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Uncertainty: Probability and Markov chains - Coggle Diagram
Uncertainty: Probability and Markov chains
Uncertainty
Uncertainty simply means the lack of certainty or sureness of an event.
Belief state
A belief state encapsulates the beliefs an agent has about its current state
Belief state = representation of all possible world states
Make contingency plans
Problem-solving and logical agents keep track of belief state
Probability
Sample space = all possible outcomes
The probability of an outcome e in a sample space S is a number P between 1 and 0 that measures the likelihood that e will occur on a single trial of the corresponding random experiment.
Domain of “total for 2 dice” = {2, ...... , 12}
Diagnosis involves uncertainty
Typical of judgemental domains
Eg. law, business, design, car repair, gardening, dating
DENTAL DIAGNOSIS
Toothache => Cavity
Could be abcess, gum disease, punch in mouth
LOGIC FAILS
Laziness
- too much work for complete set of antecedents or consequences
Theoretical ignorance
- no complete theory in medical science
Practical ignorance
- lack of tests on patient
Be aware
80% chance of cavity
In real world, either there is one or not It’s TRUE or FALSE
Probability is only in respect of our current state of knowledge
References
Belief state:
https://www.igi-global.com/dictionary/belief-state/2341
Probability theory:
https://www.geeksforgeeks.org/probability-theory/
Markov Chains:
https://brilliant.org/wiki/markov-chains/
Decision theory:
https://corporatefinanceinstitute.com/resources/career-map/sell-side/capital-markets/decision-theory/
Rational decision:
https://www.pitchlabs.org/library/operations/project-management-tools/rational-decision-making?
Uncertainty:
https://corporatefinanceinstitute.com/resources/career-map/sell-side/risk-management/uncertainty/
Hidden Markov Model:
https://scholar.harvard.edu/files/adegirmenci/files/hmm_adegirmenci_2014.pdf
Chaos Theory:
https://www.investopedia.com/terms/c/chaostheory.asp
Monte Carlo Methods:
https://www.ibm.com/think/topics/monte-carlo-simulation
Phylo
:
https://phylo.cs.mcgill.ca/
What could go wrong?
Petrol runs out
Plane delayed
Traffic jam
Accident on road
Make a plan
Could plan to leave 2h ahead of schedule
Could plan to leave 2h ahead of schedule
Optimal timing avoids a long wait at the airport
avoids speeding fine
Personal Issue
Rational decision
The rational decision-making process is often employed when making choices or decisions.
Depends on
Logic
Reason
Carefully considering available information
Relative importance of various goals
The likelihood of achieving them
Probability theory
Probability theory is an advanced branch of mathematics that deals with measuring the likelihood of events occurring.
It provides tools to analyze situations involving
uncertainty
and helps in determining how likely certain outcomes are.
Logic and probability
SAME
world composed of facts
DIFFERENT
Logic says TRUE/FALSE
Back to the airport
To make choices, agent must have PREFERENCES with regard to OUTCOMES
Utility theory - every state has a degree of usefulness
Decision theory
Decision theory is the study of a person or agents’ choices.
Types
Normative Decision Theory
Optimal Decision Theory
Decision theory =
probability theory + utility theory
Markov Chains
A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules.
Property :
Markov Property or Memorylessness
State space,transition Probability & transition matrix
Absorbing Markov Chains
Garkov
Andrey Markov
born in 1856 and grew up first in Ryazan, Russia and later in St. Petersburg,
mathematical system that undergoes transitions in state space
RANDOM (memoryless)
Chaos Theory
Chaos theory is a mathematical concept that explains that it is possible to get random results from normal equations.
Fractals
Based on deterministic process
PHYLO
Aligning genetic sequence through a puzzle game
3000 regular players
produced 350,000 solutions to various problems
The comparison of the genomes from various species is one of the most fundamental and powerful technique in molecular Biology.
Monte Carlo Methods
Monte Carlo Simulation is a type of computational algorithm that uses repeated random sampling to obtain the likelihood of a range of results of occurring.
Monte Carlo integration is a technique for numerical integration using random numbers.
Markov Chain Monte Carlo
(MCMC)
Gibbs sampling
Metropolis-Hasting algorithm
Application of Monte Carlo Methods
Game playing & optimization
Bayesian inference
Risk analysis & decision making
Markov Chains Modeling
Stationary distribution & long-term behaviour
Absorbing & non-absorbing states
Calculation of steady-state probabilities
Examples
PageRank algorithm
Modeling weather patterns
Random walk
Stochastic process
Collection of random variables
Indeterminate outcome
Represents evolution of system over time
Applications - statistical models of real world
Ex Hidden Markov model
A hidden Markov model is a tool for representing probability distributions over sequences of observations.
Forward-Backward Algorithm
Examples
Part-of-speech tagging
Speech recognition
Bioinformatics
