Bayesian Data Analysis
Revision as of 16:16, 25 September 2019 by Colettace (talk | contribs) (Colettace moved page Bayesian Networks to Bayesian Data Analysis)
Contents
Bayesian Network
- Reasoning under uncertainty
set of events some causally related to others with certain probability
- certain factors unknowns or partially known factors
- model systems with probability
- interested in joint probability distributions
- also given certain sub variable is true, what is prob of macro var
- random vars: x1 x2, x3
- know nothing about intercausal relationships
- if you are provided the probability distribution P( x1, x2, x3 ), want to try to compute the following:
- P[x1 = 0, x2 = 1, x3 = 0]
- P[x1, x2] = P(x1, x2, x3) + P(x1, x2, x3bar)
- P[x1 | x2, x3 ] = read "the probability of x1, given x2 and x3"
bayesian network
- Bayesian network is way to reduce size of representation, a "succinct way" of representing distribution
- store probability distribution explicitly in a table
- x1 .. x10 are booleans
- what is size of table for set of vars P[ x1 ... x10] = 2^n
- how can rewrite joint pdf P[x1, x2, ..., x10]= P[x1| x2, ..., x10] * P[x2, ..., x10]
- = P[x1| x2, ..., x10] * P[x2 | x3, ..., x10] ... P[Xn-1|Xn]*P[Xn]
- P[Xi|Xi+1, ..., Xn] = P[Xi] if Xi is totally independent of the others
- sometime can also be conditionally independent, only dependent on a subset of the other variables
- the variable on which P[Xi] depends "subsumes" the other variables
- belief network - order of variables matters when setting up dependencies in belief network.
- Count parents of each node to figure out size of conditional probability tables
- If use improper ordering, results in valid representation of joint probabilty funtion, but would require producing conditional probability tables which aren't natural/difficult to obtain experimentally. could also result in inflation of conditional tables / size of table representation is large compared to others
Incremental Network Construction
- Choose the set of relevant set of variables X that describe the domain
- Choose an ordering for the variables (very important step)
- While there are variables left:
- dequeue variable X off the queue and add node
- Set Parents(X) to some minimal set of existing of existing nodes such that the conditional independence is satisfied
- Define the conditional probability table
inferences using belief networks
- diagnostic inferences (from effects to causes
- causal inferences (given symptoms, what is probability of disease)
- intercausal inferences
- mixed inferences