Artificial Intelligence - Intelligent agents paradigm

[email protected] - Zunda Krastmala 10 room 410

Course work

laboratory works as hometasks - 40%
video lectures
online tests
exam 60%
Online tests sometimets to earn more points. (~5 tests)

grade = 0.4*course_work+06*exam+bouns only if 0.4*course_work+0.6*exam>=4

lectures:

12:30-14:05
14:30-16:05

mandatory tasks:

5 labs - min 2
- large set of practical problems
- tasks
  - do the task with the packaged software, create a report
laboratory works:
- intelligent agents
  - 4 points
  - .net
- search
  - 4 points
  - java
- planning agents
  - java
- inductive learning
  - java
- complex decision making
  - .net

points scored   grade
at least 12     10
10 to 12        9
9 to 10         8
8 t 9           7
...             ...

deadlines

fst deadline: april 3
all other semester work: may 22
extended deadline, without feedback: may 30

text books:

Russel S and Norvig: Artifcial Intelligence

objective strengthen theoretical knowledge about modern approaches of artificial intelligence acquired at the lectures of the course

What is AI

those unknown pocesses with which our brains solve problems ~ Minskiy
the definition change through the years, which makes it difficult to explain

roots of AI:

philosophy
- how the world works
- what we think about something
mathematics
- formal logic
- algorithms -
psychology
- behaviorism
- cognitive science
neuroscience
economics
- decision theory, utility theory
- game theory
computer science (1940- about this part of computing)
- efficient computer
linguistics
- natural language processing

The mind is something behind the machine

If you look at the mind as a machine then you can replicate it. If you look the emotional beeing behind the mind then you will see emotions, feelings, free will, etc, and we can't replicate them.

Turing test
rational agent aproach
- the agent is just something that acts
- a rational agent acts like the best outcome

view of intelligence
collective behaviors
agennt -oriented and emergent view of intelligence
- autonoumus systems
- agents situted in enviroment

Agents and their structures - introduction to search

presentation: slidetodoc.com

Agent is just something that perceives and acts. Functions: mapping percepts to actions.

An agent is anything that can be viewed as perceiving its anviroment through sensors and acting upon that enviroment through effectors

based on this definitoin a termostat is an intelligent agent

properties of intelligent agents:

reactivity

the capability to follow the enviroment all the time, including during the reasoning and acting

the agent must take an action when the action is meaningful

proactivity

autonoumus

the agent coióntinues to learn uppon its experience

one whose actions are not based completly on:

built-in knowledge

own experience

truly autonoumus agent can adapt to wide variety of enviroments

social capability

can interact with other agents

agents can work together => multi agent system

other aproaches of defining

viewed as consisting of mental components such as: beliefs, capabilities, choices, commitments

an entity is a software agent if and only if it communicates correctly in an agent communication language

rational agent

does the right thing
the best available action
rational: performande, percept sequence, knowledge about the enviroment, actions
ideal rational
omniscient <- knows everything

performance measure

to determine if the agent does right things
outside the agent <- to be more objective valuation
is defined by some authority
how and wen to evaluate agents success

percept sequnece

knowledge about the enviroment

actions

An agent is a proram and an architecture

percepts
actions
goals
enviroment

PEAS/PAGE description - Performance enviroment actuators and sensors / percepts actions goals environment

Agent	Performance	Enviroment	Actions	Sensors
Interactive English tutor	Maximie scored in the test	Student group	Witten tasks	written words
automated taxi driver	safe, fast, legal trip, maximal profit	Roads, other traffic, pedestrians, customers	Steering, accelerator, brakes, signal	cameras sonar, speedometer
Medical diagnosis system	Healthy patioents minimal costs	patients, hospital staff	questions, tests, treatments	symptoms, patent answers

enviroments

the range can be vast
there are some dimensions that categorize enviroments
characteristics:
- observability
  - fully observale: agents sensors give access to compplete state of the enviroment
  - effectively fully observe: detect everthing which is relevant to me now
  - partially observable: noisy or inaccurate sensors or parts of the state are missing ins sensors sdata *ex: vacuum cleaner can not perceivve dirt in another room
  - unobservable: agent has no sensors at all
- deterministics vs stochastic:
  - deterministic: next state the enviroment is completly determinated by current state and action executed by the agent
  - stochastic: uncertainity about outcomes is quantified in terms of probabilities
- episodic vs sequnetial
  - episodic: the experience is divided in atomic episodes
    - episode: an agent recieves a percept and performs a single action
    - epsodes do not depend on actions taken in prev episodes
    - calssification tasks
  - sequnetial: current decision can be larger
- single vs multi-agent
  - single: only on agent ex: agent solving crosswqord puzzle
  - multi-agent: there are other agents maximizin a performance measure that depends on the behavior of other one ex: chess
- static vs dynamic
- known vs unknown:
  - in known enviroment the designe knows the laws of physics ex: the outcome probability of actios is known
  - unknown: the agent has to learn how the enviroment works: ex an unkown video game, wher don't know how to solve/use it

the skeleton of agent:

choose best action
action: update memory
return: action

table driven agent:

percept sequneces: look-up percepts, table
return action

rule based agent:

percept: interpretation
rule match: interpreted percept, if pattern
rule firing: then pattern
action: return actrion

applications in lgoical reasoning systems

Intelligent agent

model based agent

goal based agent

goal
interface
search and planning

utility agent

utility function assigs a real number starting how good the state is for the agent
the higher the utility is the better the state is for the agent

lab1

compare performance of differen computer agent types with human agents that have the same information as the computer agents

The objective of the laboratory work is to study behaviour of different types of intelligent agents on the example of vacuum cleaning agent. The following types of agents are used in experiments: global optimization agent, local optimisation agent and random choice agent. Results achieved by human agent and computer agent must be compared based on experiments with the software “AgSim – 05”. The answers to the control tasks and conclusions must be given in the report.

Search

problem solving agent

goal formulation
problem formulation - process of decideing what actions and states to consider
search: systematic exploratio of the sequnece of alternative states that appear in a problem solving
current state
action
return

problem types:

single state problem
- percepts: give enough info to tell exactly which state is it
- agent knows exactly what each of its actions
multiple state poblem
- percepts give not enough info to tell which state is it
- agent knows all the effects of its actions
- agent must reason about f states that it might get to

problem: isa collection of information that the agent can use to decide what to do. specialization for single state probelm

state space s the set of all states reeacheble from the initial state by any sequence of actions

initial state: is the state that the agent knows itself to be in

goal statet corresponds to the solution of the problem

operator describes an action in terms od which state will be reached by carrying out the action

representation: four tuple: [N, A, IS, GS]

N - nodes - states

A - edges

IS - Initial state

GS - goal states: agent can apply if explicit state or measurable property of the states encountered in the search, property of the path developed in the search.

representation with raphs:

nodes: particluar problem splution states
arcs: problem solving process
initial states: from root of the graph and correspond to the given information ina problem instance

Branching factor: number of edges of a node.

how to mesure effectiveness:

does a search find a solution at all?
is a solution good?
what is the search cost associated with the time and memory required to fin a solution

total cost = pth cost + search cost

search process:

builds up a search tree
start drom search node - initial state
dettermining the childre of state is called expanding the state

at each step the search algorythm chhooses one leaf node to excpand:

they have not been expanded
expanded but not the correct way

Uninformed search

Backtracking

systematic move foreward and not coming back

starts at initial and end at particular solution

there is no solution at any time

but how will machine implement it?

Classification of Search strategies

there are cases where data driven is better then goal driver, and vice versa

data driven

like above

goal driven:

search strarts from the goal, applies operators that could produce the goal, and chains backward to given facts of the problem
searching backward means generating
is better when there is more branching factor

informed search

problem specific knowledge is used, so we have to know the problem

uninformed search

no information is available about:

the number of steps
the cost
how good is the current state

in order which nodes are expended is:

breath first search
uniform cost search
depth-first search
depth limited search
iterative deepening search
bidirectional search

breath first search

this search expands the shallowest node in the search tree first

YouTube - Breadth First Search Algorithm | Shortest Path | Graph Theory

uniform cost search

this search expands the lowesst cost node on the frontier first

the path cost must never decrease g(successor(n)) >= g(n)

depth-first search

works the same way as the backtracking

depth limited search

same as depth-first search, but have a maximum trashold what can achive in a direction

iterative deepening search

has a limit

if fails then increase the limit

each time restart the search algo from the root

bidirectional search

iterative deepening search and depth-first from the other direction at the same time

saving path

Implementing search algorithms

two lists are used to implement algo:

open: the nodes which we didn't found yet
closed: which we already found it

to implement breadth-first and depth-first

initially open contains the initial node
states are excluded from the left side of the open list
new states are included in the right side of open in case of breadth first search and in the left side in case of depth first search
algo terminates when the goal state is either in the first position of the list open or is inserted into the list closed
states that already are open or closed are not added to the list open

evaluation of uniformed search strategies

completeness - is a solution guaranteed?
time complexity - time needed for solution
space complexity - memory needed for solution
optimality - is highest quality solution quaranteed when there are several different solutions?

criterion	breadth first	uniform	depth first	dept limited	iterative deepening	bidirectional
complete	+	+	-	+ if l>= d	+	+
time	b^d	b^d	b^m	b^l	b^d	b^(d/2)
space	b^d	d^b	bm	bl	b^(d/2)
optimal	+	+	-	-	+	+

b branching factor (every state can be expanded to yield b new states)
d is the depth of solution
m is the maximum depth of the search tree
l is the depth limit

NEXT TIME TEST

open zoom
open camera
turn off microphone
open test
take it
send it in MSWord or Pdf

heuristics

experience bassed
empirically found
allows process the most promising branches first

to define heuristics:

costs: lower value is better
*quality of the node path: higher values is better

depend on the domain/state space and not the algorithms/search strategy

classification of search strategies

greedy search - minimizing the estimate cost to real search
A*
IDA* - iterative deepening A*
SMA* search - simpilfied memory bounded A*
iterative improvement algorithms
- hill climbing
- beam search
- random restart hill climbing
- stochastic hill climbing
- simulated annealing

Best first search

this search expands thew node appears to be the best according to the evaluation fuinction

greedy search

https://www.geeksforgeeks.org/greedy-algorithms/

reduces the time and pace complexity with a good heuristic function

informed search strategies

more in: https://github.com/gabboraron/MI-EA

A*

f(n) = g(x) + h(x)

memory bounded

condservs memory

iterative deepening A* - IDA*

-each iteration is depth first search - modified to use an f-cost expands all nodes inside the contour for the current f-cost peeping over the contour to find out where the next contour is, where all nodes have f(n) less than or equal to f-cost

only goes through two or three restarts since typically f only increases several times along the heuristic route

iterative improvement algorithms

cheap - fast
this search explores the state space trying to find the best value of estimated cost
take to reach the goal very fast and with very low resources

SMA

more complicated tasks can be handled with this

beam search

uses only the best W nodes at each level the other nodes are ignored
uses lot of memory
breaadth first level by level
all nodes from the list OPEN are expeaded
expect only the the best weight

beam means how many nodes can we store

simulated annealing

uses local maxima
picks a randomly seleted successor of the current node
if the move improves the situation always executed
probability decrases eponentially with the badness of the move calculated as difference of values between next and current state

hill climbing

when we are creating path it should be good all the time
if local choices are perfectly then the algo is it too

stochastic hill climbing

randomy choses on of sucessors that imprves the quality of the state
depends on the improvement
lowest heuristic: allwys going to lower value in the graph!
highest heuristic: allwys going to hisgher value in the graph!

weakness of informed search strategies

because heuristics use limited information they can lead a search algorithm to a suboptimal solution or fail to find any solution at all

best first: is not an answer to all searching needs because at the worst case exponentional growt will occur.

greedy star: is optimal, finding the shortest path, but in worth case is neither complete

A*:

IDA*: it can expand N^2 nodes in N iteration

SMA* not complete and not optimal, huge memory usage

Hill climbing: will halt even thought the solution may be far from satisfactory

different use cases needs different algorithms

two-player games

two player (human or computer)
conflict games
players use heuristics

like in chess:

both players know where the pieces are
both know the possible moves

like in card games:

players do not know the other's pieces

in each situation each player has full information:

chess
tick-tack-toe
checkers
chess

are not:

poker
bridge
dice games

game tree

game trees consists of nodes and arcs
nodes represent a possible state of game
edges represent a the way how we occure there
root node is the starting position
internal node
leaf nodes are the possible termination places of the game
each level in gme tree is a turn of a player

Mini-Max algorithm

dividing into minimizer and maximizer levels of the game tree
the idea is to maxximise your result in the game by minimizing opponents result
the algorithm comes down to the leaf nodes and assigns the heuristic values that denote how good the state is for the maximizer
the evaluations are transferred upwards
heuristic evaluations:
- the heuristic evaluations are given as numbers
- they characterise how good the state is for the given player
- 1 if the maximizer wins
- -1 if minimizer wins
- 0 if draw

steps of mimiax algorithm

javatpoint.com description

build full gam tree

divide the gam tree into the levels (MIN-MAX)

assign heuristi evaluations to the leaf nodes

transfer the evaluations upwards till the root node

MINIMAX

in building the game tree choosing the parameters is the key

Complex games

it is hard to create the game tree

looking forward by n moves

F(s) = a*B + b*R+c*M+d*C+e*P+f*A where

B total value of pieces

R protection of the king

M mobility of the pieces

C contorl over the center

P pawn structure

A possibel attack

a;b;c;d;e;f - weights

using knowladge

what is knowladge?

an individual understanding of given subject

for us only the domain specific knowledge is intersting!

domain is a well-focused area!

in general a representation is a set of conventions about how to describe a class of things

a description makes use of the conventions of a represenation to describe some particular thing the function of any representation scheme is to capture essential features of a problem domain => make this available for the machine

the represenation language must allow the programmer to use the knowladgebase

wumpus world

https://www.javatpoint.com/the-wumpus-world-in-artificial-intelligence

cave explored by agent consist from rooms connected with passageways

knewledge representation

declarative knowledge

concepts
facts
objects

procedureal knowledge: describes how a problem is solved

rules
strategies
agendas
procedures

heuristic knowledge (shallow knowledge): describe a rule of thumb that guides the reasoning process; it is empirical and represents the knowledge compiled by an expert throught the experience of solving past problems

rules of thumb

meta knowledge: describes knowledge about knowledge. This type of knowlede is used to pick other knowledge that is best suited for solving a problem. experts use this type of knowledge to enhance the efficiency of problem solving by directing their reasoning in the most promising area.

knowledge about the other types of knowledge and how to use them

structural knowledge: describes knowledge structures. this type of knowledge describes an expert's overall mental model of the problem. The expert's mental model of concepts sub-concepts and objects is typical of rhis type of knowledge

rule sets
concepts relationships
concept to object relationships

knowledge representation language:

syntax of the language describes the possible config

semantics determines the facts in the world which the sentences refer

if yntacs and semantics are defines precisely the language is called logic

Sentences => semantics => facts

representation must be encoded some way that can be physically stored within an intelligent system

it is imposible to put the worlld inside a computer so all reasoning mechanisms must operate on representations of facts, rather than on facts themselves

representation:

kexical

structural

procedural

semantic

the role of logic in artificial intelligenc

logical formalism suggests a powerful way of deriving new knowledge from olf mathematifal deduction; this conclusion can be made that a new statement is true by proving that it follows from the statements that are already known.

sound
complete

AI programming requires means of capturing and reasoning about qualitative aspects of a problem

logic provides ai progrmmers with a well-definied language => create automation for us

logical axioms, and "problmes as theorems" representations help us attack a problem

=> AI programming languages: PROLOG

not all problems can be solved with logic
some knowledge resits embodiment in axioms
logic is weak as a representation for certain kinds of knowledge
the notation of pure logic does not allow to express such notations as heuristic distances
difficult to develop a system like that

Kinnds of logic:

general lang:
- propositional calculus
- predicate calculus
based on general lang:
- temporal
- probability
- fuzzy

propositional logic

symbols: P, Q, R, S, T
truth symbols: True, False
connectivities: ^; ˇ; =>, ≅, !

where

every propositinoal symbo and truth symbol is a sentence
negation of a sentance is a sentance
conjunction or AND of two sentences is a sentence

how to test truth validity:

truth table method:
- + any propositional calculus expression can be tested
- - time consuming in case of large number of propositions
complete proof procedures
- + can procedure any expression that logically folllows from a set of experiences
- inference mechanism <-methods by the conclusion is reached

a |= b meaning:b can be derived from a by inference: a/b

an interfence rule is sound if the conclusion is true, in all cases where the premises are true
to prove the soundness the truth table must be constructed with one line for each possible model of the proposition symbols in the premises, in all model where the premise is true, the conclusion must be also true

intelligent agents

predictive calculus

B=It rained on Firday
will it rain on Monday?

irreducible syntactic elements
symbols may represent constants tom represents Tom. the true and false are reserved as truth symbols
a fact or proposition is divided into two parts: predicate and argument. The arguments athe object or objecxt of the proposition

Tom owns the car => owns(tom,car)
- owns - predicate
- tom, car - arguments and symbols

variables: represent general classes of objects or poperties
functions: mapping one or more leemnt of a set (domain) to a unique elemnt of another set (range)
- elements of the domain and range are objects in the world of discourse
- every function symbol has an associated arity (number od elements in domain which are used in the function)
functions can be used within predicates
atomic sentences

if x and y is a syntax then x and y and x=>y and x or y and ∀x and ∃x is also a syntax.

∀X (cat(X) => animal(X)) means: All cats are animals
∀X (likes(X, mary) and nice(mary) => nice(X)) mens: all instances of X who like Mary, and if Mary is nice, then it is implied that those who like Mary are also nice.
∀X, Y (parent(X, Y) => child(Y, X)) is “For all persons, if X is a parent of Y, then Y is a child of X”
∀X ∃Y loves(X, Y) means Everybody loves somebody

Predicate calculus semantics provide a formal basis, for determining the truth value of well-formed expressions

The truth of expressions depends on the mapping of constants, variables, predicates, and functions into objects and relations in the domain of discourse

ex: owns(john, ford); friends(john, bill)

Let the domain D be a nonempty set.

An interpretation over D is an assignment of the entities of D to each of the constant, variable, predicate and function symbols of a predicate calculus expression

Each constant is assigned an element of D

Each variable is assigned to a nonempty subset of D.

Each function f of arity m is defined on m arguments of D and defines a mapping from D^m into D

Each predicate p of arity n is defined on n arguments from D and defines a mapping from D^n into {T, F}

Given an interpretation, the meaning of an expression is a truth value assignment over the interpretation.

Let E be an expression and I an interpretation for E over a nonempty domain D. The truth value for E is determined by:

The value of constant is the element of D it is assigned to by I

The value of a variable is the set of elements of D it is assigned to by I

The value of a function expression is that element of D obtained by evaluating the function for the parameter values assigned by the interpretation

The value of truth symbol “true” is T and “false” is F

The value of an atomic sentence is either T or F, as determined by the interpretation I

The value of the negation of a sentence is T if the value of the sentence is F and is F if the value of the sentence is T

***If the domain of interpretation is infinite, exhaustive testing of all substitutions to a universally quantified variable is computationally impossible: the algorithm may never halt

That is why the predicate calculus is said to be undecidable***

Evaluating the truth value of a sentence containing an existentially quantified variable may not be easier

If the domain of the variable is infinite and the sentence is false under all substitutions, the algorithm will never halt

In the language defined, universally and existentially quantified variables may refer only to objects (constants) in the domain of discourse

This language is called the first-order logic (first-order predicate calculus)

any natural language may be expressed in first-order logic using the symbols, connectives, and variable symbols

Example:

“All basketball players are tall”. Mappings:

∀X(basketball_player(X) and tall(X))

∀XX(basketball_player(X) => tall(X))

In propositional calculus two expressions match if and only if they are syntactically identical

In predicate calculus a decision process based on universal instantiation is needed for determining the variable substitutions under which two or more expressions can be made identical

Unification is an algorithm for determining the substitutions needed to make two predicate calculus expressions match

All variables must be universally quantified

Existentially quantified variables may be eliminated by replacing them with the constants

all quantifiers can be dropped

Skolemization replaces each existentially quantified variable with a function that returns the appropriate constant *ex: ∀X ∃Y father(X,Y) become ∀X father(X, f(X))

Unification is complicated because a variable may be replaced by any term, including other variables and function expressions of arbitrary complexity

STRIPS operators consist from:

the action description
the precondition: it is a conjuction of predicates that must be true before the operator can be applied
the effect of an operator: it isa conjuction of predicates that describes how the situation changes when the operator is applied

operator(ACTION: action's name,
PRECONDITION: p(x) and q(y) and ...
EFFECT: r(y) and w(x) and ...)

progression planner: searches forward
regression planner: searches backward

plans:

pratially ordered plans
totally ordered plans

POP algorithm:

starts with minimal partial plan
extends the plan by achieving a precondition of needed step
records a causal link for the newly achieved precondition
if the new step threatens an existing causal link or an existing step threatens the new causal link the threat resolving is applied
if at any point the algorithm fails to find a relevant operator or resolve a threat it backtracks to previos choice point

execution monitoring: the agent monitors what is happening during the execution of the plan. The agent can do replanning. the execution monitoring agent is deferring the job of dealing with those conditions until they actually arise

Intorduction of machine learning

Learning element is responsible for making imporvements and for efficiency of the performance element. Performance element is responsible for sleecting exrternal actions. Critic is designed to tell the learning elemeent how well the agent is doing. Problem generator is responsible for suggesting actions

decision tree as performance element

decision trees are a method for learning boolean functions.

internal nodes response to a test of the value of the attributes

The tree can be expaussed as a confunction of inividual implications to the paths through the tree ending in YES nodes

The decision tree must answer in expressions.

the DT can't represent any set - implicity limiter
any boolean function can be written in a DT

construction of a DT

Machine learning

Neural neworks

artificial neural network is composed of a number of a sinple arithmetic computing elements or nodes connected by links - synoyms: connectionism, parallel distributed processing, and neural copmutation.
mathematical model of operation of brain
brains information capacity is thought to emerge primarily from networks of such neurons

math model of neuron

1943 McCulloch and Pitts - simple math model of neuron is devised
neural network is composed of a number of units connected by links
each link has a numeric weight associated with

first linear component called input ifunction: $in_i=\sum_{j=0}^{n}W_{ji}\cdot a_j$ wehere w0i connected to fixed input a0 =-1

second is a nonliner component called the activation function, g the ttransforms the weighted sum into the final value $a_i=g(in_i)=g(\sum_{j=0}^{n}W_{ji}\cdot a_j)$

activation function

the activation function g is designed to meet two wishes

network structures

feed forward networks
recurrent network - the network structure is cyclic
- feeds its outputs back into its own inputs
- the activation levels of recurrent networks form a dynamic system that may reach a stable state or exhibit oscillations
hopfield network are the best-understood class of recurrent networks
- bidirectional connections with symmetric weights

feed forward neural networks

single layered feed-forward neural network
with a treshold activation function the percetron represents a boolean function
a treshold perceptron returns 1 if and only if the weighted sum of its input is positive $\sum_{j=0}^{n}W_{j}\cdot a_j)>0$
this eq defines a hyperplane in the input space, perceptron returns 1 if and only if the input is on one side of that hyperplane
for this reason the threshold perceptron is called a liner separator
the error = T-O where O is the output of the perceptron on the example and Tis the true output value.
- if the error is positive then O must be increased
- if error is negative
- the perceptron learning rate is alpha and $W_{j}=W_j+\alpha\cdot I_j\cdot Err$

how do networks learn?

edible 1
not edible 0
good fruit 1
not good fruit 0

multilayer network

back-prpagation learning: at the output layer in weight update rule the activation of the hidden unit aj is used instead of the input value: and the rule contains a term for the gradient of the activation function

$W_{ji}=W_{ji}+\alpha\cdot a_j\cdot Err\cdot g'(in_i)$

where Erriis the erro (Ti-Oi) at the output node: g'(ini) is the derivative of the activation function g -> only sigmouid function can e used in multilayer network

new error term is which for output nodes is $\Delta_i = Err_i\cdot g'(in_i)$

update rule now is the following $W_{ji}=W_{ji}+\alpha \cdot a_j \cdot \Delta_i$

repeat this calculation the following for each layer in the network, until the earliest hidden layer is reached: propagate the delta value

divide example set
- how many units
- kind of units
- units are connected
- initialize the weights
- encode the examples in terms of inputs
train model - train the weights
test model

uncertain knowledge and reasoning

agents almost never have access to the whole truth about network
utility theory is used to represent and reason with prefereneces
termutility is used in sense of the quality of being useful
every state has a degree of usefulness or utility to an agent and that the agent will preffer states with higher utility
the utility of state is realtive o the aent whose preferneces the utility function is supposed to represent

joint pobability

$P(a)=\sum_{e_i=e(a)}P(e_i)$

an atomic snetence is a full specification of the state of the world

values for all variables world cosist of

The probability of both events: $P(cavity |Toothache) = \frac{P(Cavity \wedge Toothache)}{P(Toothache)}$

the false version: $P(\not cavity |Toothache) = \frac{P(\not Cavity \wedge Toothache)}{P(Toothache)}$

the multiplication rule: $P(A\wedge B) = P(A|B)\cdot P(B)$

Belief Bayesian network: a data structure to represent the dependence between variables and to give a concise specification of the joint probability distribution

a set of random variables makes up nodes of the network

set of directed links

the probability of all events: $P(x_i,...x_n) = \prod_{i=1}^n{x_i}$

transtion model

used to refer ro the set

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backtrack_search.jpg		backtrack_search.jpg
breadt fst.jpg		breadt fst.jpg
depth limited.jpg		depth limited.jpg
saving path.jpg		saving path.jpg

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