how to determine which of the important missing variables to ignore I understand there are two general approaches when dealing with missing variables: Deletion and estimation.
However, what if we were given a hint that one of the variables is super important? How would we deal with missing values for the important variable then?


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*Deletion of the row/observation where that important variable is missing.
Problem: How would I deal with bias in this one? What if the missing values actually provides some sort of value? (Like people not answering a certain question on a survey on purpose?)

*Estimation of the missing values. Problem: Estimating missing values based on existing data just.... doesn't feel right to me. Wouldn't this make the model less accurate? 
 A: Dropping the incomplete cases, called ''complete case analysis'' almost always leads to biased and inefficient estimates.  This can only be avoided under very stringent assumptions namely when the missing is completely at random (missing completely at random or MCAR): see this link for details. This is because you lose information that is available in the incomplete cases.  
Replacing the missing values by other another value, the so-called ''single value imputation method'', has similar shortcomings.  If e.g. you replace the missing value by the average, then the ''overall'' average will remain OK, but the variability of the data (standard error) will be underestimated. 
The state of the art methods are (1) maximum likelihood estimation and (2) multiple imputation methods.  
If your estimates are obtained by maximum likelihood using all (complete and incomplete) cases then under less stringent assumptions (missing at random or MAR, see link supra) you will get unbiased and efficient estimates.  This is because under certain assumptions (MAR) the likelihood function is not impacted by the missingness (see link supra for details). 
Multiple imputation replaces a missing value with more than one alternative (see link for the details) and then ''averages out'' the results.  It also leads to less biased and efficient estimates when the missings are MAR. This is because the imputation by multiple values ''re-introduces'' additional variability compared with imputation by only one value (see link supra for details). 
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[...] Like people not answering a certain question on a survey on purpose?

This sounds like you would want to transform the information instead of discarding it, e.g. by adding "answered"/"not answered" as variable to your dataset. Doing so is similar to variable binning/censoring a variable, so is necessarily connected to some information loss - but the advantage is that "leaving out a question" itself thereby becomes information easily usable by the model.

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Estimation of the missing values. [...] Wouldn't this make the model less accurate? 

Your assumptions are true, you will necessarily cause some inaccuracies in your data if you impute/estimate data. But how much this effects the models you apply to the data depends on the model and the feature-target relation represented with the affected variables - so it might or might not be a degradation of prediction performance. What you could do is try both approaches and compare their performance: when properly evaluating them (e.g. repeated cross validation with a held-back test set, for which you need to do imputation as well if you relied on it during training the model).
