Why class stratified sampling is not compatible for naïve bayesian modeling, if sampling is used?

Question

I read a book recently and it mentioned related to prior probability of naïve bayesian:

"Since the probability of an outcome is calculated from the data set, it is important that the data set used for data mining is representative of the population, if sampling is used. A class-stratified sampling of data from the population will not be compatible for naïve Bayesian modeling." -- Page 114-115 and the book-website friendly offers entire eBook view.

Within the context of the book, the first term of probability (in italic font style) means the prior probability of certain outcome $P(Y)$, and the two times of "data set" mean training data set, from my understandings. Let's say we have training data for naïve bayesian modeling:

id	credit	income	default-on-mortgages
1	100	\$100	yes
2	1000	\$1000	no
3	50	\$50	yes

Prior probability of $P(Y=yes)=2/3$, $P(Y=no)=1/3$.

I cannot figure out why stratified sampling does not suit for bayesian modeling?

$P(Y)$ is one of the essential probabilities to solve questions. The essence and smartness of naïve bayesian are computing the prior probability $P(Y)$ and class conditional probability $P(X\mid Y)$ and they will lead to result of the unknown $P(Y\mid X)$. The former two probabilities are relatively easier to get compared with the latter one.

If sampling method is used, I will expect the $P(Y)$ of subset will be close to $P(Y)$ of the original training set. So that the training results of applying NB algorithms on two data sets would not be much different. I assumed, stratified sampling could ensure subset and original data set have similar class contents, for example, the original data set has 150 samples with 100 "default=yes" and 50 "default=no". In this case, $P(Y=yes)=100/150=2/3$. If I used stratified sampling which subsetting data set down to its 1/50, I could get the subset data set (shown as above small data set) of original 150 data set. And these two have similar prior probability (here they are same). However, simple random sampling cannot guarantee. Therefore, I support stratified sampling will suit for NB modeling.

In sum, is it possible that this is just a typo? The "not" should be wiped out? Looking forward to your suggestions.

Answer 1

IMHO, I think although Naive Bayesian assumes independence between features, but in reality this is only an approximation. That means there exists correlation between features somehow. If the correlations differs in your subgroup, any subgroup's correlation end up being different from the overall population, which leads to bias. On the other hand, stratified sampling imposed exclusiveness between sample points, which destroyed independence between data points. Actually, I doubt it can be used in any inference about the population if you don't know exact parameters in subgroup.

Answer 2

I realized that my assumption might be wrong, which states "I will expect the P(Y) of subset will be close to P(Y) of the original training set."

The parameters of probability distribution (or underlying data structure) is unknown. Bayesian methods begin with a prior probability (prior degree of belief). After training model by learning more observation data, we can update the belief, i.e. post probability.

Training data set composes of features (attributes) and classes (or labels). Stratified sampling would introduce bias by the labels.

Answer 3

I am a bit unsure if I fully understand the question. From what I understand, stratified random sample in general is not an independent random sample of the target population. Therefore, it is not a representative sample in the population level, but is only representative within each sampling stratum (when there is no non-response).