Probability Theory: Univariate Models

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Questions and Answers

What does the binomial coefficient represent mathematically?

  • The number of ways to choose k items from N (correct)
  • The number of ways to arrange N items
  • The expected value in a binary distribution
  • The probability of success in a Bernoulli trial

Which distribution does the binomial distribution reduce to when N equals 1?

  • Poisson distribution
  • Geometric distribution
  • Bernoulli distribution (correct)
  • Normal distribution

In the logistic function, what does the output range from?

  • -1 to 1
  • 0 to ∞
  • 0 to 1 (correct)
  • -∞ to ∞

What role does the parameter ω play in the conditional probability distribution p(y|x, ω)?

<p>It alters the probability calculation (B)</p> Signup and view all the answers

What is the Heaviside function primarily used to represent?

<p>Thresholds in binary decisions (B)</p> Signup and view all the answers

What characteristic do all datasets in the Datasaurus Dozen share?

<p>They all have the same low order summary statistics. (D)</p> Signup and view all the answers

Which visualization technique can better distinguish differences in 1d data distributions?

<p>Violin plot (A)</p> Signup and view all the answers

What is a key limitation mentioned regarding the violin plot visualization?

<p>It is limited to visualizing 1d data. (C)</p> Signup and view all the answers

Bayes' theorem is compared to which theorem in geometry?

<p>Pythagoras's theorem (D)</p> Signup and view all the answers

In the context of Bayesian inference, what does the term 'inference' refer to?

<p>Generalizing from sample data. (C)</p> Signup and view all the answers

What is the purpose of the simulated annealing approach as mentioned in the content?

<p>To optimize the shape of datasets. (D)</p> Signup and view all the answers

What do the central shaded parts of the box plots indicate?

<p>Median and inter-quartile range of the datasets. (C)</p> Signup and view all the answers

What kind of data is Bayes' rule primarily applied to?

<p>Probabilistic data. (C)</p> Signup and view all the answers

What is a random variable?

<p>An unknown quantity that can change and has different outcomes. (C)</p> Signup and view all the answers

Which of the following best describes a discrete random variable?

<p>It can only take on a finite or countably infinite number of outcomes. (C)</p> Signup and view all the answers

What does the probability mass function (pmf) compute?

<p>The probabilities of events corresponding to setting the rv to each possible value. (A)</p> Signup and view all the answers

Which of the following statements is true regarding the properties of the pmf?

<p>All probabilities must be between 0 and 1, inclusive. (C)</p> Signup and view all the answers

In the context of rolling a dice, which of the following represents the sample space?

<p>The set of numbers {1, 2, 3, 4, 5, 6}. (D)</p> Signup and view all the answers

What is an example of a degenerate distribution?

<p>A distribution that assigns all probability mass to a single outcome. (A)</p> Signup and view all the answers

How is a continuous random variable defined?

<p>It can take on any value within a given range of real numbers. (D)</p> Signup and view all the answers

What does the event of 'seeing an odd number' represent if X is the outcome of a dice roll?

<p>X = {1, 3, 5} (A)</p> Signup and view all the answers

What does the variable $Y$ represent in the context of univariate Gaussians?

<p>A mixture component indicator variable (A)</p> Signup and view all the answers

In the formulas provided, what does $V[X]$ represent?

<p>The variance of the random variable $X$ (A)</p> Signup and view all the answers

Which of the following statements about Anscombe’s quartet is true?

<p>All datasets in Anscombe's quartet have the same low order summary statistics. (A)</p> Signup and view all the answers

What do the terms $ heta_y$ and $ u_y$ likely refer to in the distribution $N(X| heta_y, u_y)$?

<p>The variance and mean of the Gaussian respectively (B)</p> Signup and view all the answers

What does the notation $E[Y|X]$ represent?

<p>The expected value of $Y$ given $X$ (C)</p> Signup and view all the answers

Which equation highlights the relationship between the variance and the expectations of random variables?

<p>$V[X] = E[X^2] - (E[X])^2$ (C)</p> Signup and view all the answers

What is likely the role of the hidden indicator variable $Y$ in the mixture model?

<p>To determine which mixture component generates the observation (D)</p> Signup and view all the answers

What can we infer if the datasets in Anscombe's quartet appear visually different?

<p>They can produce different correlation results despite similar statistics. (A)</p> Signup and view all the answers

How can the joint distribution of two random variables be represented when both have finite cardinality?

<p>As a 2D table where entries sum to one. (D)</p> Signup and view all the answers

What is the mathematical expression for obtaining the marginal distribution of variable X?

<p>$p(X = x) = \sum p(X = x, Y = y)$ (C)</p> Signup and view all the answers

What does it mean if two random variables, X and Y, are independent?

<p>Their joint distribution can be represented as the product of their marginal distributions. (C)</p> Signup and view all the answers

How is the conditional distribution of Y given X defined mathematically?

<p>$p(Y = y|X = x) = \frac{p(X = x, Y = y)}{p(X = x)}$ (A)</p> Signup and view all the answers

What is the purpose of using the sum rule in probability?

<p>To compute marginal distributions from joint distributions. (B)</p> Signup and view all the answers

Which of the following correctly summarizes the joint distribution in probabilistic terms?

<p>$p(x, y) = p(x)p(y)$ when X and Y are independent. (B)</p> Signup and view all the answers

In the context of joint distributions, what does the term 'marginal' refer to?

<p>The sum of all probabilities at the edge of a table. (C)</p> Signup and view all the answers

How can the joint distribution table be restructured if the variables are independent?

<p>As two separate 1D vectors representing each variable. (B)</p> Signup and view all the answers

What is the output of the sigmoid function when applied to a > 0?

<p>Ï‘(a) (A)</p> Signup and view all the answers

How is the log-odds 'a' defined in relation to the probability 'p'?

<p>a = log(p / (1 - p)) (D)</p> Signup and view all the answers

Which function maps the log-odds 'a' back to probability 'p'?

<p>Sigmoid function (B)</p> Signup and view all the answers

In binary logistic regression, what form does the linear predictor take?

<p>f(x; ω) = w^T x + b (A)</p> Signup and view all the answers

What does the function p(y = 1|x, ω) represent in the context of the sigmoid function?

<p>The probability of y = 1 given x (C)</p> Signup and view all the answers

What is the output of the logit function when applied to probability 'p'?

<p>log(p / (1 - p)) (C)</p> Signup and view all the answers

Which of the following correctly describes the inverse relationship between the sigmoid and logit functions?

<p>The sigmoid function maps log-odds to probability, while the logit function does the opposite. (A)</p> Signup and view all the answers

What represents the probability distribution in binary logistic regression?

<p>Bernoulli distribution (D)</p> Signup and view all the answers

Flashcards

Random Variable

A quantity whose value is unknown and can vary.

Sample Space

The set of all possible values that a random variable can take.

Event

A specific outcome or set of outcomes from the sample space.

Discrete Random Variable

A random variable whose values can be counted or listed.

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Probability Mass Function (PMF)

A function that assigns probabilities to each possible value of a discrete random variable.

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Continuous Random Variable

A random variable whose values can fall anywhere within a continuous range.

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Uniform Distribution

A specific type of discrete distribution where each possible value has an equal probability.

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Degenerate Distribution

A specific type of discrete distribution where all the probability mass is concentrated on a single value. It is also called a degenerate distribution.

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Joint Distribution

A function that maps each possible pair of values (x, y) for random variables X and Y to a probability. It represents the likelihood of observing those specific values together.

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Marginal Distribution

A function that represents the probability of a specific value of a random variable occurring, regardless of the values of other random variables. It is obtained by summing the probabilities of all joint events involving the specific value.

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Conditional Distribution

A function that describes the probability of a specific value (y) of the random variable Y occurring given that the random variable X takes a specific value (x). It tells you how likely Y is to occur given knowledge about X.

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Independence of Random Variables

If the probability of observing a specific value (y) of Y is independent of the value of X, then Y and X are said to be independent. In other words, knowing X doesn't alter the probability of Y occurring.

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Product Rule for Independent Variables

The joint probability of two independent random variables is equal to the product of their individual marginal probabilities. This means their joint behaviour can be understood simply by multiplying their individual probabilities.

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2D Joint Probability Table

A representation of joint probability distributions for discrete random variables. It's a table where each cell represents the probability of a specific combination of values for the variables.

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1D Marginal Probability Vector

A representation of probabilities for discrete random variables. It's a 1D vector where each element corresponds to the probability of a specific value for the variable.

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Rule of Total Probability

The rule of total probability states that the probability of an event can be calculated by summing the probabilities of all mutually exclusive events that lead to that event. It's a way of breaking down complex events into simpler ones.

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Expected Value (E[X])

The expected value of a random variable X is the weighted average of all possible values of X, where the weights are the probabilities of each value.

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Variance (Var[X])

The variance of a random variable X measures how much the values of X are spread out around the expected value.

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Standard Deviation (SD[X])

The standard deviation of a random variable X is the square root of the variance.

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Conditional Expectation (E[X|Y])

The conditional expectation of X given Y is the expected value of X when we know the value of Y.

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Conditional Variance (Var[X|Y])

The conditional variance of X given Y is the variance of X when we know the value of Y.

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Law of Total Expectation

The law of total expectation states that the expected value of a random variable X can be calculated as the average of the conditional expectations of X given each value of Y, weighted by the probabilities of each value of Y.

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Mixture Distribution

A mixture distribution combines multiple distributions together, weighted by their proportions.

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Sigmoid function

A function used in predicting binary outcomes (0 or 1) by calculating the probability of an event based on an input value and parameters. It returns a probability between 0 and 1, representing the likelihood of the event occurring.

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Binomial Distribution

A statistical distribution that models the probability of getting exactly k successes from n trials, where each trial has only two possible outcomes (success or failure) with a constant probability of success (p).

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Bernoulli Distribution

A distribution that describes the probability of success or failure in a single event, where the probability of success is p and the probability of failure is 1-p. It's the simplest case of the binomial distribution with only one trial.

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Heaviside Function

The function that takes a real number 'a' and returns 1 if 'a' is greater than 0, otherwise it returns 0. It represents a step function.

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Conditional Probability

The process of determining the probability of an event based on the values of other related variables. It uses conditional probabilities to quantify the impact of one variable upon another.

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Violin Plot

A visual representation of data where the width of the violin shape represents the density of data points at each value, providing a richer understanding of the distribution compared to a box plot.

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Simulated Annealing

A method used in machine learning to find the best solution by making small, incremental changes to the parameters of a model, aiming to minimize a specific cost function.

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Inter-Quartile Range (IQR)

The statistical measure of the middle 50% of data, represented by the distance between the first and third quartiles.

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Box Plot

A type of data visualization that uses a rectangular box to summarize the distribution of data. The box shows the median, quartiles, and outliers.

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Bayesian Inference

A statistical method that uses Bayes' theorem to update prior beliefs about an event based on new evidence.

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Bayes’ Theorem

A mathematical theorem that allows us to calculate the probability of an event occurring based on prior knowledge and new evidence.

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Kernel Density Estimate

A one-dimensional graphical representation of a probability distribution, showing the density of data points at different values.

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p(y = 1|x, ω)

The probability of an event, given specific input features and model parameters.

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Log-Odds (a)

The log-odds of an event, representing the ratio of probabilities for the event occurring versus not occurring.

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Logistic Function

A function that transforms log-odds (a) into probability (p), often used in logistic regression models.

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Logit Function

The inverse of the logistic function, mapping probability (p) back to log-odds (a).

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Binary Logistic Regression

A statistical model used to predict the probability of a binary outcome (e.g., success or failure) based on input features.

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Linear Predictor (f(x; ω))

A linear combination of input features and model parameters, used to predict an outcome in linear models and logistic regression.

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Conditional Bernoulli Model

A conditional Bernoulli distribution, where the probability of success is determined by the sigmoid function applied to the linear predictor.

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Study Notes

Probability: Univariate Models

  • Probability theory is common sense reduced to calculation.
  • Two interpretations of probability exist: frequentist and Bayesian.
  • Frequentist interpretation: probability represents long-run frequencies of events.
  • Bayesian interpretation: probability quantifies uncertainty or ignorance about something.
  • Bayesian interpretation models uncertainty about one-off events.
  • Basic rules of probability theory remain consistent despite differing interpretations.
  • Uncertainty can stem from ignorance (model uncertainty) or intrinsic variability (data uncertainty).

Probability as an Extension of Logic

  • Probability extends Boolean logic.
  • An event (A) can either hold or not hold.
  • Pr(A) represents the probability of event A being true.
  • Values range from 0 to 1 (inclusive).
  • Pr(A) = 0 means event A will not happen, Pr(A) = 1 means event A will happen.

Probability of Events

  • Joint probability: Pr(A, B) or Pr(AB) is the probability of both A and B occurring.
  • If A and B are independent, then Pr(A, B) = Pr(A) x Pr(B).
  • Conditional probability: Pr(B|A) is the probability of B happening given A has occurred.
  • Pr(B|A) = Pr(A, B)/Pr(A)
  • Conditional independence: events A and B are conditionally independent given C if Pr(A, B|C)= Pr(A|C) x Pr(B|C).

Random Variables

  • Random variables (r.v.) are unknown or changeable quantities.
  • Sample space: the set of possible values of a random variable.
  • Events are subsets of outcomes in a given sample space,
  • Discrete random variables have finite or countably infinite sample spaces.
  • Continuous random variables take on any value within a given range.

Cumulative Distribution Function (CDF)

  • Cumulative distribution function (CDF) of a random variable X, denoted by P(x), is the probability that X takes on a value less than or equal to x.
  • P(x) = Pr(X ≤ x)
  • Pr (a ≤ X ≤ b) = P(b) – P(a)

Probability Density Function (PDF)

  • Probability density function is derived from the CDF.
  • PDF is the derivative of the CDF.
  • Pr (a ≤ X ≤ b) = integral of p(x) dx from a to b

Quantiles

  • Quantile function is the inverse of the CDF.
  • P-¹ (q) is the value x such that Pr (X ≤ xq) = q

Moments of a Distribution

  • Mean (μ): the expected value of a distribution.
  • E [X] = integral(x * p(x) dx) for continuous rv's.
  • E [X] = Σ (x * p(x)) for discrete rv's.
  • Variance (σ²): the expected squared deviation from the mean.
  • V [X] = E [(X - μ)²]
  • Standard Deviation (σ): the square root of the variance.
  • Mode: the value with the highest probability or probability density.

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