tmt said:Given $$P(B \land C)$$ will it always be true that $$P(B \land C | A) P(A) + P(B \land C | \lnot A) P( \lnot A)$$ (regardless what $P(A)$ would be)?
How can I prove this?
Marginalizing a probability is the process of summing over all possible values of one or more variables in a probability distribution in order to obtain the probability distribution of the remaining variables.
Marginalizing a probability allows us to focus on the variables of interest and ignore any irrelevant variables. It also allows us to simplify complex probability distributions and make them more manageable to work with.
Conditioning a probability involves fixing the value of one or more variables and calculating the probability of the remaining variables. Marginalizing a probability involves summing over all possible values of one or more variables to obtain the probability distribution of the remaining variables. In other words, conditioning focuses on a specific scenario, while marginalizing takes into account all possible scenarios.
No, marginalizing a probability does not change the values of the remaining variables. It only changes the way we look at the probability distribution by summing over all possible values of some variables.
Marginalizing a probability is used in a variety of applications, including data analysis, machine learning, and statistical modeling. For example, in healthcare research, marginalizing probabilities can help determine the likelihood of a patient having a certain disease based on their symptoms and other risk factors.