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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.

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1 Answer 1

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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.

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