Podcast
Questions and Answers
In the context of density estimation, which statement best differentiates between parametric and nonparametric approaches?
In the context of density estimation, which statement best differentiates between parametric and nonparametric approaches?
- Parametric methods are instance-based, using data instances directly, whereas nonparametric models rely on aggregating data to estimate parameters.
- Parametric methods optimize a predetermined number of parameters by fitting a model to the dataset, while nonparametric methods determine the density function entirely from the data without assuming a specific model. (correct)
- Parametric methods do not require any prior assumptions about the data, unlike nonparametric methods which are heavily reliant on distributional assumptions.
- Parametric methods estimate the probability density function directly from the data, whereas nonparametric methods assume an underlying parameterized model.
In Maximum Likelihood Estimation (MLE), the likelihood function, $p(D|\theta)$, represents the joint probability of the parameters $\theta$ given the observed dataset $D$, assuming independent and identically distributed (i.i.d.) observations.
In Maximum Likelihood Estimation (MLE), the likelihood function, $p(D|\theta)$, represents the joint probability of the parameters $\theta$ given the observed dataset $D$, assuming independent and identically distributed (i.i.d.) observations.
False (B)
In the context of maximum likelihood estimation (MLE) for a Bernoulli distribution, given a dataset $D = {x^{(1)}, x^{(2)}, ..., x^{(N)}}$ where $m$ represents the number of heads (1) and $N-m$ represents the number of tails (0), what is the MLE estimate for the probability of heads, denoted as $\theta_{ML}$? Express your answer in terms of $m$ and $N$.
In the context of maximum likelihood estimation (MLE) for a Bernoulli distribution, given a dataset $D = {x^{(1)}, x^{(2)}, ..., x^{(N)}}$ where $m$ represents the number of heads (1) and $N-m$ represents the number of tails (0), what is the MLE estimate for the probability of heads, denoted as $\theta_{ML}$? Express your answer in terms of $m$ and $N$.
$\theta_{ML} = \frac{m}{N}$
In Maximum A Posteriori (MAP) estimation, the goal is to find the parameter value that maximizes the ______ probability of the parameters given the data.
In Maximum A Posteriori (MAP) estimation, the goal is to find the parameter value that maximizes the ______ probability of the parameters given the data.
Match the estimation method with its respective optimization objective.
Match the estimation method with its respective optimization objective.
Consider a scenario where you are using Maximum Likelihood Estimation (MLE) to estimate the parameter $\mu$ of a Gaussian distribution with known variance $\sigma^2$. Given a dataset of $N$ independent samples, which of the following statements correctly describes the behavior of the MLE estimate as $N$ approaches infinity?
Consider a scenario where you are using Maximum Likelihood Estimation (MLE) to estimate the parameter $\mu$ of a Gaussian distribution with known variance $\sigma^2$. Given a dataset of $N$ independent samples, which of the following statements correctly describes the behavior of the MLE estimate as $N$ approaches infinity?
In Bayesian inference, the predictive distribution $p(x|D)$ is computed by integrating over the posterior distribution $p(\theta|D)$, effectively averaging the predictions of all possible parameter values weighted by their posterior probabilities.
In Bayesian inference, the predictive distribution $p(x|D)$ is computed by integrating over the posterior distribution $p(\theta|D)$, effectively averaging the predictions of all possible parameter values weighted by their posterior probabilities.
For a Bernoulli likelihood with a Beta prior, if the prior distribution is given by $Beta(\theta | \alpha_1, \alpha_0)$ and we observe a dataset $D$ with $m$ heads and $N-m$ tails, what is the form of the posterior distribution $p(\theta|D)$? Specify the parameters of the Beta distribution in terms of $\alpha_1$, $\alpha_0$, $m$, and $N$.
For a Bernoulli likelihood with a Beta prior, if the prior distribution is given by $Beta(\theta | \alpha_1, \alpha_0)$ and we observe a dataset $D$ with $m$ heads and $N-m$ tails, what is the form of the posterior distribution $p(\theta|D)$? Specify the parameters of the Beta distribution in terms of $\alpha_1$, $\alpha_0$, $m$, and $N$.
In the context of Bayesian inference, a prior distribution is considered a ______ prior if the posterior distribution belongs to the same family as the prior distribution.
In the context of Bayesian inference, a prior distribution is considered a ______ prior if the posterior distribution belongs to the same family as the prior distribution.
Match each term with its corresponding definition in the context of Maximum A Posteriori (MAP) estimation.
Match each term with its corresponding definition in the context of Maximum A Posteriori (MAP) estimation.
Assume you are estimating parameters using Maximum Likelihood Estimation (MLE) for a dataset drawn from a distribution. Which of the following considerations is most critical for ensuring that the resulting parameter estimates are statistically sound and generalizable?
Assume you are estimating parameters using Maximum Likelihood Estimation (MLE) for a dataset drawn from a distribution. Which of the following considerations is most critical for ensuring that the resulting parameter estimates are statistically sound and generalizable?
In Maximum A Posteriori (MAP) estimation, if the prior distribution $p(\theta)$ is uniform (constant) across all possible values of $\theta$, then the MAP estimate is equivalent to the Maximum Likelihood Estimation (MLE) estimate.
In Maximum A Posteriori (MAP) estimation, if the prior distribution $p(\theta)$ is uniform (constant) across all possible values of $\theta$, then the MAP estimate is equivalent to the Maximum Likelihood Estimation (MLE) estimate.
In the context of Bayesian inference, what fundamental principle guides the updating of beliefs about model parameters given observed data?
In the context of Bayesian inference, what fundamental principle guides the updating of beliefs about model parameters given observed data?
In the Bayesian approach, the ______ distribution represents the probability of observing new data points given the observed dataset, integrating over all possible parameter values weighted by their posterior probabilities.
In the Bayesian approach, the ______ distribution represents the probability of observing new data points given the observed dataset, integrating over all possible parameter values weighted by their posterior probabilities.
Associate the following concepts with their relevance to either Maximum Likelihood Estimation (MLE), Maximum A Posteriori (MAP), or Bayesian inference.
Associate the following concepts with their relevance to either Maximum Likelihood Estimation (MLE), Maximum A Posteriori (MAP), or Bayesian inference.
In a complex Bayesian hierarchical model, which of the following techniques is most rigorously applied to approximate the posterior distribution when analytical solutions are intractable?
In a complex Bayesian hierarchical model, which of the following techniques is most rigorously applied to approximate the posterior distribution when analytical solutions are intractable?
In Bayesian inference, as the size of the observed dataset increases, the influence of the prior distribution on the posterior distribution diminishes, causing the posterior distribution to converge towards the Maximum Likelihood Estimate (MLE).
In Bayesian inference, as the size of the observed dataset increases, the influence of the prior distribution on the posterior distribution diminishes, causing the posterior distribution to converge towards the Maximum Likelihood Estimate (MLE).
Given a dataset $D$ and a prior distribution $p(\theta)$, write the formula for the posterior distribution $p(\theta|D)$ using Bayes' theorem. Express your answer in terms of the likelihood $p(D|\theta)$ and any necessary normalizing constants.
Given a dataset $D$ and a prior distribution $p(\theta)$, write the formula for the posterior distribution $p(\theta|D)$ using Bayes' theorem. Express your answer in terms of the likelihood $p(D|\theta)$ and any necessary normalizing constants.
When using Maximum A Posteriori (MAP) estimation, incorporating a well-chosen prior distribution can serve as a form of ______, which helps to prevent overfitting, especially when dealing with limited data.
When using Maximum A Posteriori (MAP) estimation, incorporating a well-chosen prior distribution can serve as a form of ______, which helps to prevent overfitting, especially when dealing with limited data.
Match the term with its purpose in Bayesian inference.
Match the term with its purpose in Bayesian inference.
In the context of Bayesian model comparison, which of the following statements presents the most accurate and nuanced interpretation of the Bayes factor?
In the context of Bayesian model comparison, which of the following statements presents the most accurate and nuanced interpretation of the Bayes factor?
The marginal likelihood, also known as the evidence, is calculated by integrating (or summing) the likelihood function over the prior distribution, and it represents the probability of observing the data given the model, averaging over all possible parameter values.
The marginal likelihood, also known as the evidence, is calculated by integrating (or summing) the likelihood function over the prior distribution, and it represents the probability of observing the data given the model, averaging over all possible parameter values.
In Maximum Likelihood Estimation (MLE), what is the significance of the gradient of the log-likelihood function? Specifically, what does setting the gradient to zero accomplish?
In Maximum Likelihood Estimation (MLE), what is the significance of the gradient of the log-likelihood function? Specifically, what does setting the gradient to zero accomplish?
In the context of Bayesian inference, the term 'evidence' refers to the ______ likelihood, which is the probability of the observed data given the model, marginalized over all possible parameter values.
In the context of Bayesian inference, the term 'evidence' refers to the ______ likelihood, which is the probability of the observed data given the model, marginalized over all possible parameter values.
Match the distribution with scenario to apply MLE or MAP estimation
Match the distribution with scenario to apply MLE or MAP estimation
Consider a scenario where you are applying Maximum A Posteriori (MAP) estimation to estimate the parameters of a complex non-linear model with high dimensionality. Which computational challenges are most pertinent, and what strategies can be effectively employed to address them?
Consider a scenario where you are applying Maximum A Posteriori (MAP) estimation to estimate the parameters of a complex non-linear model with high dimensionality. Which computational challenges are most pertinent, and what strategies can be effectively employed to address them?
Both Maximum Likelihood Estimation (MLE) and Maximum A Posteriori (MAP) estimation always provide a probability distribution as an estimate for the unknown parameters.
Both Maximum Likelihood Estimation (MLE) and Maximum A Posteriori (MAP) estimation always provide a probability distribution as an estimate for the unknown parameters.
In Bayesian inference, what is the role of the 'burn-in' period when using Markov Chain Monte Carlo (MCMC) methods, and why is it important to discard samples from this period?
In Bayesian inference, what is the role of the 'burn-in' period when using Markov Chain Monte Carlo (MCMC) methods, and why is it important to discard samples from this period?
When comparing models using Bayesian methods, a higher ______ indicates stronger evidence for the model given the observed data, accounting for both its accuracy and complexity.
When comparing models using Bayesian methods, a higher ______ indicates stronger evidence for the model given the observed data, accounting for both its accuracy and complexity.
Match following methods for their purpose regarding parameter estimation:
Match following methods for their purpose regarding parameter estimation:
Which of the following statements best describes the relationship between Maximum Likelihood Estimation (MLE), Maximum A Posteriori (MAP), and Bayesian inference as the amount of observed data approaches infinity?
Which of the following statements best describes the relationship between Maximum Likelihood Estimation (MLE), Maximum A Posteriori (MAP), and Bayesian inference as the amount of observed data approaches infinity?
In Bayesian inference, using a non-informative prior guarantees that the posterior distribution will accurately reflect the true underlying distribution of the data, regardless of the sample size.
In Bayesian inference, using a non-informative prior guarantees that the posterior distribution will accurately reflect the true underlying distribution of the data, regardless of the sample size.
Consider a scenario involving parameter estimation for a model with a known likelihood function and a conjugate prior. Explain how the posterior distribution is updated sequentially as new data becomes available.
Consider a scenario involving parameter estimation for a model with a known likelihood function and a conjugate prior. Explain how the posterior distribution is updated sequentially as new data becomes available.
In the context of applying Bayesian methods to real-world problems, the selection of a ______ prior can introduce biases and assumptions that may not be justified by the available data, leading to potentially misleading conclusions.
In the context of applying Bayesian methods to real-world problems, the selection of a ______ prior can introduce biases and assumptions that may not be justified by the available data, leading to potentially misleading conclusions.
Match the following concepts with their corresponding expressions or formulas in the context of Maximum A Posteriori (MAP) estimation, Maximum Likelihood Estimation (MLE) and Bayesian Inference.
Match the following concepts with their corresponding expressions or formulas in the context of Maximum A Posteriori (MAP) estimation, Maximum Likelihood Estimation (MLE) and Bayesian Inference.
Considering a scenario where one must choose between Maximum Likelihood Estimation (MLE), Maximum A Posteriori (MAP) estimation, and full Bayesian inference for a complex machine learning task, what factors should most critically influence the choice?
Considering a scenario where one must choose between Maximum Likelihood Estimation (MLE), Maximum A Posteriori (MAP) estimation, and full Bayesian inference for a complex machine learning task, what factors should most critically influence the choice?
In Bayesian model comparison, the model with the highest posterior probability is always the best model for prediction and generalization.
In Bayesian model comparison, the model with the highest posterior probability is always the best model for prediction and generalization.
Suppose you are using Maximum Likelihood Estimation (MLE) to estimate the parameter $\lambda$ of a Poisson distribution. Derive the MLE estimator $\hat{\lambda}_{MLE}$ given a set of $N$ independent samples ${x_1, x_2, ..., x_N}$.
Suppose you are using Maximum Likelihood Estimation (MLE) to estimate the parameter $\lambda$ of a Poisson distribution. Derive the MLE estimator $\hat{\lambda}_{MLE}$ given a set of $N$ independent samples ${x_1, x_2, ..., x_N}$.
In Maximum A Posteriori (MAP) estimation, increasing the ______ of the prior distribution reduces its impact on the posterior estimate, allowing the likelihood function to dominate.
In Maximum A Posteriori (MAP) estimation, increasing the ______ of the prior distribution reduces its impact on the posterior estimate, allowing the likelihood function to dominate.
Match key aspects of MAP and MLE to their mathematical formulation
Match key aspects of MAP and MLE to their mathematical formulation
Flashcards
MLE (Maximum Likelihood Estimation)
MLE (Maximum Likelihood Estimation)
A method of estimating the parameters of a statistical model given data.
Likelihood
Likelihood
The conditional probability of observations given the value of parameters.
Parametric density estimation goal
Parametric density estimation goal
Estimate parameters of a distribution from a dataset D = {x(1),...,x(N)} containing N independent, identically distributed training samples.
Density estimation
Density estimation
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Parametric Approach
Parametric Approach
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Nonparametric Approach
Nonparametric Approach
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MAP estimation
MAP estimation
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Bayesian inference
Bayesian inference
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Bayesian estimation
Bayesian estimation
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Conjugate Priors
Conjugate Priors
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Beta distribution
Beta distribution
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Choosing a Prior
Choosing a Prior
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Study Notes
Relation of Learning & Statistics
- Target models in learning problems can be considered statistical models
- Estimation methods estimate the target from available data for a fixed dataset and underlying target
Density Estimation
- Process involves estimating the probability density function p(x) given a set of data points {x(i)}^N_i=1 drawn from it
- Main approaches to density estimation include parametric and nonparametric methods
- Parametric Methods: Assume a parameterized model for the density function and optimize parameters by fitting the model to the dataset
- Nonparametric Methods: (Instance-based) Do not assume a specific parametric model; the form of the density function is determined entirely by the data
Parametric Density Estimation
- Process estimates the probability density function p(x) using data points {x(i)}^N_i=1
Parametric density estimation (Goal)
- Goal is to estimate parameters of a distribution from a dataset D = {x(1),...,x(N)}
- D contains N independent/identically distributed training samples
- Need to determine θ given {x(1), ..., x(N)} represented as θ* or p(θ)
Maximum Likelihood Estimation (MLE) Defined
- MLE estimates parameters of a statistical model given data
- Likelihood is the conditional probability of observations D = {x(1), x(2), ..., x(N)} for a given parameter θ
- Assuming i.i.d. observations: p(D|θ) = ∏[i=1 to N] p(x(i)|θ)
- MLE Formula -> θ_ML = argmax[θ] p(D|θ)
Maximum Likelihood Estimation (MLE) Formula Explained
- L(θ) = ln p(D|θ) = ln ∏[i=1 to N] p(x(i)|θ) = ∑[i=1 to N] ln p(x(i)|θ)
- θ_ML = argmax[θ] L(θ) = argmax[θ] ∑[i=1 to N] ln p(x(i)|θ)
- Solving ∇θL(θ) = 0 finds the global optimum
MLE Bernoulli
- Given D = {x(1), x(2), ..., x(N)}, m heads (I), N – m tails (0), then p(x|θ) = θ^x (1 – θ)^(1-x)
- p(D|θ) = ∏^{N}{i=1} p(x^{(i)}|θ) = ∏^{N}{i=1} θ^{x^{(i)}} (1 − θ)^{1−x^{(i)}}
- ln p(D|θ) = ∑^{N}{i=1} ln p(x^{(i)}|θ) = ∑^{N}{i=1} {x^{(i)} ln θ + (1 − x^{(i)}) ln(1 – θ)}
- (∂ ln p(D|θ))/∂θ = 0 => θML = (∑^{N}_{i=1} x^{(i)})/N = m/N
MLE Bernoulli Example
- D = {1,1,1}, θ_ML = 3/3 = 1
- Prediction: all future tosses will land heads up, overfitting to D
MLE: Multinomial Distribution
- Multinomial distributions occur on variables with K states
- Parameter space includes θ = [θ_1, ..., θ_K] where θ_i ε [0,1]
- ∑^{K}_{k=1} θ_k = 1
- Distributions use P(x|θ) = ∏^{K}_{k=1} θ_k^{x_k} where P(x_k = 1) = θ_k
MLE: Multinomial distribution Formula
- D = {x⁽¹⁾, x⁽²⁾, ..., x⁽ᴺ⁾}
- P(D|θ) = ∏[i=1 to N] P(x⁽ⁱ⁾|θ)
- L(θ) = ln p(D|θ) + λ(1 −∑[k=1 to K] θ_k)
- θ̂_k = (∑[i=1 to N] x_k^(i)) / N = N_k / N
MLE Gaussian: Unknown μ
- ln p(x(i)|μ) = - ln{√(2πσ)} - (1/(2σ^2)) (x(i) - μ)^2
- (∂L(μ))/∂μ = 0 => ∂/∂μ (∑[i=1 to N] ln p(x(i)|μ)) = 0
- ∑[i=1 to N] (1/σ^2) (x(i) - μ) = 0 => μ_ML = (1/N) ∑[i=1 to N] x(i)
- MLE corresponds to many well-known estimation methods
MLE Gaussian: unknown μ and σ
- θ = [μ, σ]
- ∇θL(θ) = 0
- (∂L(μ, σ))/(∂μ) => μ_ML = (1/N)∑[i=1 to N] x(i)
- (∂L(μ, σ))/(∂σ) => σ_ML = (1/N)∑[i=1 to N] (x(i) − μ_ML)^2
Maximum a Posteriori (MAP) Estimation
- θ_MAP = argmax[θ] p(θ|D)
- Since p(θ|D) ∝ p(D|θ)p(θ) -> θ_MAP = argmax[θ] p(D|θ)p(θ)
MAP Estimation: Example of Prior Distribution
- p(θ) = N(θ_0, σ^2)
MAP Estimation: Gaussian w/ Unknown μ
- μ is the only unknown parameter while μ_0 and σ_0 are known
- Given: p(x|μ) ~ N(μ, σ^2) and p(μ|μ_0) ~ N(μ_0, σ_0^2)
- (d/dμ) ln (p(μ) ∏[i=1 to N] p(x(i)|μ)) = 0 -> ∑[i=1 to N] (1/σ^2) (x(i) - μ) - (1/σ_0^2) (μ - μ_0) = 0
- Thus: μ_MAP = (μ₀ + (σ₀²/σ²)∑[i=1 to N] x⁽ⁱ⁾) / (1 + (σ₀²/σ²)N)
- If σ₀²/σ² >> 1 or N → ∞, then μ_MAP = μ_ML =∑[i=1 to N] x(i) / N
Maximum a Posteriori (MAP) Estimation
- Given a set of observations D and a prior distribution p(θ) on parameters, the parameter vector that maximizes p(D|θ)p(θ) is found
MAP estimation: Gaussian with unknown μ (known σ)
- p(μ|D) ∝ p(μ)p(D|μ)
- p(μ|D) = N(μ|μ_N, σ_N)
- μ_N = (μ₀ + (σ₀²/σ²)∑[i=1 to N] x⁽ⁱ⁾) / (1 + (σ₀²/σ²)N)
- N = 1 / σ₀² + N / σ²
- More Samples: sharper p(μ|D), giving higher confidence
Conjugate priors
- A form of prior distribution with simple interpretations and useful analytical properties
- The posterior distribution proportional to p(D|θ)p(θ) will have the same functional form as the prior
- ∀α, D ∃α′ P(θ|α′) ∝ P(D|θ)P(θ|α)
Prior for Bernoulli Likelihood
- Beta distribution occurs of θ∈ [0,1]
- Beta(θ|α₁, α₀) ∝ θ^(α₁−1)(1 − θ)^(α₀−1)
- Beta(θ|α₁, α₀) = (Γ(α₀ + α₁) / (Γ(α₀)Γ(α₁))) * θ^(α₁−1)(1 – θ)^(α₀−1)
- Beta distribution is the conjugate prior of Bernoulli
- P(x|θ) = θ^x(1 – θ)^(1-x)
Bernoulli Likelihood: Posterior
- Given: D = {x^(1), x^(2), ..., x^(N)}, m heads (1), N – m tails (0)
- p(θ|D) ∝ p(D|θ)p(θ)
- p(θ|D) ∝ (∏[i=1 to N] θ^(x^(i)) (1-θ)^(1-x^(i))) * Beta(θ|α₁, α₀)
- ∝ θ^(m+α₁-1) (1–θ)^(N-m+α₀-1)
- p(θ|D) ∝ Beta(θ|α₁′, α₀′)
- α₁′ = α₁ + m
- α₀′ = α₀ + N − m
Toss example with Conjugate priors
- MAP estimation reduces overfitting
- where D = {1,1,1}, θ_ML = 1, then θ_MAP = 0.8 (with prior p(θ) = Beta(θ|2,2))
Bayesian Inference Defined
- Parameters become random variables with a priori distribution.
- Bayesian estimation uses available prior information.
- Unlike ML and MAP, it does not seek a specific point estimate of the unknown parameter vector θ.
- Observed samples, D, convert prior densities, p(θ), into a posterior density, p(θ|D).
- Approach tracks beliefs about θ's values and uses these beliefs for reaching conclusions.
- Within a Bayesian approach: Specify p(θ|D) before computing the predictive distribution p(x|D).
Bayesian Estimation: Predictive Distribution Defined
- Given samples D = {x⁽ⁱ⁾} from i=1 to N, a prior distribution on the parameters P(θ), and the form of the distribution P(x|θ).
- P(θ|D) is used to specify an estimate P̂(x) = P(x|D) of P(x). where:
- P(x|D) = ∫ P(x, θ|D)dθ = ∫ P(x|D, θ)P(θ|D)dθ = ∫ P(x|θ)P(θ|D)dθ
- When the parameters of θ are known, the distribution of x is known as P.
- Analytical solutions exist for involved functions.
Benoulli Likelihood: Prediction Formula
- Training Samples: D = {x(1), ..., x(N)}
- P(θ) = Beta(θ|α₁, α₀) ∝ θ^(α₁−1)(1 − θ)^(α₀−1)
- P(θ|D) = Beta(θ|α₁ + m, α₀ + N − m) ∝ θ^(α₁+m−1)(1 – θ)^(α₀+(N-m)-1) P(x|D) = ∫P(x|θ) ⋅ P(θ|D)dθ = Ε_P(θ|D)[P(x|θ)] P(x = 1|D = E_P(θ|D)[θ] = (α₁ + m) / (α₀ + α₁ + N)
Relationships between ML, MAP, and Bayesian Estimation
- If p(θ|D) has a sharp peak at θ = θ̂ (i.e., p(θ|D) ≈ δ(θ, θ̂)), then p(x|D) ≈ p(x|θ)
- Bayesian estimation will be approximately equal to the MAP estimation
- If p(D|θ) is concentrated around a sharp peak and p(θ) is broad enough around this peak, the ML, MAP, and Bayesian estimations yield approximately the same result
- All three methods asymptotically (N → ∞) result in the same estimate
Summary of ML, MAP, and Bayesian Estimation
- ML and MAP lead to a single point estimate of the unknown parameter vectors
- More interpretable and simpler thant Bayesian estimation
- The Bayesian approach utilizes all available information to find a predictive distribution
- Expected to give accurate results needing higher computational complexity
- Bayesian methods gained popularity due to computer technology advances
- All three methods converge to the same estimate asymptotically (N → ∞).
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