## Reminders

### Conditional Probability

$$
p(x \mid y) = \frac{p(x, y)}{p(y)}
$$

### Bayes' Rule

$
\begin{align}
p(x \mid y) &= \frac{p(y \mid x)p(x)}{p(y)} \\
            &= \frac{p(y \mid x)p(x)}{\int_x p(y \mid x)p(x)}
\end{align}
$

Posterior, likelihood and prior:

$
Posterior = \frac{Likelihood \cdot Prior}{Normalizer}
$

### Frequentist Perspective on Probability
If we were to repeat the same experiment many times, in the limit, the frequency of the event would approach the given probability.

### Bayesian Perspective on Probability
Probability is a reasonable guess based on our degree of belief about the environment.

### Independence
$$
\text{x and y are independent} \iff p(x, y) = kf(x)g(y)
$$

### Conditional Independence

$$
X \ci Y \mid Z
$$

denotes that random variables $X$ and $Y$ are independent provided that we know the state of $Z$.

$$
X \ci Y \mid Z \iff p(X, Y \mid Z) = p(X \mid Z)p(Y \mid Z)
$$

1. If $Z = \varnothing$, then $X$ and $Y$ are (unconditionally) independent.
2. It may be the case that $X$ and $Y$ are not independent, but $X$ and $Y$ are conditionally independent given $Z$.

See https://www.eecs.qmul.ac.uk/~norman/BBNs/Independence_and_conditional_independence.htm for examples on conditional independence.

cde211692a89bec79b032c4718a50419a5bfaaf7
ipynb
Jupyter Notebook
BRML/notebooks/chapter1.ipynb
eozd/brml-notes
46a14ae7ea22e9786a750b99293bea70c8a11af9
MIT
BRML/notebooks/chapter1.ipynb
eozd/brml-notes
46a14ae7ea22e9786a750b99293bea70c8a11af9
MIT
BRML/notebooks/chapter1.ipynb
eozd/brml-notes
46a14ae7ea22e9786a750b99293bea70c8a11af9
MIT
2020-03-23T00:44:06.000Z
2020-03-23T00:44:06.000Z
Qwen/Qwen-72B
1. YES
2. YES