# Classification based on several marginal probabilities

In my data, each instance has several attributes as TRUE or FALSE. For example:

Instance1: X1=TRUE, X2=TRUE, X3=FALSE, ...
Instance2: X1=FALSE, X2=TRUE, X3=FALSE, ...


I need to classify a TRUE/FALSE attribute, named Y, of each instance. So far, what I know is the conditional probability $P(Y|X_1), P(Y|X_2), ...$, and the marginal probability $P(X_1), P(X_2), ...$

Is there an existent model to guide the inference? Thanks.

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Are you saying you don't have the original data - you ONLY have the conditional probabilities and the marginals? – Corone Feb 27 '13 at 10:37
The more common kinds of analysis would be some form of logistic or loglinear model, but it kind of sounds like you don't have the Y's to go with each full instance, which you'd need for that. – Glen_b Feb 27 '13 at 11:05

Only $P(Y|X_n)$ and $P(X_n)$ are available in practice.

For example, the $X_n$ are "whether a user has visited a specific website", and $Y$ is the gender. We have data of $X_n$, and we can buy the distribution of gender for some specific websites.

Now I'll try the following approach:

1. Assume that $P(X_1, X_2, ..., X_n|Y) = P(X_1|Y) P(X_2|Y) ... P(X_n|Y)$, i.e. $X_1, X_2, ..., X_n$ are independent given $Y$
2. Evaluate $P(X_n|Y)$ according to $P(Y|X_n)$ and $P(X_n)$

The following steps are straightforward.

Maybe there are better assumptions for this case.

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I believe this is called "Naive Bayes". (en.wikipedia.org/wiki/Naive_Bayes_classifier) – Wayne Feb 26 '14 at 22:56