Accounting for uncertainty is a crucial component in decision making (e.g., classification) because of ambiguity in our measurements.
Probability theory is the proper mechanism for accounting for uncertainty.
The document provides an overview of key concepts in probability theory and stochastic processes. It defines fundamental terms like sample space, events, probability, conditional probability, independence, random variables, and common probability distributions including binomial, Poisson, exponential, uniform, and Gaussian distributions. Examples are given for each concept to illustrate how it applies to modeling random experiments and computing probabilities. The three main axioms of probability are stated. Key properties and formulas for expectation, variance, and conditional expectation are also summarized.
1. The document covers probability axioms and rules including the additive rule, conditional probability, independence, and Bayes' rule. It also defines discrete and continuous random variables and their probability distributions.
2. Important discrete distributions discussed include the Bernoulli distribution for a binary outcome experiment and the binomial distribution for repeated Bernoulli trials.
3. Techniques for counting permutations, combinations, and sequences of events are presented to handle probability problems involving counting.
This document provides a concise probability cheatsheet compiled by William Chen and others. It covers key probability concepts like counting rules, sampling tables, definitions of probability, independence, unions and intersections, joint/marginal/conditional probabilities, Bayes' rule, random variables and their distributions, expected value, variance, indicators, moment generating functions, and independence of random variables. The cheatsheet is licensed under CC BY-NC-SA 4.0 and the last updated date is March 20, 2015.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, expectations, independence, and more. The cheatsheet is designed to summarize essential concepts in probability.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, moments, and more. The cheatsheet is regularly updated with comments and suggestions submitted through a GitHub repository.
Probability formula sheet
Set theory, sample space, events, concepts of randomness and uncertainty, basic principles of probability, axioms and properties of probability, conditional probability, independent events, Baye’s formula, Bernoulli trails, sequential experiments, discrete and continuous random variable, distribution and density functions, one and two dimensional random variables, marginal and joint distributions and density functions. Expectations, probability distribution families (binomial, poisson, hyper geometric, geometric distribution, normal, uniform and exponential), mean, variance, standard deviations, moments and moment generating functions, law of large numbers, limits theorems
for more visit http://tricntip.blogspot.com/
STOMA FULL SLIDE (probability of IISc bangalore)2010111
This document provides an overview of a course on stochastic models and applications. It includes:
- A list of reference materials for the course.
- Background prerequisites in calculus, matrix theory, and basic probability concepts.
- Details on course grading with midterm exams, assignments, and a final exam comprising the overall grade.
- An introduction to probability theory and examples of applications in engineering, statistics, and algorithms.
- A review of basic probability concepts like sample space, events, axioms, and examples of finite and countably infinite sample spaces.
The document discusses random processes and probability theory concepts relevant to communication systems. It defines key terms like random variables, sample space, events, probability, and distributions. It describes different types of random variables like discrete and continuous, and their probability mass functions and density functions. It also discusses statistical measures like mean, variance, and covariance that are used to characterize random signals and compare signals. Specific random variables discussed include binomial and uniform. The document provides foundations for analyzing random signals in communication systems.
The document provides an overview of key concepts in probability theory and stochastic processes. It defines fundamental terms like sample space, events, probability, conditional probability, independence, random variables, and common probability distributions including binomial, Poisson, exponential, uniform, and Gaussian distributions. Examples are given for each concept to illustrate how it applies to modeling random experiments and computing probabilities. The three main axioms of probability are stated. Key properties and formulas for expectation, variance, and conditional expectation are also summarized.
1. The document covers probability axioms and rules including the additive rule, conditional probability, independence, and Bayes' rule. It also defines discrete and continuous random variables and their probability distributions.
2. Important discrete distributions discussed include the Bernoulli distribution for a binary outcome experiment and the binomial distribution for repeated Bernoulli trials.
3. Techniques for counting permutations, combinations, and sequences of events are presented to handle probability problems involving counting.
This document provides a concise probability cheatsheet compiled by William Chen and others. It covers key probability concepts like counting rules, sampling tables, definitions of probability, independence, unions and intersections, joint/marginal/conditional probabilities, Bayes' rule, random variables and their distributions, expected value, variance, indicators, moment generating functions, and independence of random variables. The cheatsheet is licensed under CC BY-NC-SA 4.0 and the last updated date is March 20, 2015.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, expectations, independence, and more. The cheatsheet is designed to summarize essential concepts in probability.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, moments, and more. The cheatsheet is regularly updated with comments and suggestions submitted through a GitHub repository.
Probability formula sheet
Set theory, sample space, events, concepts of randomness and uncertainty, basic principles of probability, axioms and properties of probability, conditional probability, independent events, Baye’s formula, Bernoulli trails, sequential experiments, discrete and continuous random variable, distribution and density functions, one and two dimensional random variables, marginal and joint distributions and density functions. Expectations, probability distribution families (binomial, poisson, hyper geometric, geometric distribution, normal, uniform and exponential), mean, variance, standard deviations, moments and moment generating functions, law of large numbers, limits theorems
for more visit http://tricntip.blogspot.com/
STOMA FULL SLIDE (probability of IISc bangalore)2010111
This document provides an overview of a course on stochastic models and applications. It includes:
- A list of reference materials for the course.
- Background prerequisites in calculus, matrix theory, and basic probability concepts.
- Details on course grading with midterm exams, assignments, and a final exam comprising the overall grade.
- An introduction to probability theory and examples of applications in engineering, statistics, and algorithms.
- A review of basic probability concepts like sample space, events, axioms, and examples of finite and countably infinite sample spaces.
The document discusses random processes and probability theory concepts relevant to communication systems. It defines key terms like random variables, sample space, events, probability, and distributions. It describes different types of random variables like discrete and continuous, and their probability mass functions and density functions. It also discusses statistical measures like mean, variance, and covariance that are used to characterize random signals and compare signals. Specific random variables discussed include binomial and uniform. The document provides foundations for analyzing random signals in communication systems.
Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.
Recursive State Estimation AI for Robotics.pdff20220630
1) Recursive state estimation uses probabilistic methods like Bayes filters to estimate states of a dynamic system from sensor measurements over time. Bayes filters involve prediction of state based on motion model and correction of prediction based on sensor observations using Bayes' rule.
2) An example of applying a Bayes filter to estimate the state of a door being open or closed is given. The robot's belief is updated as it takes actions like pushing the door and receives sensor feedback.
3) Key concepts discussed include belief distributions, probabilistic generative models relating state transitions and measurements, and the Bayes filter algorithm involving prediction and correction steps.
This document outlines the key concepts that will be covered in Lecture 2 on Bayesian modeling. It introduces the likelihood function and how it can be used to determine the most likely parameter values given observed data. It provides examples of applying Bayesian modeling to proportions, normal distributions, linear regression with one predictor, and linear regression with multiple predictors. The lecture aims to give students a basic understanding of how Bayesian analysis works and prepare them for fitting linear mixed models.
This document provides an introduction to Bayesian analysis and probabilistic modeling. It begins with an overview of Bayes' theorem and common probability distributions used in Bayesian modeling like the Bernoulli, binomial, beta, Dirichlet, and multinomial distributions. It then discusses how these distributions can be used in Bayesian modeling for problems like estimating probabilities based on observed data. Specifically, it explains how conjugate prior distributions allow the posterior distribution to be of the same family as the prior. The document concludes by discussing how neural networks can quantify classification uncertainty by outputting evidence for different classes modeled with a Dirichlet distribution.
Statement of stochastic programming problemsSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 1.
More info at http://summerschool.ssa.org.ua
This document provides an overview of information theory and coding concepts including:
1) Definitions of information, entropy, joint entropy, conditional entropy, and mutual information are introduced along with examples of calculating these quantities for discrete memoryless sources and channels.
2) Shannon's theorem for channel capacity is discussed and the channel capacity of a discrete memoryless channel is defined as the maximum mutual information over all possible input distributions.
3) Properties of entropy such as it being a measure of uncertainty, having a minimum of 0 and maximum of log2K, and being maximized when probabilities are equal are proven.
The document discusses Bayes' rule and entropy in data mining. It provides step-by-step derivations of Bayes' rule from definitions of conditional probability and the chain rule. It then gives examples of calculating entropy for variables with different probability distributions, noting that maximum entropy occurs with a uniform distribution where all outcomes are equally likely, while minimum entropy occurs when the probability of one outcome is 1.
1) This document introduces concepts of probability and statistics that will be covered in the course, including defining probability, discrete vs. continuous probability distributions, and how to calculate mean, mode, median, and variance.
2) It discusses the differences between accuracy and precision in measurements and explains common sources of measurement errors like statistical and systematic uncertainties.
3) Key points on experimental results are presented, such as how statistical and systematic errors are combined and various ways of quoting measurements and their uncertainties.
This document contains permissions and copyright information for Chapter 2 of the Handbook of Applied Cryptography. It grants permission to retrieve, print, and store a single copy of this chapter for personal use, but does not extend permission to bind multiple chapters, photocopy, produce additional copies, or make electronic copies available without prior written permission. Except as specifically permitted, the standard copyright from CRC Press applies and prohibits reproducing or transmitting the book or any part in any form without prior written permission.
Probability
Random variables and Probability Distributions
The Normal Probability Distributions and Related Distributions
Sampling Distributions for Samples from a Normal Population
Classical Statistical Inferences
Properties of Estimators
Testing of Hypotheses
Relationship between Confidence Interval Procedures and Tests of Hypotheses.
The document summarizes parameter estimation methods: 1) The method of moments estimates parameters by equating sample and population moments. 2) The maximum likelihood method estimates parameters by maximizing the likelihood function. 3) Mean-squared error compares estimators based on their variance and bias, preferring those with the smallest error.
This document provides an introduction to key concepts in probability and statistics for machine learning. It covers topics such as sample spaces, events, axioms of probability, permutations, combinations, conditional probability, Bayes' rule, random variables, probability distributions, expectations, variance, transformations of random variables, jointly distributed random variables, parameter estimation, and the central limit theorem.
This document presents a new model of decision making under risk and uncertainty called the Harmonic Probability Weighting Function (HPWF) model. The HPWF model incorporates mental states using a weak harmonic transitivity axiom and an abstract harmonic representation of noise. It explains phenomena like the conjunction fallacy and preference reversal. The HPWF uses a harmonic component controlled by a phase function to characterize how a decision maker's mental states influence probability weighting. Maximum entropy methods can be used to derive a coherent harmonic probability weighting function from the HPWF model.
1. The document discusses basic concepts in probability and statistics, including sample spaces, events, probability distributions, and random variables.
2. Key concepts are explained such as independent and conditional probability, Bayes' theorem, and common probability distributions like the uniform and normal distributions.
3. Statistical analysis methods are introduced including how to estimate the mean and variance from samples from a distribution.
Accelerating Metropolis Hastings with Lightweight Inference CompilationFeynman Liang
This document summarizes research on accelerating Metropolis-Hastings sampling with lightweight inference compilation. It discusses background on probabilistic programming languages and Bayesian inference techniques like variational inference and sequential importance sampling. It introduces the concept of inference compilation, where a neural network is trained to construct proposals for MCMC that better match the posterior. The paper proposes a lightweight approach to inference compilation for imperative probabilistic programs that trains proposals conditioned on execution prefixes to address issues with sequential importance sampling.
2 Review of Statistics. 2 Review of Statistics.WeihanKhor2
This document provides an overview of discrete probability distributions, including the binomial and Poisson distributions.
1) It defines key concepts such as random variables, probability mass functions, and expected value as they relate to discrete random variables. 2) The binomial distribution describes independent Bernoulli trials with a constant probability of success, and is used to calculate probabilities of outcomes from events like coin flips. 3) The Poisson distribution approximates the binomial when the number of trials is large and the probability of success is small. It models rare, independent events with a constant average rate and can be used for problems involving traffic accidents or natural disasters.
The document defines and discusses random variables. It begins by defining a random variable as a function that assigns a real number to each outcome of a random experiment. It then discusses the conditions for a function to be considered a random variable. The document outlines the key types of random variables as discrete, continuous, and mixed and introduces the cumulative distribution function (CDF) and probability density function (PDF) as ways to describe the distribution of a random variable. It provides examples of CDFs and PDFs for discrete random variables and discusses properties of distribution and density functions. The document also introduces important continuous random variables like the Gaussian random variable.
Delayed acceptance for Metropolis-Hastings algorithmsChristian Robert
The document proposes a delayed acceptance method for accelerating Metropolis-Hastings algorithms. It begins with a motivating example of non-informative inference for mixture models where computing the prior density is costly. It then introduces the delayed acceptance approach which splits the acceptance probability into pieces that are evaluated sequentially, avoiding computing the full acceptance ratio each time. It validates that the delayed acceptance chain is reversible and provides bounds on its spectral gap and asymptotic variance compared to the original chain. Finally, it discusses optimizing the delayed acceptance approach by considering the expected square jump distance and cost per iteration to maximize efficiency.
In this presentation we describe the formulation of the HMM model as consisting of states that are hidden that generate the observables. We introduce the 3 basic problems: Finding the probability of a sequence of observation given the model, the decoding problem of finding the hidden states given the observations and the model and the training problem of determining the model parameters that generate the given observations. We discuss the Forward, Backward, Viterbi and Forward-Backward algorithms.
Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.
Recursive State Estimation AI for Robotics.pdff20220630
1) Recursive state estimation uses probabilistic methods like Bayes filters to estimate states of a dynamic system from sensor measurements over time. Bayes filters involve prediction of state based on motion model and correction of prediction based on sensor observations using Bayes' rule.
2) An example of applying a Bayes filter to estimate the state of a door being open or closed is given. The robot's belief is updated as it takes actions like pushing the door and receives sensor feedback.
3) Key concepts discussed include belief distributions, probabilistic generative models relating state transitions and measurements, and the Bayes filter algorithm involving prediction and correction steps.
This document outlines the key concepts that will be covered in Lecture 2 on Bayesian modeling. It introduces the likelihood function and how it can be used to determine the most likely parameter values given observed data. It provides examples of applying Bayesian modeling to proportions, normal distributions, linear regression with one predictor, and linear regression with multiple predictors. The lecture aims to give students a basic understanding of how Bayesian analysis works and prepare them for fitting linear mixed models.
This document provides an introduction to Bayesian analysis and probabilistic modeling. It begins with an overview of Bayes' theorem and common probability distributions used in Bayesian modeling like the Bernoulli, binomial, beta, Dirichlet, and multinomial distributions. It then discusses how these distributions can be used in Bayesian modeling for problems like estimating probabilities based on observed data. Specifically, it explains how conjugate prior distributions allow the posterior distribution to be of the same family as the prior. The document concludes by discussing how neural networks can quantify classification uncertainty by outputting evidence for different classes modeled with a Dirichlet distribution.
Statement of stochastic programming problemsSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 1.
More info at http://summerschool.ssa.org.ua
This document provides an overview of information theory and coding concepts including:
1) Definitions of information, entropy, joint entropy, conditional entropy, and mutual information are introduced along with examples of calculating these quantities for discrete memoryless sources and channels.
2) Shannon's theorem for channel capacity is discussed and the channel capacity of a discrete memoryless channel is defined as the maximum mutual information over all possible input distributions.
3) Properties of entropy such as it being a measure of uncertainty, having a minimum of 0 and maximum of log2K, and being maximized when probabilities are equal are proven.
The document discusses Bayes' rule and entropy in data mining. It provides step-by-step derivations of Bayes' rule from definitions of conditional probability and the chain rule. It then gives examples of calculating entropy for variables with different probability distributions, noting that maximum entropy occurs with a uniform distribution where all outcomes are equally likely, while minimum entropy occurs when the probability of one outcome is 1.
1) This document introduces concepts of probability and statistics that will be covered in the course, including defining probability, discrete vs. continuous probability distributions, and how to calculate mean, mode, median, and variance.
2) It discusses the differences between accuracy and precision in measurements and explains common sources of measurement errors like statistical and systematic uncertainties.
3) Key points on experimental results are presented, such as how statistical and systematic errors are combined and various ways of quoting measurements and their uncertainties.
This document contains permissions and copyright information for Chapter 2 of the Handbook of Applied Cryptography. It grants permission to retrieve, print, and store a single copy of this chapter for personal use, but does not extend permission to bind multiple chapters, photocopy, produce additional copies, or make electronic copies available without prior written permission. Except as specifically permitted, the standard copyright from CRC Press applies and prohibits reproducing or transmitting the book or any part in any form without prior written permission.
Probability
Random variables and Probability Distributions
The Normal Probability Distributions and Related Distributions
Sampling Distributions for Samples from a Normal Population
Classical Statistical Inferences
Properties of Estimators
Testing of Hypotheses
Relationship between Confidence Interval Procedures and Tests of Hypotheses.
The document summarizes parameter estimation methods: 1) The method of moments estimates parameters by equating sample and population moments. 2) The maximum likelihood method estimates parameters by maximizing the likelihood function. 3) Mean-squared error compares estimators based on their variance and bias, preferring those with the smallest error.
This document provides an introduction to key concepts in probability and statistics for machine learning. It covers topics such as sample spaces, events, axioms of probability, permutations, combinations, conditional probability, Bayes' rule, random variables, probability distributions, expectations, variance, transformations of random variables, jointly distributed random variables, parameter estimation, and the central limit theorem.
This document presents a new model of decision making under risk and uncertainty called the Harmonic Probability Weighting Function (HPWF) model. The HPWF model incorporates mental states using a weak harmonic transitivity axiom and an abstract harmonic representation of noise. It explains phenomena like the conjunction fallacy and preference reversal. The HPWF uses a harmonic component controlled by a phase function to characterize how a decision maker's mental states influence probability weighting. Maximum entropy methods can be used to derive a coherent harmonic probability weighting function from the HPWF model.
1. The document discusses basic concepts in probability and statistics, including sample spaces, events, probability distributions, and random variables.
2. Key concepts are explained such as independent and conditional probability, Bayes' theorem, and common probability distributions like the uniform and normal distributions.
3. Statistical analysis methods are introduced including how to estimate the mean and variance from samples from a distribution.
Accelerating Metropolis Hastings with Lightweight Inference CompilationFeynman Liang
This document summarizes research on accelerating Metropolis-Hastings sampling with lightweight inference compilation. It discusses background on probabilistic programming languages and Bayesian inference techniques like variational inference and sequential importance sampling. It introduces the concept of inference compilation, where a neural network is trained to construct proposals for MCMC that better match the posterior. The paper proposes a lightweight approach to inference compilation for imperative probabilistic programs that trains proposals conditioned on execution prefixes to address issues with sequential importance sampling.
2 Review of Statistics. 2 Review of Statistics.WeihanKhor2
This document provides an overview of discrete probability distributions, including the binomial and Poisson distributions.
1) It defines key concepts such as random variables, probability mass functions, and expected value as they relate to discrete random variables. 2) The binomial distribution describes independent Bernoulli trials with a constant probability of success, and is used to calculate probabilities of outcomes from events like coin flips. 3) The Poisson distribution approximates the binomial when the number of trials is large and the probability of success is small. It models rare, independent events with a constant average rate and can be used for problems involving traffic accidents or natural disasters.
The document defines and discusses random variables. It begins by defining a random variable as a function that assigns a real number to each outcome of a random experiment. It then discusses the conditions for a function to be considered a random variable. The document outlines the key types of random variables as discrete, continuous, and mixed and introduces the cumulative distribution function (CDF) and probability density function (PDF) as ways to describe the distribution of a random variable. It provides examples of CDFs and PDFs for discrete random variables and discusses properties of distribution and density functions. The document also introduces important continuous random variables like the Gaussian random variable.
Delayed acceptance for Metropolis-Hastings algorithmsChristian Robert
The document proposes a delayed acceptance method for accelerating Metropolis-Hastings algorithms. It begins with a motivating example of non-informative inference for mixture models where computing the prior density is costly. It then introduces the delayed acceptance approach which splits the acceptance probability into pieces that are evaluated sequentially, avoiding computing the full acceptance ratio each time. It validates that the delayed acceptance chain is reversible and provides bounds on its spectral gap and asymptotic variance compared to the original chain. Finally, it discusses optimizing the delayed acceptance approach by considering the expected square jump distance and cost per iteration to maximize efficiency.
In this presentation we describe the formulation of the HMM model as consisting of states that are hidden that generate the observables. We introduce the 3 basic problems: Finding the probability of a sequence of observation given the model, the decoding problem of finding the hidden states given the observations and the model and the training problem of determining the model parameters that generate the given observations. We discuss the Forward, Backward, Viterbi and Forward-Backward algorithms.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Assessment and Planning in Educational technology.pptxKavitha Krishnan
In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
2. 2
Why Bother About Probabilities?
• Accounting for uncertainty is a crucial component
in decision making (e.g., classification) because of
ambiguity in our measurements.
• Probability theory is the proper mechanism for
accounting for uncertainty.
• Need to take into account reasonable preferences
about the state of the world, for example:
"If the fish was caught in the Atlantic ocean, then
it is more likely to be salmon than sea-bass
3. 3
Definitions
• Random experiment
– An experiment whose result is not certain in advance
(e.g., throwing a die)
• Outcome
– The result of a random experiment
• Sample space
– The set of all possible outcomes (e.g., {1,2,3,4,5,6})
• Event
– A subset of the sample space (e.g., obtain an odd
number in the experiment of throwing a die = {1,3,5})
4. 4
Intuitive Formulation
• Intuitively, the probability of an event a could be defined
as:
• Assumes that all outcomes in the sample space are equally
likely (Laplacian definition)
Where N(a) is the number that event a happens in n trials
6. 6
Prior (Unconditional) Probability
• This is the probability of an event prior to arrival
of any evidence.
P(Cavity)=0.1 means that “in the absence of any
other information, there is a 10% chance that the
patient is having a cavity”.
7. 7
Posterior (Conditional) Probability
• This is the probability of an event given some
evidence.
P(Cavity/Toothache)=0.8 means that “there is an
80% chance that the patient is having a cavity
given that he is having a toothache”
8. 8
Posterior (Conditional) Probability
(cont’d)
• Conditional probabilities can be defined in terms of
unconditional probabilities:
• Conditional probabilities lead to the chain rule:
P(A,B)=P(A/B)P(B)=P(B/A)P(A)
( , ) ( , )
( / ) , /
( ) ( )
P A B P A B
P A B P B A
P B P A
9. 9
Law of Total Probability
• If A1, A2, …, An is a partition of mutually exclusive events
and B is any event, then
• Special case :
• Using the chain rule, we can rewrite the law of total
probability using conditional probabilities:
1 1 2 2
1
( ) ( / ) ( ) ( / ) ( ) ... ( / ) ( )
( / ) ( )
n n
n
j j
j
P B P B A P A P B A P A P B A P A
P B A P A
B
A1
A2
A3
S
A4
( ) ( , ) ( , ) ( / ) ( ) ( / ) ( )
P A P A B P A B P A B P B P A B P B
10. 10
Law of Total Probability: Example
• My mood can take one of two values
– Happy, Sad
• The weather can take one of three values
– Rainy, Sunny, Cloudy
• We can compute P(Happy) and P(Sad) as follows:
P(Happy)=P(Happy/Rainy)+P(Happy/Sunny)+P(Happy/Cloudy)
P(Sad)=P(Sad/Rainy)+P(Sad/Sunny)+P(Sad/Cloudy)
11. 11
Bayes’ Theorem
• Conditional probabilities lead to the Bayes’ rule:
where
• Example: consider the probability of Disease given Symptom:
where
( / ) ( )
( / )
( )
P B A P A
P A B
P B
( / ) ( )
( / )
( )
P Symptom Disease P Disease
P Disease Symptom
P Symptom
( ) ( / ) ( )
( / ) ( )
P Symptom P Symptom Disease P Disease
P Symptom Disease P Disease
( ) ( , ) ( , ) ( / ) ( ) ( / ) ( )
P B P B A P B A P B A P A P B A P A
12. 12
Bayes’ Theorem Example
• Meningitis causes a stiff neck 50% of the time.
• A patient comes in with a stiff neck – what is the probability
that he has meningitis?
• Need to know two things:
– The prior probability of a patient having meningitis (1/50,000)
– The prior probability of a patient having a stiff neck (1/20)
P(M/S)=0.0002
( / ) ( )
( / )
( )
P S M P M
P M S
P S
13. 13
General Form of Bayes’ Rule
• If A1, A2, …, An is a partition of mutually exclusive events
and B is any event, then the Bayes’ rule is given by:
where
( / ) ( )
( / )
( )
i i
i
P B A P A
P A B
P B
1
( ) ( / ) ( )
n
j j
j
P B P B A P A
14. 14
Independence
• Two events A and B are independent iff:
P(A,B)=P(A)P(B)
• From the above formula, we can show:
P(A/B)=P(A) and P(B/A)=P(B)
• A and B are conditionally independent given C iff:
P(A/B,C)=P(A/C)
e.g., P(WetGrass/Season,Rain)=P(WetGrass/Rain)
15. 15
Random Variables
• In many experiments, it is easier to deal with a summary
variable than with the original probability structure.
• Example: in an opinion poll, we ask 50 people whether
agree or disagree with a certain issue.
– Suppose we record a "1" for agree and "0" for disagree.
– The sample space for this experiment has 250 elements.
– Suppose we are only interested in the number of people who agree.
– Define the variable X=number of "1“ 's recorded out of 50.
– Easier to deal with this sample space (has only 51 elements).
16. 16
Random Variables (cont’d)
• A random variable (r.v.) is the value we assign to the
outcome of a random experiment (i.e., a function that
assigns a real number to each event).
X(j)
17. 17
Random Variables (cont’d)
• How is the probability function of the random variable is
being defined from the probability function of the original
sample space?
– Suppose the sample space is S=<s1, s2, …, sn>
– Suppose the range of the random variable X is <x1,x2,…,xm>
– We observe X=xj iff the outcome of the random experiment is an
such that X(sj)=xj
P(X=xj)=P( : X(sj)=xj)
j
s S
j
s S
e.g., What is P(X=2) in the previous example?
18. 18
Discrete/Continuous Random
Variables
• A discrete r.v. can assume only a countable number of values.
• Consider the experiment of throwing a pair of dice
• A continuous r.v. can assume a range of values (e.g., sensor
readings).
19. 19
Probability mass (pmf) and
density function (pdf)
• The pmf /pdf of a r.v. X assigns a probability for each
possible value of X.
• Warning: given two r.v.'s, X and Y, their pmf/pdf are denoted as
pX(x) and pY(y); for convenience, we will drop the subscripts and
denote them as p(x) and p(y), however, keep in mind that these
functions are different !
20. 20
Probability mass (pmf) and
density function (pdf) (cont’d)_
• Some properties of the pmf and pdf:
21. 21
Probability Distribution Function
(PDF)
• With every r.v., we associate a function called probability
distribution function (PDF) which is defined as follows:
• Some properties of the PDF are:
– (1)
– (2) F(x) is a non-decreasing function of x
• If X is discrete, its PDF can be computed as follows:
( ) ( )
F x P X x
0 ( ) 1
F x
0 0
( ) ( ) ( ) ( )
x x
k k
F x P X x P X k p k
23. 23
Probability mass (pmf) and
density function (pdf) (cont’d)_
• If X is continuous, its PDF can be computed as follows:
• Using the above formula, it can be shown that:
( ) ( )
x
F x p t dt for all x
( ) ( )
dF
p x x
dx
24. 24
Probability mass (pmf) and
density function (pdf) (cont’d)_
• Example: the Gaussian pdf and PDF
0
25. 25
Joint pmf (discrete r.v.)
• For n random variables, the joint pmf assigns a
probability for each possible combination of
values:
p(x1,x2,…,xn)=P(X1=x1, X2=x2, …, Xn=xn)
Warning: the joint pmf /pdf of the r.v.'s X1, X2, ..., Xn and Y1, Y2, ...,
Yn are denoted as pX1X2...Xn(x1,x2,...,xn) and pY1Y2...Yn(y1,y2,...,yn); for
convenience, we will drop the subscripts and denote them p(x1,x2,...,xn)
and p(y1,y2,...,yn), keep in mind, however, that these are two different
functions.
26. 26
Joint pmf (discrete r.v.) (cont’d)
• Specifying the joint pmf requires an enormous number of
values
– kn assuming n random variables where each one can assume one
of k discrete values.
– things much simpler if we assume independence or conditional
independence …
P(Cavity, Toothache) is a 2 x 2 matrix
Toothache Not Toothache
Cavity 0.04 0.06
Not Cavity 0.01 0.89
Joint Probability
Sum of probabilities = 1.0
27. 27
Joint pdf (continuous r.v.)
For n random variables, the joint pdf assigns a probability
for each possible combination of values:
1 2
( , ,..., ) 0
n
p x x x
1
1 2 1
... ( , ,..., ) ... 1
n
n n
x x
p x x x dx dx
29. 29
Interesting Properties
• The conditional pdf can be derived from the joint
pdf:
• Conditional pdfs lead to the chain rule (general
form):
( , )
( / ) ( , ) ( / ) ( )
( )
p x y
p y x or p x y p y x p x
p x
1 2 1 2 2 3 1
( , ,..., ) ( / ,..., ) ( / ,..., )... ( / ) ( )
n n n n n n
p x x x p x x x p x x x p x x p x
30. 30
Interesting Properties (cont’d)
• Knowledge about independence between r.v.'s is
very powerful since it simplifies things a lot,
e.g., if X and Y are independent, then:
• The law of total probability:
( , ) ( ) ( )
p x y p x p y
( ) ( / ) ( )
x
p y p y x p x
31. 31
Marginalization
• From a joint probability, we can compute the probability of
any subset of the variables by marginalization:
– Example - case of joint pmf :
– Examples - case of joint pdf :
( ) ( , )
p x p x y dy
( ) ( , )
y
p x p x y
1 2 1 1 1 2
( , ,..., , ,..., ) ( , ,..., )
i i n n i
p x x x x x p x x x dx
3
1 2 1 2 3
( , ) ... ( , ,..., ) ...
n
n n
x x
p x x p x x x dx dx
34. 34
Probabilistic Inference
• If we could define all possible values for the
probability distribution, then we could read off any
probability we were interested in.
• In general, it is not practical to define all possible
entries for the joint probability function.
• Probabilistic inference consists of computing
probabilities that are not explicitly stored by the
reasoning system (e.g., marginals, conditionals).
37. 37
Normal (Gaussian) Distribution
(cont’d)
• Mahalanobis distance:
• If variables are independent, the multivariate normal
distribution becomes:
2 1
( ) ( )
t
r
x x
2
2
2
( )
1
( ) exp[ ]
2
2
i i
i
i
i
x
p x