# Notation: Posterior distribution $k(\theta | x_1, ... ,x_n)$ in Bayesian estimation.

I am having trouble with Bayesian statistics and would like to understand it a little more.

And I would like to ask if the following is true.

Suppose observations $$X_1, ... , X_n \sim_{iid} f(x;\theta)$$ and prior of the parameter $$\theta$$ is a random variable $$\Theta \sim \pi(\theta)$$.

I know that by definition the posterior distribution of $$\theta$$ is

$$k(\theta|x_1,...,x_n)= \frac{f(\theta|x_1,...,x_n)\pi(\theta)}{\int_{-\infty}^{\infty}f(\theta|x_1,...,x_n)\pi(\theta)d\theta}$$ *(when continuous)

Since the Xs are iid, I want to say that the likelihood function can be substituted in instead of $$f(\theta|x_1,...,x_n)$$ as such

$$k(\theta|x_1,...,x_n)=\frac{ L(\theta)\pi(\theta)}{\int_{-\infty}^{\infty}L(\theta)\pi(\theta)d\theta}$$

Is this an acceptable notation or is it incorrect?

There aren't many examples that I am able to understand in my notes and I would really like to overcome the fear of Bayesian statistics.

I appreciate your help.

## 1 Answer

You say:

$$k(\theta|x_1,...,x_n)= \frac{f(\theta|x_1,...,x_n)\pi(\theta)}{\int_{-\infty}^{\infty}f(\theta|x_1,...,x_n)\pi(\theta)d\theta}$$

But this is not the definition. In actual fact it is:

$$k(\theta|x_1,...,x_n)= \frac{f(x_1,...,x_n|\theta)\pi(\theta)}{\int_{-\infty}^{\infty}f(x_1,...,x_n|\theta)\pi(\theta)d\theta}$$

Where $$f(x_1,...,x_n|\theta)$$ is the likelihood of obtaining the data given the parameter value $$\theta$$. The notation here is really the same as saying $$L(x_1,...,x_n|\theta)$$, it is just different notation. When looking at maximum likelihood estimates, we maximise our likelihood function under the parameter $$\theta$$. In this case we write $$L(\theta)$$ in place of $$L(x_1,...,x_n|\theta)$$. Again, just notation.