"A Discussion on Bayes" This is a discussion about Bayes. The left-hand side represents the posterior distribution, while the right-hand side is the product of the prior distribution and the likelihood, divided by the marginal likelihood. Here, y is the observed value and θ is the parameter. Essentially, we are estimating the parameter from the observed values.

"A Discussion on Bayes" This is a discussion about Bayes. The left-hand side represents the posterior distribution, while the right-hand side is the product of the prior distribution and the likelihood, divided by the marginal likelihood. Here, y is the observed value and θ is the parameter. Essentially, we are estimating the parameter from the observed values.

It's about inverting conditional probability. Regarding the right-hand side, the numerator is the product of the prior distribution and the likelihood, which is called the kernel. In practice, for MCMC (Markov Chain Monte Carlo) or Bayesian estimation, we deal with this kernel. We set the prior distribution to, say, a binomial or beta distribution, and the observed value y fits into it. We then sample from this to obtain the result.

The denominator is the marginal likelihood, written as an integral and also called the denominator. Since this becomes nearly impossible to compute in high dimensions, it's often conveniently ignored. Once the prior distribution and the form of the likelihood function are fixed, we can sample from that distribution, allowing us to estimate the posterior distribution.