A conditional probability is the probability that an event has occurred, taking into account additional information about the result of the experiment. Two events A and B are independent if the probability P(A∩B) of their intersection A ∩ B is equal to the product P(A)·P(B) of their individual probabilities.

Specifically, you learned: Joint probability is the probability of two events occurring simultaneously. Marginal probability is the probability of an event irrespective of the outcome of another variable. Conditional probability is the probability of one event occurring in the presence of a second event.

Conditional percentages are calculated for each value of the explanatory variable separately. They can be row percents, if the explanatory variable “sits” in the rows, or column percents, if the explanatory variable “sits” in the columns.

Events A and B are independent if the equation P(A∩B) = P(A) · P(B) holds true. You can use the equation to check if events are independent; multiply the probabilities of the two events together to see if they equal the probability of them both happening together.

You don’t need to have any conditional probability at hand. If you have one conditional probability P(A|B), then you can get the opposite conditional probability P(B|A) directly by Bayes’ rule (my first equation). Interestingly, you can also get P(B|A) without having anything else that occurs in Bayes’ rule.

Since this is the most probable value among all possible target values v, the Optimal Bayes classifier maximizes the performance measure e(ˆf). As we always use Bayes classifier as a benchmark to compare the performance of all other classifiers.

The Bayes approach is an average-case analysis by considering the average risk of an estimator over all θ ∈ Θ. Concretely, we set a probability distribution (prior) π on Θ. Then, the average risk (w.r.t π) is defined as Rπ(ˆθ) = Eθ∼πRθ(ˆθ) = Eθ,Xl(θ, ˆ θ).

Naive Bayes assumes conditional independence, P(X|Y,Z)=P(X|Z), Whereas more general Bayes Nets (sometimes called Bayesian Belief Networks) will allow the user to specify which attributes are, in fact, conditionally independent.

The Bayes Optimal Classifier is a probabilistic model that makes the most probable prediction for a new example. Bayes Theorem provides a principled way for calculating conditional probabilities, called a posterior probability.

Bayesian classification uses Bayes theorem to predict the occurrence of any event. P(Y/X) is a conditional probability that describes the occurrence of event Y is given that X is true. P(X) and P(Y) are the probabilities of observing X and Y independently of each other. This is known as the marginal probability.