Note for Bayesian inference

Note for learning Bayesian inference

1. conjugate prior （共轭先验）

1.1 Posterior Probability

$\theta$ is the model, $x$ is obeserved data.
Marginal likelihood:

$$P(x) = \int P(x \vert \theta)P(\theta) d\theta$$

When face a new point $x_{new}$.

$$P(x_{new} \vert x) = \int P(x_{new} \vert \theta) P(\theta \vert x) d\theta$$

1.2 How to select prior

If prior is Uniform. The posterior is:

$$P(\theta \vert x) = \frac{P(x \vert \theta)}{\int P(x \vert \theta) C d\theta}$$