Bayesian Inference in Machine Learning

Bayesian inference is a powerful tool in machine learning that allows us to update our beliefs about a parameter as we gather more data. It is based on Bayes' theorem, which is a mathematical formula that describes how to update the probability of a hypothesis as more evidence or information becomes available.

Key Concepts

Prior Probability: The probability of a hypothesis before new evidence is taken into account.
Likelihood: The probability of the observed data given the hypothesis.
Posterior Probability: The probability of a hypothesis after new evidence is taken into account.

Bayes' Theorem

Bayes' theorem is expressed as:

$$ P(H|D) = \frac{P(D|H) \cdot P(H)}{P(D)} $$

Where:

( P(H|D) ) is the posterior probability of the hypothesis ( H ) given the data ( D ).
( P(D|H) ) is the likelihood of the data given the hypothesis.
( P(H) ) is the prior probability of the hypothesis.
( P(D) ) is the marginal likelihood of the data.

Applications

Bayesian inference has many applications in machine learning, such as:

Classification: Predicting the class of an instance.
Regression: Predicting the value of a continuous variable.
Clustering: Grouping similar instances together.

Example

Let's say we have a coin that we believe is fair. We flip the coin 10 times and get 8 heads. Using Bayesian inference, we can update our belief about the fairness of the coin.

Prior Probability

Before seeing the data, we might think that the coin is fair with a prior probability of 0.5.

Likelihood

The likelihood of getting 8 heads out of 10 flips with a fair coin is:

$$ P(D|H) = \binom{10}{8} \cdot (0.5)^8 \cdot (0.5)^2 = 0.2051 $$

Posterior Probability

Using Bayes' theorem, we can calculate the posterior probability:

$$ P(H|D) = \frac{P(D|H) \cdot P(H)}{P(D)} $$

Where ( P(D) ) is the marginal likelihood, which is the sum of the likelihoods over all possible values of ( H ):

$$ P(D) = \sum_{H} P(D|H) \cdot P(H) $$

After calculating, we find that the posterior probability of the coin being fair is approximately 0.6.