ease if some of it doesn't stick with you. The main point here is that you'll multiply each model's predicted probabilities by a weight 0 < w < 1 and then sum (sum results shall be in $$[0, 1]$$). You can attribute weights empirically at first and see where it gets you.

## Edit:

Due to the added complexity of my proposed solution, as stated in this (also useful) answer, you could opt to implement Naive Bayes in pure Python, since it's not complicated (and there are plenty tutorials to base upon). That'd make it a lot easier to bend the algorithm to your needs.