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

this is a direct result of the probability chain rule.

Typically, we name \(p(y|x)\) the “likelihood”, \(p(x)\) the “prior”.

Better normalization

What if you don’t fully know \(p(y)\), say it was parameterized over \(x\)?

\begin{align} p(x|y) &= \frac{p(y|x) \cdot p(x)}{p(y)} \\ &= \frac{p(y|x) \cdot p(x)}{\sum_{X_{i}} p(y|X_{i})} \end{align}

just apply law of total probability! taad

\begin{equation} P(B=b | D=d) = P(D=d|B=b) P(B=b) k \end{equation}

where, \(P(B=b | D=d)\) is your “posterior”; \(P(D=d|B=b)\) is your likelyhood; and \(P(B=b)\) is your prior.

A Baysian Network is composed of:

1. a directed, acyclic graph
2. a set of conditional probabilities acting as factors.

You generally want arrows to go in the direction of causality.

Via the chain rule of Bayes nets, we can write this equivalently as:

\begin{equation} (P(B) \cdot P(S)) \cdot P(E \mid B,S) \cdot P(D \mid E) \cdot P(C \mid E) \end{equation}

generally, for \(n\) different variables,

\begin{equation} \prod_{i=1}^{n} p(X_{i} \mid pa(x_{i})) \end{equation}

where, \(pa(x_{i})\) are the parent values of \(x_{i}\).

Representing conditional dependencies.

Semiparametric Expert Bayes Net

We use a Semiparametric Expert Bayes Net to learn the structure of the dynamics…. of medicine somewhere?

Atienza et al. 2022

• learns semiparemetic relations in expert basian networks
• uses gaussian rocesses for modeling no-linear performances + horseshoe regularization

2401.16419

Results

UCI Liver Disorder Dataste

• what is the oracle graph?
• what specific dynamics did the model learn?

We treat this as an inference problem in Naive Bayes: observations are independent from each other.

Instead of trying to compute a \(\theta\) that works for Maximum Likelihood Parameter Learning, what we instead do is try to understand what \(\theta\) can be in terms of a distribution.

That is, we want to get some:

“for each value of \(\theta\), what’s the chance that that is the actual value”

To do this, we desire: