Bayesian Learning

Bayesian learning allows us to incorporate prior knowledge, and Empirical Data.

Source of Prior Knowledge The notion that prior knowledge must be received from a divine or transcendent source, such as God, touches on epistemological and metaphysical questions:

Epistemological Perspective:

From an epistemological standpoint, prior knowledge can be seen as originating from various sources, including

• intuition

• previous experience

• expert opinion

• theoretical considerations

God

The idea of this knowledge being divinely inspired or received from a higher power adds a layer of metaphysical depth.

It suggests that some forms of knowledge are beyond empirical verification and are grounded in a belief in a higher, possibly omniscient, source.

Metaphysical Perspective:

Metaphysically, considering prior knowledge as divinely inspired of God.

Risk minimization in the Bayesian perspective

Steps for Bayesian Risk Minimization

1. Determine the Posterior Distribution: Compute the posterior distribution \(p(\theta | X)\) using Bayes’ theorem.

\[ p(\theta | X) = \frac{p(X | \theta) p(\theta)}{p(X)} \]

2. Define the Loss Function: Choose an appropriate loss function \(L(\theta, \theta^{*})\) based on the problem context.

3. Compute the Expected Posterior Loss: Integrate the loss function over the posterior distribution to get the expected loss for each possible action.

\[ R(\theta^{*} | X) = \int L(\theta, \theta^{*}) p(\theta | X) \, d\theta \]

4. Minimize the Expected Loss: Select the action \(a^*\) that minimizes the expected posterior loss.

\[ \theta^{optimal} = \arg\min_\theta R(\theta | X) \]