Image Processing Operations​

Point Operation​

$b[m, n] = f(a[m, n])$

• It only depends on the value of the pixel itself, not on the values of its neighbors.
• e.g. current pixel + 20.
• to increase the brightness of an image, adjust contrast, or apply a threshold to create a binary image.

Local Operation​

$b[m, n] = f(a[m - 1, n - 1], a[m - 1, n], a[m - 1, n + 1], a[m, n - 1], a[m, n], a[m, n + 1], a[m + 1, n - 1], a[m + 1, n], a[m + 1, n + 1])$

• It depends on the values of the pixel and its neighbors.
• e.g. current pixel + average of 8 neighbors.
• to blur an image, sharpen an image, detect edges, or convolution with a kernel.
• The most common type of neighborhoods are:
  • 4-neighbors:
    • top, bottom, left, right.
  • 8-neighbors:
    • top, bottom, left, right, and the 4 diagonal neighbors.

Global Operation​

$b[m, n] = f(a[0, 0], a[0, 1], ..., a[M - 1, N - 1])$

• It depends on the values of all pixels in the image.
• e.g. current pixel + average of all pixels in the image.
• to compute the histogram equalization, apply a global threshold, or perform a Fourier transform.

Image Histogram​

• It is a graph showing how many pixels in an image have each possible intensity value.
  • Intensity value: the brightness of a pixel.
• e.g. 8-bit grayscale image has 256 possible intensity values (0-255).
  • The histogram will graphically display 256 numbers showing the distribution of pixels among those gray-scale values.

Histogram Equalization​

0                     255
|------████████--------|
       80~140에 몰림

0                     255
|--██--██--██--██--██--|

• It spreads out the intensity values that are concentrated in a narrow range, increasing the contrast of the image.
• It is useful when the images have been acquired under poor lighting conditions or have low contrast (different circumstances).

Noise​

• Any undesired information that contaminatest the image.
• During the analog-to-digital conversion process, it is a side effect of the physical conversion of patterns of light energy into electrical patterns.
• The shape of distribution of noise types used to describe many of them and is related closely to the histogram.

frequency
 ^
 |             █
 |          █████
 |       █████████
 |    █████████████
 +----------------------> noise gray level
        -20  0  +20

• Gaussian noise: the most common type of noise, with a bell-shaped distribution.

frequency
 ^
 | █                       █
 | █                       █
 | █                       █
 +----------------------------> gray level
   0                         255

• Salt-and-pepper noise: the most common type of impulse noise, with a distribution that has two spikes at the minimum and maximum intensity values.

1.0: Mechanisation​

• 1780
• Industrial production based on machines
• Powered by water and steam

2.0: Electrification​

• 1870
• Mass-production using assembly lines

3.0: Automation​

• 1970
• Automation using electronics and computers

3.5: Globalisation​

• 1980
• Offshoring of production to low-cost economies

4.0: Digitalisation​

• Today
• Introduction of connected devices
• Data analytics
• Artificial intelligence technologies
• Further automation of processes

5.0: Personalisation​

• Future
• The fifth industrial revolution, or Industry 5.0, will be focused on the co-operation between man and machine
• Human intelligence works in harmony with cognitive computing
• Humans are put back into industrial production with collaborative robots
• Workers will be upskilled to provide value-added tasks in production
• Leading to mass customisation and personalisation for customers

Ref​

• What will Industry 5.0 mean for manufacturing?

Moral Machines​

Anatomy of an AI System​

Questions​

• Why is this labor so frequently rendered invisible to the end user?
  • 왜 이러한 노동은 최종 사용자에게 그토록 자주 보이지 않게 되는가?
• How does the short lifespan of a smart device compare to the geological time needed to form its raw minerals and the centuries required for it to decay as e-waste?
  • 스마트 기기의 짧은 수명은 원료 광물이 형성되는 데 필요한 지질학적 시간, 그리고 전자폐기물로 분해되는 데 걸리는 수백 년의 시간과 어떻게 대비되는가?
• What ethical and intergenerational obligations do technology companies bear for the imbalance between timescales?
  • 이러한 시간 규모의 불균형에 대해 기술 기업은 어떤 윤리적 책임과 세대 간 책임을 져야 하는가?
• How is the word 'extraction' a good bridge between the ethical issues related to mining on one hand and capitalism's thirst for data on the other?
  • ‘추출(extraction)’이라는 단어는 한편으로는 채굴과 관련된 윤리적 문제를, 다른 한편으로는 자본주의의 끝없는 데이터 욕구를 연결하는 개념으로서 왜 적절한가?
• What legislative, regulatory, or architectural interventions could hold technology companies accountable for these hidden environmental and social costs?
  • 어떤 입법적, 규제적 또는 설계·구조적 개입을 통해 기술 기업이 숨겨진 환경적·사회적 비용에 대해 책임을 지도록 할 수 있는가?
• Should companies be allowed to sell devices with very short lifespans when the materials used to make them take millions of years to form and may remain as waste for generations?
  • 기기를 만드는 데 사용되는 원료는 형성되는 데 수백만 년이 걸리고 폐기물로는 여러 세대 동안 남을 수 있는데, 기업이 수명이 매우 짧은 기기를 판매하도록 허용해야 하는가?
• How much responsibility should current technology companies have for environmental harms that may only become fully visible decades from now?
  • 수십 년이 지나야 완전히 드러날 수 있는 환경 피해에 대해 현재의 기술 기업은 어느 정도까지 책임을 져야 하는가?
• If both natural resources and human data can be extracted for profit, what makes data extraction ethically different from mining — and does that difference matter?
  • 천연자원과 인간의 데이터가 모두 이윤을 위해 추출될 수 있다면, 데이터 추출은 채굴과 윤리적으로 무엇이 다르며, 그 차이는 중요한가?
• Should technology companies be legally required to account for the full social and environmental cost of an AI system, even if doing so makes the technology more expensive?
  • 기술 비용이 더 비싸지더라도, 기술 기업은 인공지능 시스템의 모든 사회적·환경적 비용을 법적으로 산정하고 책임지도록 요구받아야 하는가?

Sequential Decision Making​

• A sequence is a series of events or actions that occur in a specific order over time.
• Goal: select actions that maximize total expected future reward.
• It may require balancing immediate and long-term rewards.
• It may be better to sacrifice short-term gains for long-term benefits.
• It may require strategic behavior to achieve high rewards.

Example of SDM​

• Web Ads
  • Agent AdBot
    1. Choose and Ad $A_t$.
    2. Receives View time $O_t$
    3. Receives Click on Ad $R_t$
  • Environment
    1. Receives the Chosen Ad $A_t$.
    2. Provides the View time $O_{t+1}$.
    3. Provides the Click on Ad $R_{t+1}$.
• Robot picking trash
  • Agent Robot
    1. Moves arms to pickup trash $A_t$.
    2. Receives Camera image of the room $O_t$.
    3. Receives Reward if no trash on the floor $R_t$
  • Environment
    1. Receives robots action/arm movements $A_t$.
    2. Provides Camera Image of the room $O_{t+1}$.
    3. Provide reward of +1 $R_{t+1}$
• Robot making Pizza.
  • Agent Robot
    1. Makes an action from Action space $A_t$ (a range of possible actions, Arm Movements, Get Pizz from Oven, Place Pizza on Tray, Cut Pizza...).
    2. Receives Camera image of the kitchen $O_t$.
    3. Receives Reward if pizza is made correctly $R_t$
  • Environment
    1. Receives robots action/arm movements $A_t$.
    2. Provides Camera Image of the kitchen $O_{t+1}$.
    3. Provide reward appropriate to the action $R_{t+1}$.
  • Reward space
    • +10 Successfully made pizza.
    • +1 Get Pizza from oven.
    • +1 Place Pizza on tray.
    • +1 Get Packing box.
    • +1 Cut Pizza successfully.

Rewards​

• a scalar signal that indicates how well the agent is doing at time step $t$. (agent's performance)
• Examples of rewards:
  • Robot play soccor:
    • $+ve$ reward for scoring a goal.
    • $-ve$ reward for kicking the ball out of bounds.
  • Chess
    • $+ve$ reward for winning the game.
  • Autonomous drone flying stunt:
    • $+ve$ reward for following the intended trajectory.
    • $-ve$ reward for crashing into an obstacle.

History​

$H_t = \{O_1, A_1, R_1, O_2, A_2, R_2, ..., O_t, A_t, R_t\}$

• a sequence of past Observations, Actions, and Rewards up to time step $t$.
• Agent selects actions base on history.
• Environment selects observations and reward.

State​

$S_t = f(H_t)$

• The information used to determine what happens next.
• A function of the history that captures all relevant information for decision-making.

Environment State​

• $S_t^e$: the environment's private representation of the current state.
• It used to generate next observation and reward.
• It is not directly accessible to the agent (Invisible to the agent).
• If it is visible to the agent, it may contain information not required by the agent to make decisions (e.g., the internal state of a robot's motors).

Agent State​

• $S_t^a$: the agent's private representation of its state.
• Information used to pick next action and used by RL algorithms to learn from experience.
• It can be a function of the history: $S_t^a = f(H_t)$

Information State​

$P[S_{t+1} | S_t] = P[S_{t+1} | S_1, \cdots, S_t]$

• It contains all useful information from the history as known as the Markov State.
• Probability[next state | current state] = Probability[next state | given the whole history]
• The future is independent of the past given the present.
• The dog example illustrates a situation where the agent needs to remember past signals in order to receive a food reward.
• Possible definitions of the agent state:
  • Last 3 events in the sequence
    • Example: bell ring → light → bell press
    • Useful when recent observations are enough to predict the reward
  • Counts of events
    • Example: number of lights, bell rings, and bell presses
    • Useful for compressing information, but it loses the order of events
  • Complete sequence
    • Example: bell ring → light → bell press → bell ring → ...
    • Contains all past information, but can become too large and difficult to learn from
• Each state definition involves a trade-off between the amount of information stored and the complexity of learning.
• A good state should contain enough information to predict the next state and reward.

Markov assumption​

• Markov assumption is popular because it simplifies decision-making in RL.
• A Markov state can always be created by defining the state as the full history:
  • $S_t = H_t$
  • This includes all past observations, actions, and rewards.
  • It satisfies the Markov property because no extra past information is needed.
• However, using the full history creates a very large state space.
  • More computation
  • More memory
  • More data required
  • Harder learning
• Ideally, the current observation is enough:
  • $S_t = O_t$
  • This means the current observation is a sufficient statistic of the history.
  • The agent does not need to remember the full past.
• Smaller state spaces are preferred because they reduce computational complexity and data requirements.
• The choice of state representation affects:
  • Computational complexity
  • Amount of data required
  • Final performance
• Main idea: A good state should be small, but still contain enough information to predict the next state and reward.

Markov Property​

• The current state contains all relevant information from the history needed to predict the future.
• The future is independent of the past given the present state.

Full Observable Environments, MCPs​

$O_t = S_t^a = S_t^e$

• Full observable: Agent is able to directly observe the environment.
• Agent State = Environment State = Information State
• Known as Markov Decision Process

Partially Observable Environments, POMDPs​

• Partially observable: Agent only observes partial information about the environment.
• Agent State != Environment State
• Agent must construct its own internal state (S_t^a)
• Ways to construct agent state:
  • Complete history:
    • (S_t^a = H_t)
  • Belief state:
    • (S_t^a = b(S_t^e))
    • Probabilities over possible environment states
  • RNN hidden state:
    • previous hidden state + current observation → new hidden state
• Known as Partially Observable Markov Decision Process, POMDP.
• Examples:
  • Soccer-playing robot with a camera and limited information about its exact location.
  • Share trading agent that observes current share price but not full market trends or company history.
  • Card games where opponent cards and deck order are hidden.

Exploration vs Exploitation​

• Exploration: Learn more about the environment, may lose immediate reward
• Exploitation: Exploit known information to maximize rewards
• Discover a good policy, while interacting with the environment, the agent must balance exploration and exploitation.
• These are key aspects of RL learning, due to trial-and-error learning and the need to maximize long-term rewards.
• Examples:
  • Playing Chess
    • Exploitation: Use familiar opening lines that have won in the past.
    • Exploration: Try a new opening sequence.
  • Online Advertisements
    • Exploitation: Show ads with the highest past click rate.
    • Exploration: Show different ads and collect feedback such as clicks or view time.
  • Gold Mining
    • Exploitation: Mine at the best-known location.
    • Exploration: Mine at a new location.

Bandits​

• Actions that are taken has no influence on next observations. (Slot machines)
• No delayed rewards.
• RL problem with one state.
• Examples:
  • Design of Clinical trials
  • Online ad suggestion, placement
  • Games
  • Web page personalization
• Action now, Reward now.
• Single State $(S_t = S_1)$.
• Set of Actions = $\{A_1, A_2, ..., A_n\}$.
• Reward space = $[0, 1]$.
• Learn a Stochastic Reward Function: Reward probabilities for actions are unknown in advance, so they must be learned stochastically through trial and error.

Online Ad suggestion​

$\text{PayOff} = \text{CTR} \times \text{Payment Rate}$

• How to maximize PayOff among hundreds thousands of ads?
• CTR, ClickThroughRate: Probability that users click on the ad.
• Payment Rate: Money paid by the advertiser for each click.
• Arms: $\{Ad_1, Ad_2, ..., Ad_n\}$
• Rewards: $\{ 0: \text{No Click}, 1: \text{Click} \}$
  • Assuming a uniform payment of $1 for all ads, maximizing revenue simplifies to accurately estimating the Click-Through Rate (CTR).
• Exploration vs. Exploitation: Should we display new ads with unknown CTRs to test them (Exploration), or continuously show top-performing ads with the highest historical CTR (Exploitation)?
  • No known optimal solution, but there are many heuristic approaches to balance exploration and exploitation.

$\epsilon$-Greedy​

• a common exploration strategy to balence getween exploiting the current best action and exploring new actions in order to learn an optimal policy.
• Select the Arm with highest average reward so far
• May get struck with no exploration.
• $\epsilon$: The probability of selecting a random arm and used a greedy selection.

Init, for Arm = 1 to k:
  Q(Arm) = 0
  N(Arm) = 0

Loop:
  # Explore
  A = Random action with probability 𝜖
  # Exploit
  A = argmax(Q(Arm)) with probability (1 - 𝜖)

  R = bandit(A)
  N(A) += 1
  Q(A) = Q(A) + (1 / N(A)) * (R - Q(A))

• Q: Estimated reward for each Arm
• N: Number of time each Arm is pulled.
• A: Action
• R: Reward

NewEstimate = OldEstimate + StepSize[Target - OldEstimate]

UCB​

$UCBValue = Q(A) + \sqrt{\frac{2 \ln(t)}{N(A)}}$

• used to balance between exploiting known good actions and exploring potentially better actions.
• Optimism in the face of uncertaintiy: It will choose actions systematically that have a high potential for being optimal based on both their estimated value and their uncertatity.
  • The uncertainty bonus represents the value of exploring that arm.

Init, for Arm = 1 to k:
  Q(Arm) = 0
  N(Arm) = 0

Loop:
  UCBValue = Q(A) + sqrt((2 * log(t)) / N(A))
  A = argmax(UCBValue)
  R = bandit(A)

  N(A) += 1
  Q(A) = Q(A) + (1 / N(A)) * (R - Q(A))

• t: Time step
• Q: Estimated reward for each Arm
• N: Number of time each Arm is pulled
• A: Action
• R: Reward

General Update Rule same as $\epsilon$-Greedy.

Reinforcement Learning​

• Learn how to make a good sequence of decisions by interacting with the environment.

Characteristics of RL​

• Trial and Error: Based learning approach.
• Optimization: Find good sequences of actions or decisions.
• Delayed Consequences/Rewards: Takes time to relize the actions or decisions are good or bad.
• Exploration: Learn by making decisions or forming actions or through experiences. Trying new actions to discover their effects.
• Exploitation: Choosing actions with the highest expected reward, based on current knowledge.
• Generalization: Use previous experiences or knowledge to new or unseen situations effectively.

Framwork of RL​

• OpenAI Gym
• Torch RL
• AWS DeepRacer

RL Math​

Probability​

• Sample Space ($\Omega$): The set of all possible outcomes of a random experiment.

Bonferroni's Inequality​

$P(A \cap B) \geq P(A) + P(B) - 1$ $P(\cap^{n}_{i=1} A_i) \geq \sum_{i=1}^{n} P(A_i) - (n-1)$

• Gives a lower bound on the intersection probability which is useful when this probability is difficult to compute directly.
• It is useful when the probabilities of individual events are sufficiently large.

Boole's Inequality​

$P(\cup^{n}_{i=1} A_i) \leq \sum_{i=1}^{n} P(A_i)$

for any sets $A_1, A_2, ..., A_n$.

• It is useful when finding an upper bound for the probabilities of the union of events.

Bayes' Rule​

$P(A|B) = \frac{P(A \cap B)}{P(B)}$ $P(A\cap B) = P(B|A) P(A) = P(A|B) P(B)$ $P(A|B)P(B) = P(B|A)P(A)$ $P(A|B) = \frac{P(B|A) P(A)}{P(B)}$

• It allows us to compute the conditional probability $P(A|B)$ from the inverse conditional probability $P(B|A)$.
• Let $A_1, A_2, ..., A_n$ be a partition of the sample space $S$. Then let $B$ be any subset of $S$ we have:

$P(A_i|B) = \frac{P(B|A_i) P(A_i)}{\sum_{j=1}^{\infty} P(B|A_j) P(A_j)}$

Independent Events​

$P(A \cap B) = P(A) P(B)$

• A family $A_i :i \in I$ of events is independent if for every finite subset $J \subseteq I$ we have: $P\left(\bigcap_{i \in J} A_i\right) = \prod_{i \in J} P(A_i)$

• The pair-wise independence of events does not imply their mutual independence.

Conditional Independence​

$P(A \cap B|C) = P(A|C) P(B|C)$

• where $P(C) > 0$.
• or equivalently, $P(A|B \cap C) = P(A|C)$ and $P(B|A \cap C) = P(B|C)$.

Induced Probability Function​

$P_X(x_i) = P\big(\{w_j \in \Omega : X(w_j) = x_i\}\big)$

• $\Omega = \{w_1, w_2, ..., w_m\}$ is the sample space.
• $X$ is a random variable with range $X = \{x_1, x_2, ..., x_n\}$.
• The result set $\{w_j \in \Omega : X(w_j) = x_i\}$ is the set of all outcomes in the sample space that map to the value $x_i$ under the random variable $X$.

Cumulative Distribution Function (CDF)​

$F_X(x) = P_X(X \leq x) \quad \forall x \in \mathbb{R}$

• $F_X(x)$ is a cdf $\iff$ the following conditions hold:
  • Monotonicity: $F_X(x)$ is non-decreasing.
  • Limiting values: $\lim_{x \to -\infty} F_X(x) = 0$ and $\lim_{x \to \infty} F_X(x) = 1$.
  • Right-Continuity: $\lim_{y \downarrow x} F_X(y) = F_X(x) \quad \forall x \in \mathbb{R}$.

F_X(t)

1.0 |                         ●────────────
    |                         ↑  ↑  ↑
    |                      3.001 3.01 3.1
0.7 |              ●─────────○
    |              ↑  ↑  ↑
    |           2.001 2.01 2.1
0.3 |     ●────────○
    |     ↑  ↑  ↑
    |  1.001 1.01 1.1
0.0 |─────○
    +-------------------------------------- t
          1         2         3
          x         x         x

Continuous & Discrete Random Variables​

• if $X$ is continuous, then $F_X(x)$ is continuous and differentiable almost everywhere. The probability density function (pdf) is defined as: $f_X(x) = \frac{d}{dx} F_X(x)$
• if $X$ is discrete, then $F_X(x)$ is a step function and the probability mass function (pmf) is defined as: $p_X(x_i) = P(X = x_i)$

1 ────────────────────────━━━━━━━━
                        ╱
                      ╱
                    ╱
                  ╱
0 ━━━━━━━━━━━━━━━───────────────── x

1 ─────────────────────────●━━━━━━
                           │
0.7 ─────────────●━━━━━━━━━○
                 │
0.3 ─────●━━━━━━━○
         │
0 ━━━━━━━○──────────────────────── x
         1         2         3

1 ──────────────────────────━━━━━━
                          ╱
                        ╱
0.7 ───────────●━━━━━━━
               │
0.5 ───────────○
             ╱
           ╱
0 ━━━━━━━━──────────────────────── x
               a

Probability Mass Function (PMF)​

• A discrete random variable $X$ is given by $f_X(x) = P(X=x)$ for $x \in \mathbb{R}$.
• It represents the probability that the first success occurs exactly on the $x$-th trial in a sequence of independent trials.

Probability Density Function (PDF)​

• A continuous random variable $X$ is given by $f_X(x)$ for $x \in \mathbb{R}$.
• The probability is calculated as the area under the probability density function over a specific interval.

Expectation​

$E[X] = \sum x P(X=x) \quad \text{if } X \text{ is discrete}$

$E[X] = \int_{-\infty}^{\infty} x f_X(x) dx \quad \text{if } X \text{ is continuous}$

• Linearity: $E[aX + bY + c] = aE[X] + bE[Y] + c$ for any constants $a$, $b$, and $c$.
• Non-negativity: If $X \geq 0$ then $E[X] \geq 0$ because it is a weighted average of non-negative values.
• Monotonicity: If $X \geq Y$ then $E[X] \geq E[Y]$ because it is a weighted average of values that are greater than or equal to the corresponding values of $Y$.
• Boundedness: If $a \leq X \leq b$ then $a \leq E[X] \leq b$ because it is a weighted average of values that are bounded by $a$ and $b$.

a ●──────────────●──────────────● b
       가능한 값들    평균
                   E[X]

Moments​

$\mu'_n = E[X^n] \quad \forall n \in \mathbb{N}$

• The $n^{th}$ central moment of $X$ is: $\mu_n = E[(X - E[X])^n] = E(X - \mu)^n$

• 1th central moment is the mean, $\mu_1 = E[X]$.

• 2th central moment is the variance, $\mu_2 = E[(X - E[X])^2] = \text{Var}(X)$.

  • It emphasizes the variability of the distribution.
  • $\text{Var}(aX + b) = a^2 \text{Var}(X)$ for any constants $a$ and $b$.

• 3th central moment is the skewness, $\mu_3 = E[(X - E[X])^3]$.

  • It emphasizes the skewness of the distribution.

• 4th central moment is the kurtosis, $\mu_4 = E[(X - E[X])^4]$.

  • It emphasizes the tail behavior and extreme values of the distribution.

Covariance​

$\text{cov(X, Y)} = E[(X - E[X])(Y - E[Y])]$

• It measures how muc htwo random variables change together.
• Negative Covariance
• Near Zero Covariance
• Positive Covariance

Y: 시험 점수

높음  |                ●
     |          ●  ●
     |       ●
     |    ●
낮음  | ●
     +-------------------- X: 공부 시간
       적음             많음

       Cov(X,Y) > 0

Correlation​

$p(X, Y) = \frac{\text{cov(X, Y)}}{\sqrt{\text{Var}(X)\text{Var}(Y)}}$

• Individual variances must be non-zero.
• $p(X, Y)$ lies in the range $[-1, 1]$.

Joint Distributions​

$f_{X,Y}:\mathbb{R}^2\to[0,1], \\ f_{X,Y}(x,y) = P(X=x, Y=y) \quad \text{if } X, Y \text{ are discrete}$

• Joint Probability Mass Function (PMF) for discrete random variables $X$ and $Y$.

Y
1 ┤  ● 1/4       ● 1/4
  │ (0,1)       (1,1)
0 ┤  ● 1/4       ● 1/4
  │ (0,0)       (1,0)
  └──────────────────── X
      0           1

Marginal Distributions​

$f_X(x) = \sum_{y} f_{X,Y}(x,y) \\ f_Y(y) = \sum_{x} f_{X,Y}(x,y)$

• Fixing one variable and summing over the other variable gives the marginal distribution of the fixed variable.

             Y=0       Y=1       행의 합
X=0          0.10      0.20        0.30
X=1          0.30      0.40        0.70
             ─────────────────────────
열의 합       0.40      0.60        1.00

• $f_X(0) = 0.10 + 0.20 = 0.30$: sum of the first row.
• $f_X(1) = 0.30 + 0.40 = 0.70$: sum of the second row.
• $f_Y(0) = 0.10 + 0.30 = 0.40$: sum of the first column.
• $f_Y(1) = 0.20 + 0.40 = 0.60$: sum of the second column.
• Marginalization: It sums or integrates over all possible values of an unwanted variable to obtain the distribution of the variable of interest.

Conditional Distributions​

$f_{X|Y}(x|y) = P(X = x | Y = y) = \frac{f_{X,Y}(x,y)}{f_Y(y)} \quad \text{if } f_Y(y) > 0$

• It represents the probability distribution of $X$ given that $Y$ has a specific value.
• If $f_Y(y) = 0$, then $f_{X|Y}(x|y)$ is undefined.

Bernoulli Distribution​

$X = \begin{cases} 1, & \text{with probability } p \\ 0, & \text{with probability } 1-p \end{cases}$

• $E[X] = p$
• $\text{Var}(X) = p(1-p)$
• The outcome of a Bernoulli trial is either a success (1) or a failure (0).

Binomial Distribution​

• probability of success in a single trial: $p$
• probability of failure in a single trial: $q = 1 - p$

$P(X = x|n, p) = \binom{n}{x} p^x q^{n-x} \\ \text{where} \binom{n}{x} = \frac{n!}{x!(n-x)!}, \quad 0 \leq x \leq n$

• $E[X] = np$
• $\text{Var}(X) = npq = np(1-p)$
• The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials.
• $X \sim \text{Binomial}(10, 0.7)$
  • $P(X = 7) = \binom{10}{7} (0.7)^7 (0.3)^3 \approx 0.2668$

Geometric Distribution​

• probability of success in a single trial: $p$
• probability of failure in a single trial: $q = 1 - p$

$P(X = x|p) = (q^{x-1}) p \\ \text{where } x = 1, 2, 3, ...$

• $E[X] = \frac{1}{p}$
• $\text{Var}(X) = \frac{q}{p^2}$
• The geometric distribution describes the number of trials before the first success in a sequence of independent Bernoulli trials.
• $X \sim \text{Geometric}(0.7)$
  • $P(X = 3) = (0.3)^2 (0.7) \approx 0.063$

Uniform Distribution​

$f_X(x|a,b) = \begin{cases} \frac{1}{b-a}, & a \leq x \leq b \\ 0, & \text{otherwise} \end{cases}$

• $E[X] = \frac{a+b}{2}$
• $\text{Var}(X) = \frac{(b-a)^2}{12}$
• The uniform distribution describes a continuous random variable that has an equal probability of taking any value within a specified range.
• The probability depends on the length of the interval, not its location.

Normal Distribution​

$f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \quad -\infty < x < \infty$

$X \sim N(\mu, \sigma^2)$

• Central Limit Theorem: the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution.

$\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \approx N\left(\mu, \frac{\sigma^2}{n}\right)$

Multivariate Normal Distribution​

$N(x|\mu, \Sigma) = \frac{1}{\sqrt{(2\pi)^D |\Sigma|}} e^{-\frac{1}{2} (x-\mu)^T \Sigma^{-1} (x-\mu)}$

• $\mu$ is the $D$-dimensional mean vector.
• $\Sigma$ is the $D \times D$ covariance matrix.
• $|\Sigma|$ is the determinant of the covariance matrix.

Beta Distribution​

$f(x|\alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}, \quad 0 < x < 1$

• $\Gamma(n) = (n-1)!$ is the gamma function.
• $E[X] = \frac{\alpha}{\alpha + \beta}$
• $\text{Var}(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$

α < β                  α = β                  α > β

높이                   높이                   높이
│╲                     │    ╭──╮             │        ╱
│ ╲___                 │  ╭─╯  ╰─╮           │    ___╱
└──────── x            └────────── x         └──────── x
0        1             0          1           0        1

0 쪽 강조              가운데 강조             1 쪽 강조

Beta(8,2)                 Beta(80,20)

       ╭────╮                    /\
     ╭─╯    ╰─╮                 /  \
─────╯        ╰───        ─────/────\─────
         0.8                      0.8

불확실성 큼                 불확실성 작음

• $p | \text{data} \sim \text{Beta}(\alpha + s, \beta + f)$
  • where new observations are $s$ successes and $f$ failures.
• The Beta distribution is a distribution of success probabilities for a Bernoulli or Binomial distribution.
• Application
  • Measuring uncertainty in the probability of success for a Bernoulli or Binomial distribution.
  • Conversion rate or click-through rate (CTR) in online advertising.
  • Defect rate in manufacturing.

Linear Algebra​

Axioms of Vector Spaces​

• Commutative Law: $u + v = v + u, \quad \forall u, v \in V$
• Associative Law: $(u + v) + w = u + (v + w), \quad \forall u, v, w \in V$
• Additive Identity: $\exists 0 \in V$ such that $v + 0 = v, \quad \forall v \in V$
• Additive Inverse: $\forall v \in V, \exists -v \in V$ such that $v + (-v) = 0$
• Distributive Law:
  • $a(u + v) = au + av, \quad \forall a \in F, u, v \in V$
  • $(a + b)v = av + bv, \quad \forall a, b \in F, v \in V$
• Associative Law: $(ab)v = a(bv), \quad \forall a, b \in F, v \in V$
• Unitary Law: $1v = v, \quad \forall v \in V$

Subspace​

• $x + \alpha y \in W, \quad \forall x, y \in W, \forall \alpha \in \mathbb{R}$

Norm​

$f: \mathbb{R}^n \to \mathbb{R}$

• It outputs a real number that represents the length or size of a vector in a vector space.

$x=(x_1,x_2,\ldots,x_n) \mapsto f(x)=\lVert x\rVert$

• Non-negativity: $\forall x \in \mathbb{R}^n, \quad \lVert x\rVert \geq 0$
• Definiteness: $f(x) = 0 \iff x = 0$
• Homogeneity: $\forall x \in \mathbb{R}^n, \quad \forall t \in \mathbb{R}, f(tx) = |t| f(x)$
• Triangle inequality: $\forall x,y \in \mathbb{R}^n, \quad f(x+y) \leq f(x) + f(y)$
  • $\lVert x + y \rVert \leq \lVert x \rVert + \lVert y \rVert$
• A norm is non-negative, only the zero vector has a norm of zero, scaling a vector by two doubles its norm, and the direct distance cannot be greater than the detoured distance.

$\lVert x \rVert_p = (\sum_{i=1}^{n} |x_i|^p)^{\frac{1}{p}}, \quad p \geq 1$

• L1 norm: $\lVert x \rVert_1 = \sum_{i=1}^{n} |x_i|$
  • Manhattan distance.
• L2 norm: $\lVert x \rVert_2 = \sqrt{\sum_{i=1}^{n} |x_i|^2}$
  • Euclidean distance.
• L-infinity norm: $\lVert x \rVert_\infty = \max_i|x_i|$
  • $\lVert (3, 4) \rVert_\infty = \max\{3, 4\} = 4$

$\lVert A \rVert_F = \sqrt{\sum_{i=1}^{m} \sum_{j=1}^{n} |a_{ij}|^2} = \sqrt{\text{tr}(A^T A)}$

• Frobenius norm: $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
  • $\lVert A \rVert_F = \sqrt{1^2 + 2^2 + 3^2 + 4^2} = \sqrt{30}$
  • It is the square root of the sum of the absolute squares of its elements.

Span​

$span\{x_1, x_2, ..., x_n\} = \{ \alpha_1 x_1 + \alpha_2 x_2 + ... + \alpha_n x_n : \alpha_n \in \mathbb{R}\}$

• A set of vertors $X = \{x_1, x_2, ..., x_n\}$ in a vector space $V$ is said to span $V$ if every vector in $V$ can be expressed as a linear combination of the vectors in $X$.
• Span is the entire region that can be reached by scaling and adding the given vectors.

Range​

$R(A) = \{\alpha_1 a_1 + \alpha_2 a_2 + ... + \alpha_n a_n : \alpha_n \in \mathbb{R}\}$

• The range of a matrix $A$ is the set of all possible linear combinations of its column vectors.
• Same as the columnspace of the matrix $A$.

Nullspace​

$N(A) = \{x \in \mathbb{R}^n : Ax = 0\}$

• A set of all vectors that equal 0 when multiplied by the matrix $A$.
• The dimensionality of the nullspace is called the nullity of the matrix $A$.

Linear Independence​

• A set of vectors $\{v_1, v_2, ..., v_n\}$ is said to be linearly independent if the only solution to the equation $\alpha_1 v_1 + \alpha_2 v_2 + ... + \alpha_n v_n = 0$ is $\alpha_1 = \alpha_2 = ... = \alpha_n = 0$.
• Column rank: The maximum number of linearly independent column vectors in a matrix.
• Row rank: The maximum number of linearly independent row vectors in a matrix.
• If one row vector is a combination of other row vectors, then it is linearly dependent.
• The number of rows that are removed due to redundancy is the Rank of the matrix.
• Due to the redundancy, the number of directions which is vanishing is the Nullity of the matrix.

Rank​

$Rank(A) = \text{dim}(R(A)), \quad A \in \mathbb{R}^{m \times n}$

• $rank(A) \leq \min(m, n), \quad A \in \mathbb{R}^{m \times n}$
  • if $rank(A) = min(m, n)$, then $A$ is said to be full rank.
  • Rank cannot exceed the number of rows or columns in the matrix.
• $rank(A) = rank(A^T), \quad A \in \mathbb{R}^{m \times n}$
  • The rank of a matrix is equal to the rank of its transpose.
  • The amount of independent information in the rows is equal to the amount of independent information in the columns.
• $rank(AB) \leq \min(rank(A), rank(B)), \quad A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{n \times p}$
  • The information content of the product of two matrices cannot exceed the information content of either matrix.
• $rank(A + B) \leq rank(A) + rank(B), \quad A, B \in \mathbb{R}^{m \times n}$
  • The information content of the sum of two matrices cannot exceed the sum of the information content of each matrix.

Orthogonal Matrices​

$U \in \mathbb{R}^{n \times n} \text{ is orthogonal } \iff v_i^T v_j = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$

• $UU^T = U^TU = I_n$
• $\lVert Ux \rVert_2 = \lVert x \rVert_2, \quad \forall x \in \mathbb{R}^n$
• Orthonormal: A set of vectors is orthonormal if they are all unit vectors and orthogonal to each other.
• Rotation: $R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$
• Reflection: $R = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

Quadratic Form​

$Q(x) = x^T A x, \quad A \in \mathbb{R}^{n \times n}$

• The quadratic form $x^TAx$ gives a scalar value that measures the cost, energy, or weighted magnitude of the vextor $x$, according to the quadratic surface defined by the matrix $A$.
  • Positive definite: $Q(x) > 0, \quad \forall x \neq 0$
  • Positive semi-definite: $Q(x) \geq 0, \quad \forall x \neq 0$
  • Negative definite: $Q(x) < 0, \quad \forall x \neq 0$
  • Negative semi-definite: $Q(x) \leq 0, \quad \forall x \neq 0$
  • Indefinite: $Q(x)$ can be positive or negative for different $x \neq 0$

• 행렬의 정부호성(양의 정부호, 양의 준정부호, 음의 정부호, 음의 준정부호, 부정정부호)

$A^TA = A^TA \implies Q(x) = x^T A^T A x = (Ax)^T (Ax) = \lVert Ax \rVert^2 \geq 0$

• Always positive semi-definite because it is the square of the norm of the vector $Ax$.

Eigenvalues & Eigenvectors​

$\lambda \in \mathbb{R} \text{ is an eigenvalue of } A \in \mathbb{R}^{n \times n} \iff A\vec{x} = \lambda \vec{x} , \quad \vec{x} \in \mathbb{R}^n, \vec{x} \neq 0$

• An eigenvector is a nonzero vector that remains on the same line when transformed by a matrix.
• Its eigenvalue tells us how much the vector is scaled and whether its direction is reversed.

$\text{det}(A - \lambda I) = 0$

• $A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$
• $A - \lambda I = \begin{bmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{bmatrix}$
• $\text{det}(A - \lambda I) = (4 - \lambda)(3 - \lambda) - 2 = \lambda^2 - 7\lambda + 10 = (\lambda - 5)(\lambda - 2) = 0$
• $\lambda_1 = 5, \quad \lambda_2 = 2$

$\text{tr}(A) = \sum_{i=1}^{n} \lambda_i$

• The trace is the sum of the scaling factors along all eigenvector directions.

$\text{det}(A) = \prod_{i=1}^{n} \lambda_i$

• The determinant is the product of the eigenvalues, representing the overall signed scaling factor for area or volume.

$rank(A) = \text{number of non-zero eigenvalues of } A$

• For a diagonalizable matrix, the rank equals the number of nonzero eigenvalues, counted with multiplicity.

$\lambda_i (A^{-1}) = \frac{1}{\lambda_i(A)}$

• The eigenvalues of the inverse of a matrix are the reciprocals of the eigenvalues of the original matrix.

$A \vec{x} = \lambda \vec{x} \\ \implies A^{-1} A \vec{x} = A^{-1} \lambda \vec{x} \\ \implies \vec{x} = \lambda A^{-1} \vec{x} \\ \implies A^{-1} \vec{x} = \frac{1}{\lambda} \vec{x}$

Diagonalization​

$S = \begin{bmatrix} \vdots & \vdots & \cdots & \vdots \\ \vec{v_1} & \vec{v_2} & \cdots & \vec{v_n} \\ \vdots & \vdots & \cdots & \vdots \end{bmatrix}$ $AS = \begin{bmatrix} \vdots & \vdots & \cdots & \vdots \\ \lambda_1\vec{v_1} & \lambda_2\vec{v_2} & \cdots & \lambda_n\vec{v_n} \\ \vdots & \vdots & \cdots & \vdots \end{bmatrix} \\ = \begin{bmatrix} \vdots & \vdots & \cdots & \vdots \\ \vec{v_1} & \vec{v_2} & \cdots & \vec{v_n} \\ \vdots & \vdots & \cdots & \vdots \end{bmatrix} \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{bmatrix} \\ = S\Lambda$

• $S^{-1}AS = \Lambda$
  • $AS = S\Lambda$
  • $S^{-1}AS = S^{-1}S\Lambda = \mathbb{I}\Lambda = \Lambda$
  • A matrix $A$ that appears complicated in the original coordinate system becomes the diagonal matrix $\Lambda$ when expressed in the eigenvector coordinate system.
• $A = S\Lambda S^{-1}$
  • In the eigenvector basis, the matrix $A$ becomes the diagonal matrix $\Lambda$.

Symmetric Matrices​

• A real symmetric matrix satisfies $A=A^T$.
• All eigenvalues of a real symmetric matrix are real.
• A real symmetric matrix has a complete set of orthonormal eigenvectors.
• Therefore, the eigenvector matrix $S$ is orthogonal.

• The diagonalization of a symmetric matrix is

• $S$ contains the orthonormal eigenvectors of $A$.
• $\Lambda$ contains the corresponding eigenvalues.
• $S^T$ converts $x$ into the eigenvector basis.
• $\Lambda$ scales each eigenvector direction by its corresponding eigenvalue.
• $S$ converts the result back to the original basis.

Definiteness of Symmetric Matrices​

• Using $A=S\Lambda S^T$ and $y=S^Tx$,

• $y_i$ is the component of $x$ along the $i$-th eigenvector.

• Since $y_i^2\geq0$, the sign of $x^TAx$ depends entirely on the signs of the eigenvalues.

• If all eigenvalues are positive, $A$ is positive definite.

• If all eigenvalues are non-negative, $A$ is positive semidefinite (PSD).

• If all eigenvalues are negative, $A$ is negative definite.

• If all eigenvalues are non-positive, $A$ is negative semidefinite (NSD).

• If $A$ has both positive and negative eigenvalues, $A$ is indefinite.
• The eigenvectors determine the principal directions of the quadratic surface.
• The eigenvalues determine the curvature and sign along those directions.
• The definiteness of a symmetric matrix is determined entirely by the signs of its eigenvalues.

Eigenvalues of a Positive Semidefinite Matrix​

• If $A$ is positive semidefinite, then

• For an eigenvector $\vec{x}\neq\vec{0}$ with eigenvalue $\lambda$,

• Since $\vec{x}\neq\vec{0}$,

• Therefore,

• Hence, all eigenvalues of a positive semidefinite matrix are non-negative.
• A zero eigenvalue means that the corresponding eigenvector direction is collapsed to the zero vector.

Singular Value Decomposition​

• Diagonalization is generally defined for square matrices, while singular value decomposition can be applied to any rectangular or square matrix.

• For $A\in\mathbb{R}^{m\times n}$,

• The columns of $V$ are the right singular vectors and represent orthonormal directions in the input space.
• The columns of $U$ are the left singular vectors and represent orthonormal directions in the output space.
• The diagonal entries of $\Sigma$ are the singular values.

• For each singular-vector pair,

• $\vec{v_i}$ is an input direction.

• $\sigma_i$ is the scaling factor.

• $\vec{u_i}$ is the corresponding output direction.

• The transformation $A$ can be interpreted as three steps:

  • $V^T$ expresses the input in the right singular-vector basis.
  • $\Sigma$ scales each direction by its singular value.
  • $U$ maps the scaled result into the output space.

• $U$ and $V$ are orthogonal matrices.