In the frequentist approach, probability is a limiting frequency. Thus, probabilities are always associated with random events. In Bayesian inference, on the other hand, probability is used to quantify your belief that an unobservable variable has a particular value. For example a Bayesian can ask questions such as:

What is the probability that heritability for milk yield is larger than 0.5?

What is the probability that variability in milk yield is due to more than 100 loci?

These Bayesian probabilities are not necessarily associated with a random experiment that assigns values to the variables in question.

Bayesian inference starts by specifying what you believe about the parameters or unknowns through prior probabilities. In whole-genome analyses, we will use a prior probability density to quantify our belief that the effect of most marker covariates is zero or close to zero and only a few covariates have effects that deviate from zero.

These parameters are related to the data through the model or “likelihood”, which are conditional probabilities for the data given the parameters. In whole-genome analyses, this is usually a multiple regression model with normally distributed residuals.

The prior and the likelihood are combined using Bayes theorem to obtain posterior probabilities, which are conditional probabilities for the parameters given the data.

Inferences about the parameters are based on the posterior.

Let $f(\boldsymbol{\theta)}$ denote the prior probability density for $\boldsymbol{\theta}$.

Let $f(\boldsymbol{y}|{\boldsymbol{\theta}})$ denote the likelihood

Then, the posterior probability of $\boldsymbol{\theta}$ is:

$$\begin{eqnarray} f({\boldsymbol \theta}|{\boldsymbol y}) & =\frac{f({\boldsymbol y}|{\boldsymbol \theta})f({\boldsymbol \theta})}{f({\boldsymbol y})}\\ & \propto f({\boldsymbol y}|{\boldsymbol \theta})f({\boldsymbol \theta}) \end{eqnarray}$$