https://web.stanford.edu/~montanar/RESEARCH/BOOK/partD.pdf

Belief Propagation

1. Evidence to child nodes is introduced (red circles in bottom nodes). The new information is passed to intermediate nodes who update their belief.
2. The update is passed up to the parent node (as well as down to uninstantiated child nodes)
3. Crucially, the belief update is passed back down to intermediate nodes, updating their conditional probabilities.
4. The final step updates the conditional probabilities of the child nodes based on the belief passed down.


Image Example: Belief Propogation in Tree

No matter in what order the messages are sent out, an equilibrium needs to be eventually reached which should represent exactly the same state that can also be reached by textbook methods with much more computation.

Challenge of belief propagation

Given the example of diagnostic test, likelihood ratio measures how much more likely the positive test is in the diseased people than in the general population. It is given by
$$
P(T|D) / P(T)
$$
where $D$ denotes desease, and $T$ stands for test.

Bayes' theorem can then be rewritten as 
(Updated probability of $D$ ) $=P(D | T)=$ (likelihood ratio) $\times$ (prior probability of $D$ ).

Likelihood ratio

Belief Propagation is an iterative message passing algorithm used within graphs or trees. In Bayesian Networks, bayesian likelihoods are calculated at certain nodes and passed up or down the network.

* If the message is passed downward (e.g. from parent to child), the child can update its beliefs using conditional probabilities.
* If the message is passed upward (e.g. from child to parent), the parent can update its beliefs by multiplying them by a likelihood ratio.

University of Michigan - Ann Arbor

Bayesian networks were developed to understand reasoning under conditions of uncertainty. It is based on Bayes Theorem and is a probabilistic model that represents a set of variables and their conditional dependencies by being able to find the probability of facts being true/false given already defined facts. A Bayesian network is represented by a directed acyclic graph which helps to visualize the probabilistic model for reviewing relationships between random variables and reasonings for causal probabilities given certain facts



Bayesian Networks

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

A Bayesian network's structure is made up of nodes (which can be multi-variate), edges between paired nodes, and a conditional probability distribution in each nodes. The nodes in a Bayesian network represent random variables and edges represent the conditional dependence of these nodes. We try to use this structure to model the conditional probability distribution of outcome variables (which are often causal) after observing new evidence.

How do Bayesian Networks work?

This article covers Bayesian networks and how they work by looking at different aspects of probabilistic graphical models.

[www.bayesserver.com/docs/introduction/bayesian-networks#_dynamic-bayesian-networks](https://www.bayesserver.com/docs/introduction/bayesian-networks#_dynamic-bayesian-networks) 

Helpful article about Bayesian Networks

Belief Propagation in Bayesian Networks

Looking at related information based on the value of one variable is called conditioning on a variable. This is a helpful tool when programming a machine to update its beliefs: it can focus on relevant information while ignoring the rest.

Conditioning on a Variable

The conditional probability table displays conditional probabilities of mutually dependent random variables of a single variable contingent on the others.  Not to be confused with the Contingency Table, which describes the frequencies of data points in the given category. 

Conditional Probability Table

As mentioned before, Bayesian networks encode conditional independencies, i.e. we can use them to determine "independent" events.

Bayesian Networks and Independence

"Bayesian intelligence failed to bridge the gap between artificial and human intelligence. I'm sure you can figure out the missing ingredient: causality"



Criticism of Bayesian Network from the Judea Pearl

Most of the statistical methods are entirely objective, which means reasoning exclusively from data and experiments. But causal analysis and Bayesian analysis are different in that they involves some subjectivity.

Causal analysis requires some subjective commitments based on one's knowledge in the field before the causal digram is drawn and data is explored.

Similarly, Bayesian analysis combines objective evidence with the prior subjective belief.

From objectivity to subjectivity

Given that the guest chooses door 1 and the host opens door 3, we have: 


$P(1=car|3=opened) $

$= \frac{P(3=opened|1=car) *P(1=car)} {p(3=opened)} $

$=\frac{\frac{1}{2}*\frac{1}{3}}{p(3=opened)}$


P(3=opened)  is the probability that the host opens door 3 when guest picked door 1.

$P(3=opened)$ 
$= P(3=opened |1= car)*P(1= car)+P(3=opened|2= car)*P(2 = car)+P(3=opened |3 = car)*P(3 = car)$
$=\frac{1}{2}*\frac{1}{3}+1*\frac{1}{3}+0*\frac{1}{3}$
$=\frac{1}{2}$


$P(1=car|3=opened) $
$=\frac{\frac{1}{6}}{\frac{1}{2}}$
$=\frac{1}{3}$

Hence the possibility of car is behind door 1 is $1/3$, given guest selected door 1 and host opened door 3. 

Bayesian Solution for Monty Hall’s Paradox


- The COVID-19 testings tend to focus on already hospitalized people, not accounting for the people with mild symptoms and the asymptomatics leading to selection bias.  
- The differences in death rates across countries may not be attributed to clinical, demographic, and environmental factors.  
- Together with random sampling/testing, the Bayesian Network model allows causal explanations from data for better predictions.  


Summary of "COVID-19 infection and death rates: the need to incorporate causal explanations for the data and avoid bias in testing"

Structure Learning is a field of machine learning in which data is used to learn the links of a Bayesian network or Dynamic Bayesian network. The two major approaches for structure learning are the score-based approach and the constraint-based approach.

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