Podcast
Questions and Answers
In Bayesian inference, the posterior probability is solely determined by the prior probability and does not incorporate any information from the observed data.
In Bayesian inference, the posterior probability is solely determined by the prior probability and does not incorporate any information from the observed data.
False (B)
Assuming a uniform prior, if the likelihood function is symmetric around a specific parameter value, then the posterior distribution will invariably be asymmetric.
Assuming a uniform prior, if the likelihood function is symmetric around a specific parameter value, then the posterior distribution will invariably be asymmetric.
False (B)
Given a non-informative prior, the posterior distribution will be identical to the likelihood function.
Given a non-informative prior, the posterior distribution will be identical to the likelihood function.
True (A)
In cases where the prior probability is conjugate to the likelihood function, the posterior distribution will belong to a fundamentally different family of distributions than the prior.
In cases where the prior probability is conjugate to the likelihood function, the posterior distribution will belong to a fundamentally different family of distributions than the prior.
Within the Bayesian framework, the posterior predictive distribution is calculated by integrating the likelihood function with respect to the prior distribution.
Within the Bayesian framework, the posterior predictive distribution is calculated by integrating the likelihood function with respect to the prior distribution.
The Normal Density Function is unsuitable for modelling phenomena exhibiting non-random clustering around multiple distinct mean values.
The Normal Density Function is unsuitable for modelling phenomena exhibiting non-random clustering around multiple distinct mean values.
The Normal Density Function is inherently limited to modelling continuous data types, rendering it unsuitable for discrete variables such as the number of defects in a manufacturing process.
The Normal Density Function is inherently limited to modelling continuous data types, rendering it unsuitable for discrete variables such as the number of defects in a manufacturing process.
The kurtosis of a standard Normal Density Function can differ based on the parameters inputted.
The kurtosis of a standard Normal Density Function can differ based on the parameters inputted.
The characteristic function of the Normal Density Function with mean $\mu$ and variance $\sigma^2$ is given by $e^{i\mu t - \frac{1}{2}\sigma^2 t^2}$, indicating that all moments of the distribution are independent.
The characteristic function of the Normal Density Function with mean $\mu$ and variance $\sigma^2$ is given by $e^{i\mu t - \frac{1}{2}\sigma^2 t^2}$, indicating that all moments of the distribution are independent.
In the context of Bayesian inference, employing a Normal Density Function as a prior distribution for a parameter inherently guarantees a closed-form analytical solution for the posterior distribution, irrespective of the likelihood function.
In the context of Bayesian inference, employing a Normal Density Function as a prior distribution for a parameter inherently guarantees a closed-form analytical solution for the posterior distribution, irrespective of the likelihood function.
Flashcards
Posterior Probability
Posterior Probability
Probability of an event after observing new evidence.
Bayesian Statistics
Bayesian Statistics
A statistical approach updating beliefs with new data.
Prior Probability
Prior Probability
Initial belief about an event before new evidence.
Likelihood
Likelihood
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Evidence
Evidence
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Distribution
Distribution
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Bell Curve
Bell Curve
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Distribution Mean
Distribution Mean
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Normal Distribution
Normal Distribution
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Data Distribution
Data Distribution
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Normal Density Function
Normal Density Function
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Mean (in Normal Distribution)
Mean (in Normal Distribution)
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Normal Curve
Normal Curve
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Normal Density Function Use
Normal Density Function Use
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Study Notes
- Pattern Recognition is the discipline focused on identifying patterns and regularities present in data
Statistical Pattern Recognition
- Statistical Pattern Recognition (SPR) is a machine learning and data analysis field
- SPR focuses on recognizing, classifying, and analyzing data patterns and regularities via statistical methods
Bayesian Decision Theory
- Bayesian Decision Theory represents a foundational statistical method for pattern classification
- It employs probabilities for informed decision-making
- It provides a quantitative framework for decision-making amidst uncertainty.
Bayes Theorem
- Bayes' Theorem can calculate the probability of a hypothesis
Posterior Probability
- Posterior probability assesses the probability of an event given another event's occurrence
- It is based on prior probability, data likelihood, and evidence in Bayesian statistics
- P(ωj/x) = [P(x/ωj)P(ωj)] / P(x) is the equation representing the above values
- P(ωj/x) signifies the posterior probability of class ωj given observation x
- It represents how likely observation x belongs to class ωj based on the evidence provided
- P(x/ωj) represents the likelihood of observing x given class ωj
- P(ωj) represents the prior probability of class ωj
- P(x) represents the evidence or marginal likelihood of x
- P(ωj/x) signifies the posterior probability of class ωj given observation x
Likelihood Function
- Likelihood function P(x/ωj) is termed class likelihood
- This is the conditional probability of an event belonging to class ωj with observation value x
- P(x/ωj) is the likelihood that feature vector x or data x belongs to class ωj
Prior Probabilities
- Prior probabilities can be exemplified through class distribution in a dataset
- Example dataset with the following distribution of classes:
- Class 1 (ω1) has 30 samples
- Class 2 (ω2) has 50 samples
- Class 3 (ω3) has 20 samples
- The total samples N=50+30+20=100 The prior probabilities can be computed as follows:
- P(ω1) = 30 / 100 = 0.3
- P(ω2) = 50 / 100 = 0.5
- P(ω3) = 20 / 100 = 0.2
- Example dataset with the following distribution of classes:
Evidence
- Evidence, P(x) is calculating the sum of the product of the likelihood P(x/ωj) and the prior probability P(ωj) across all classes
- P(x) = ∑ P(x/ ωj) P(ωj) is the formula
- P(x) signifies the evidence of observing the data x
- C equals the number of classes
- P(x/ωj) signifies the likelihood of the data x given class ωj
- P(ωj) signifies the prior probability of class ωj
- Example for above formula:
- Three classes exist: ω1, ω2, and ω3
- Prior probabilities of P(ω1) = 0.3, P(ω2) = 0.5, and P(ω3) = 0.2
- The likelihoods of observing x given each class are
- P(x|ω1) = 0.4, P(x|ω2) = 0.6, and P(x|ω3) = 0.3
- Then the evidence gets computed as: P(x) = (0.4×0.3) + (0.6×0.5) + (0.3×0.2) = 0.12 + 0.3 + 0.06 = 0.48
- Three classes exist: ω1, ω2, and ω3
Decision Rule
- In classification, the observation x is assigned to the class with the highest posterior probability
- This is also known as the Maximum A Posteriori (MAP) decision rule:
- ω'(x) = arg maxj P(ωj/x) is the equation to represent this
- arg maxj refers to the value of j that maximizes the following expression
- P(ωj/x) is the posterior probability of class ωj given the observed data x, and represents the likelihood of data x belonging to class ωj within Bayesian classification
- ω'(x) represents the predicted class for data x
- ω'(x) = arg maxj P(ωj/x) is the equation to represent this
Risk and Loss Function
- Potential losses and risks in decision theory correlate with making incorrect decisions
- The aim is to diminish the expected risk
- This is the weighted sum of losses based on the probability of different events
- L(wi/ωj) is used to quantify the cost of a decision
Expected Risk
- The expected risk is displayed with R
- R(ωj/x) = ∑ L(ωi/ωj) P(ωi/x)
- R(ωj/x) represents the risk or expected loss associated with choosing class ωj given the x data
- L(ωi/ωj) The loss incurred if the true class is ωi but system states it is ωj
- P(ωi/x) posterior probability of class ωi given the data x
- R(ωj/x) = ∑ L(ωi/ωj) P(ωi/x)
- Decision rule minimizes the expected risk
Decision Example
- If we assume we have a skin sample x and we need to classify it as either having cancer ω1 (class 1) or not having cancer ω0 (class 0)
- The classes and losses can be defined:
- ω0 no cancer
- ω1 is cancer
- L(ω0/ω0) = 0: No loss if system correctly classify no cancer
- L(ω1/ω1) = 0: No loss if system correctly classify cancer
- L(ω0/ω1) = 1: Loss if system incorrectly classify cancer as no cancer
- L(ω1/ω0) = 1: Loss if system incorrectly classify no cancer as cancer
- The classes and losses can be defined:
- Probability estimates:
- P(ω0/x) =0.3: Probability class ω0 that the sample does not have cancer given the data x
- P(ω1/x) = 0.7: Probability class ω1 that the sample has cancer given the data x
- The risk can be calculated for each decision:
- Risk of deciding ω0 (no cancer):
- R(ω0/x) = L(ω0/ω0) P(ω0/x) + L(ω0/ω1) P(ω1/x)
- R(ω0/x) = 0 * 0.3 + 1 * 0.7 = 0 + 0.7 = 0.7
- Risk of deciding ω1 (cancer):
- R(ω1/x) = L(ω0/ω1) P(ω0/x) + L(ω1/ω1) P(ω1/x)
- R(ω1/x) =1*0.3 + 0 * 0.7 = 0.3 + 0 = 0.3
- Risk of deciding ω0 (no cancer):
- Decision
- Risk of deciding no cancer (ω0) is 0.7
- Since the risk is lower for deciding that the sample has cancer, we classify the sample x as having cancer
- The risk of deciding cancer (ω1) is 0.3
- Risk of deciding no cancer (ω0) is 0.7
Normal Density Function
- The Normal Density Function (Gaussian Distribution or Normal Distribution) is in statistics and probability theory
- Describes how data disperses in a continuous space (bell curve shape)
Example of Normal Density Function
- If the average score is 75, most students score approximately 65 and 85 if referring to students’ tests
- Fewer students will score extremely low or high
Uses of Normal Distribution
- Heights, weights, and blood pressure are measurable in natural and social sciences
- Data analysis occurs in statistics
- Predictive models happen in economics
- Random errors are studied in engineering
Mathematical Definition of PDF
- Probability density function (PDF) of a normal distribution in one dimension:
- f(x) = 1 / (√2πσ2) * exp(-(x - μ)2 / (2σ2))
- x = variable
- μ (mu) = mean
- σ2(sigma) is the variance σ2(sigma)
- σ is the standard deviation
- exp is an exponential function.
- π is a mathematical constant with approximately 3.14159
- f(x) = 1 / (√2πσ2) * exp(-(x - μ)2 / (2σ2))
Properties of the Normal Density Function
- Symmetry: The normal distribution is symmetric around its mean (μ)
- The means the left distribution mirrors the right distribution
- Standard Deviation: Measures data spread around the mean with more spread for larger deviations
- Bell-Shaped Curve: The curve is bell-shaped and the highest point at the mean μ
- The curve's tails indefinitely extend in both directions
Normal Distribution
- The 68-95-99.7 Rule:
- Around 68% of the data is in one standard deviation from the mean
- Around 95% of the data is in two standard deviations from the mean
- Roughly 99.7% of the data is in three standard deviations from the mean
- This pertains to the normal distribution of data
- X-axis are values (Mean and standard deviations)
- Y-axis is the for probability density (Probability Density)
- This pertains to the normal distribution of data
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Description
This lesson covers multiple concepts in statistics. It explores Bayesian inference, posterior probability and likelihood function. The content also touches on normal density function and its limitations.
