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ML - Murphy > 2. Probability > Flashcards

2. Probability Flashcards

1
Q

Discrete random variables

  • discrete random variable
  • probability mass function
  • state space
A

p. 28

A discrete random variable can take on any value from a finite or countably infinite state space.

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2
Q

Fundamental rules

  • probability of a union of two events
  • sum rule
  • product rule
  • chain rule
  • conditional probability
A

p. 29

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3
Q

Bayes’ rule

- generative vs discriminative classifier

A

p. 29

Generative classifier specifies how to generate the data using the class-conditional density p(x|y) and the class prior p(y). Discriminative classifier directly fits the class posterior p(y|x).

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4
Q

Independence and conditional indepencence

  • unconditional (marginal) independence
  • conditional indepencence
A

p. 31

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5
Q

Continuous random variables

  • cumulative distribution function
  • probability density function
  • p(a less X lessequal b) in terms of cdf and pdf
A

p. 32

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6
Q

Quantiles

  • quantile
  • quartile
  • tail area probabilities
A

p. 33

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7
Q

Mean and variance

  • E[X]
  • var[X]
  • E[X^2]
  • std[X]
A

p. 34

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8
Q

The binomial and Bernoulli distributions

  • pmf, mean, var
  • binomial coefficient
A

p. 34

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9
Q

The multinomial and multinoulli distributions

  • pmf
  • dummy and one-hot encoding
A

p. 35

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10
Q

The Poisson distribution

A

p. 37

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11
Q

The empirical distribution

  • empirical distribution def
  • Dirac measure
A

p. 37

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12
Q

Gaussian (normal) distribution

  • pdf, mean, mode, var
  • precission
  • cdf
  • error function
  • cdf in terms of error function
A

p. 38

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13
Q

Degenerate pdf

  • Dirac delta function
  • sifting property
A

p. 39

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14
Q

The Student’s t distribution

  • pdf, mean, mode, var
  • Cauchy (Lorentz) distribution
A

p. 39

https: //en.wikipedia.org/wiki/Student%27s_t-distribution#Non-standardized_Student.27s_t-distribution

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15
Q

The Laplace distribution

- pdf, mean, mode, var

A

p. 41

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16
Q

The gamma distribution

  • pdf, mean, mode, var
  • gamma function
  • exponential distribution
  • Erlang distribution
  • Chi-squared distribution
  • inverse gamma distribution (pdf, mean, mode, var)
A

p. 41

17
Q

The beta distribution

  • pdf, mean, mode, var
  • beta function
A

p. 43

18
Q

Pareto distribution

- pdf, mean, mode, var

A

p. 43

19
Q

Covariance and correlation

  • covariance and covariance matrix
  • correlation coefficient and correlation matrix
A

p. 45

20
Q

The multivariate Gaussian

  • pdf
  • precision (concentration matrix)
  • number of covariance matrix parameters (full, diagonal, isotropic)
A

p. 46

21
Q

Multivariate Student t distribution

  • pdf, mean, mode
  • scale matrix
A

p. 47

22
Q

Dirichlet distribution

  • pdf, mean, mode, var
  • symmetric Dirichlet prior
A

p. 49

23
Q

Linear transformations

  • definition of linear transformation
  • expected value
  • variance
  • when only first two moments suffice to completely define the transformed distribution?
A

p. 49

24
Q

General transformations

  • change of variables formula
  • Jacobian matrix def
  • what does the |det J| measure?
  • pdf of the transformed variables using the Jacobian (eq. 2.89)
A

p. 50

25
Q

Central limit theorem

A

p. 52

26
Q

Monte Carlo approximation

  • Monte Carlo integration formula (eq. 2.98)
  • mean, variance
A

p. 53

27
Q

Accuracy of Monte Carlo approximation

- standard error

A

p. 55

28
Q

Entropy

  • definition of information entropy for a discrete variable
  • which discrete distribution has the highest entropy?
  • binary entropy function
A

p. 57

29
Q

KL divergence

  • what does the Kullback-Leibler divergence (KL divergence) or relative entropy measure?
  • definition of the KL divergence
  • definition of the KL divergence in terms of entropy
  • definition of the cross entropy
  • Jensen’s inequality
  • proof of information inequality theorem
  • principle of insufficient reason (also principle of indifference)
A

p. 58

30
Q

Mutual information

  • definition of the mutual information (MI)
  • definition of the MI in terms of joint and conditional entropies
  • definition of the conditional entropy
  • definition of the pointwise mutual information (PMI)
  • what PMI measures?
  • what’s maximal information coefficient (MIC)?
A

p. 59

ML - Murphy (5 decks)

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