Statistics is the science of creating probabilistic models from data and validating them. We will talk about the Bayesian and classical approaches to parameter estimation, confidence intervals, hypothesis testing, and reasoning about (in)dependence.

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That's a great topic! Let's delve into the Bayesian and classical approaches to parameter estimation, confidence intervals, hypothesis testing, and reasoning about independence.

Parameter Estimation:

Classical Approach: In the classical approach, parameter estimation is typically based on methods such as maximum likelihood estimation (MLE) or method of moments. MLE aims to find the parameter values that maximize the likelihood function given the observed data. It provides point estimates of the parameters.

Bayesian Approach: In the Bayesian approach, parameter estimation is based on Bayes' theorem. It involves specifying prior distributions for the parameters and updating them using the observed data to obtain posterior distributions. Bayesian estimation provides a full distribution of the parameters, allowing for uncertainty quantification.

Confidence Intervals:

Classical Approach: In the classical approach, confidence intervals are constructed based on the sampling distribution of the estimator. The confidence interval provides a range of plausible values for the parameter, typically at a chosen confidence level (e.g., 95%). The width of the interval depends on the variability of the estimator.

Bayesian Approach: In the Bayesian approach, credible intervals are constructed from the posterior distribution. A credible interval represents a range of values that contains the parameter with a certain probability (e.g., 95%). The width of the interval depends on the uncertainty captured by the posterior distribution.

Hypothesis Testing:

Classical Approach: In the classical approach, hypothesis testing involves specifying null and alternative hypotheses, choosing a test statistic, and calculating a p-value. The p-value quantifies the evidence against the null hypothesis based on the observed data. A significance level (e.g., 0.05) is often used to make a decision to reject or fail to reject the null hypothesis.

Bayesian Approach: In the Bayesian approach, hypothesis testing is performed by comparing the posterior probabilities of different hypotheses. Bayes factors or posterior odds ratios are used to quantify the strength of evidence in favor of one hypothesis over another. Decision rules can be based on thresholds or the relative probabilities of the hypotheses.

Reasoning about (In)dependence:

Classical Approach: In the classical approach, dependence or independence between variables is typically assessed using methods such as correlation analysis or tests like chi-square test for independence. These methods analyze the association between variables based on observed data and assess the statistical significance of the relationship.

Bayesian Approach: In the Bayesian approach, dependence or independence between variables can be modeled explicitly using prior distributions and joint probability distributions. Bayesian networks are often employed to represent probabilistic dependencies between variables and make inferences about their relationships.

Both the Bayesian and classical approaches have their strengths and weaknesses, and their choice depends on the problem at hand, available data, and prior knowledge. Bayesian methods provide a more coherent framework for incorporating prior beliefs and updating them with data, while classical methods often have simpler interpretations and well-established frequentist properties.

The references for our discussion are [BT] Introduction to Probability (2nd edition) by Bertsekas and Tsitsiklis (Chapters 8-9) and [DS] Probability and Statistics (4th edition) by Morris Degroot and Mark Schervish (Chapters 7-12)

Thank you for providing the references for our discussion. "Introduction to Probability" by Bertsekas and Tsitsiklis ([BT]) and "Probability and Statistics" by Degroot and Schervish ([DS]) are widely recognized textbooks in the field of probability and statistics. The specified chapters (Chapters 8-9 in [BT] and Chapters 7-12 in [DS]) cover topics related to the Bayesian and classical approaches to parameter estimation, confidence intervals, hypothesis testing, and reasoning about (in)dependence.

For a more detailed understanding of these topics, I recommend referring to the specific chapters mentioned in the books. These textbooks provide comprehensive explanations, examples, and exercises to enhance your knowledge and application of statistical concepts. They can serve as valuable resources for studying probability theory and statistical inference.

Please note that as an AI language model, I don't have direct access to the content of these books. However, I can certainly help answer questions and provide explanations based on my training and general knowledge in the field of statistics. If you have any specific questions or need further clarification on any of the topics, feel free to ask!

tell me more on Hypothesis Testing

ENGG2780 is a course offered at The Chinese University of Hong Kong (CUHK). While I don't have access to the specific content and materials of the course, I can provide a general overview of hypothesis testing, which is a common topic covered in statistics courses.

Hypothesis testing is a statistical procedure used to make inferences or draw conclusions about a population based on sample data. It involves formulating a null hypothesis and an alternative hypothesis and then performing statistical tests to determine the strength of evidence against the null hypothesis.

The general process of hypothesis testing involves the following steps:

1. Formulating the null and alternative hypotheses:

? ?- The null hypothesis (denoted as H0) represents the claim or assumption to be tested.

? ?- The alternative hypothesis (denoted as Ha or H1) represents the alternative claim or what is being hypothesized.

2. Choosing the appropriate test statistic:

? ?- The choice of test statistic depends on the nature of the data and the research question. Common test statistics include the t-statistic, z-statistic, chi-square statistic, or F-statistic.

3. Selecting the significance level:

? ?- The significance level (denoted as α) is the probability of rejecting the null hypothesis when it is actually true. It represents the threshold for making a decision.

? ?- Commonly used significance levels include α = 0.05 (5%) or α = 0.01 (1%).

4. Computing the test statistic and obtaining the p-value:

? ?- The test statistic is calculated based on the sample data and the chosen test.

? ?- The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

5. Making a decision:

? ?- If the p-value is smaller than the significance level, the null hypothesis is rejected in favor of the alternative hypothesis.

? ?- If the p-value is larger than the significance level, there is insufficient evidence to reject the null hypothesis.

It's important to note that hypothesis testing does not prove the truth or falsehood of the null hypothesis, but rather provides evidence for or against it based on the available data.

In ENGG2780, the course likely covers various aspects of hypothesis testing, such as one-sample tests, two-sample tests, tests for proportions, and possibly more advanced topics like analysis of variance (ANOVA) or chi-square tests. The lectures and accompanying materials would provide detailed explanations, examples, and exercises to reinforce understanding and application of hypothesis testing concepts.

For a more comprehensive understanding of the specific content covered in ENGG2780, I recommend referring to the course materials, lecture notes, and textbooks provided by the instructor. They will provide the most accurate and detailed information related to the course curriculum.

Certainly! Although I don't have access to the specific exercises that appeared in past papers for ENGG2780, I can provide some general advice for hypothesis testing that may help you tackle similar exercises. Here are some key tips:

1. Understand the problem and identify the hypotheses: Clearly understand the research question or problem being addressed. Identify the null hypothesis (H0) and alternative hypothesis (Ha) based on the question or claim being tested.

2. Choose an appropriate test: Determine the appropriate statistical test based on the nature of the data and the hypotheses being tested. Common tests include t-tests, chi-square tests, ANOVA, or proportion tests. Ensure that the assumptions of the chosen test are met.

3. Set the significance level: Select the significance level (α) that represents the threshold for rejecting the null hypothesis. Commonly used values are α = 0.05 or α = 0.01, but it can vary depending on the context and the level of significance desired.

4. Collect and analyze the data: Collect the relevant sample data and calculate the test statistic based on the chosen test. Compute any necessary summary statistics (e.g., mean, standard deviation, proportion) required for the test.

5. Calculate the p-value: Calculate the p-value, which represents the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. The p-value is compared to the significance level to make a decision.

6. Make a decision: Compare the p-value to the significance level. If the p-value is smaller than the significance level, reject the null hypothesis in favor of the alternative hypothesis. If the p-value is greater than the significance level, do not reject the null hypothesis.

7. Interpret the results: Clearly state the conclusion in the context of the problem. Describe what the results mean and any implications they may have. Use appropriate statistical language and avoid generalizations beyond what the data and analysis support.

Remember, practice is crucial for mastering hypothesis testing. Work through a variety of exercises, including those from past papers, to gain familiarity with different scenarios and to develop problem-solving skills. Additionally, consult relevant textbooks and resources recommended by your course instructor to deepen your understanding of hypothesis testing concepts and techniques.

Lastly, seek clarification from your course instructor or teaching assistants if you encounter any difficulties or have specific questions related to the exercises or concepts covered in ENGG2780. They will be the best resources to provide guidance and support tailored to the course material.

Based on the given example problem, here's a summary of the solution and the corresponding advice:

(a) The likelihood ratio test approach is used with a significance level (α) of 1/4. The likelihood ratio, fH1(x)/fH0(x), is determined to be 4e^(-x^2/2) for -2 < x < 2 and infinite otherwise. If |x| ≥ 2, the null hypothesis (H0) is always rejected. For |x| < 2, a positive value of ξ is sought such that if √(4/(2π))e^(-x^2/2) > ξ, the null hypothesis is rejected; otherwise, it is accepted. By rearranging the inequality, a region of x is identified: |x| < γ. Setting the false rejection probability P(|x| < γ; H0) = 1/4, it is found that γ = 1/2. Therefore, the decision rule is to reject H0 if |x| ≥ 2 or |x| < 1/2; otherwise, H0 is accepted.

Advice: In this example, the likelihood ratio test approach is employed, and the decision rule is constructed based on the observed sample data and the likelihood ratio. The advice for hypothesis testing based on this example includes:

1. Understand the likelihood ratio test approach and its role in hypothesis testing.

2. Familiarize yourself with the null and alternative hypotheses and their corresponding likelihood functions.

3. Calculate the likelihood ratio and identify the critical region based on the desired significance level (α).

4. Determine the decision rule by setting the false rejection probability and solving for the critical value or region.

5. Clearly state the decision rule and the conclusions drawn based on the observed data.

(b) The false acceptance error, or Type II error, refers to accepting the null hypothesis (H0) when the alternative hypothesis (Ha) is actually true. In this case, a false acceptance occurs when 1/2 ≤ |Z| < 2 for a standard normal random variable Z. The probability of this event is computed as P(1/2 ≤ |Z| < 2) ≈ 0.5716.

Advice: Understanding and calculating the false acceptance error is important to evaluate the performance of a hypothesis test. To calculate the false acceptance error:

1. Identify the critical region for the alternative hypothesis.

2. Compute the probability of the observed data falling within this critical region assuming the null hypothesis is true.

3. Interpret the obtained probability as the false acceptance error.

It's important to note that the advice provided is specific to the given example problem. For a more comprehensive understanding of hypothesis testing and tailored advice for your course ENGG2780 at CUHK, referring to the course materials, lecture notes, and textbooks provided by your instructor will be the most reliable and accurate source of information.

The abbreviation "s.t." most likely stands for "such that." In mathematical notation, "s.t." is often used to denote a condition or constraint that must be satisfied in a particular context or equation. It indicates that the following statement or equation is subject to the condition specified before "s.t."

Based on the given example problem, here's a summary of the solution and the corresponding advice:

(a) The paired t-test is more appropriate in this scenario because the data collected from the typists' work is paired, meaning there is a natural pairing between the morning and afternoon shifts for each typist. The assumption is that the performance of different typists is independent, but we cannot assume that the number of typos is independent of the typist. For instance, if a typist made many errors in the morning, we would expect them to make many errors in the afternoon as well.

Advice: When analyzing paired data, such as the number of typos before and after a shift, the paired t-test should be used. Ensure that you understand the concept of paired data and the assumptions associated with this type of analysis. Justify your choice of test based on the nature of the data and the research question at hand.

(b) To calculate the p-value for the chosen paired t-test, compute the differences between the number of afternoon and morning typos for each typist. Then, calculate the sample mean and adjusted sample variance of these differences. In this example, the sample mean is Z = 4/3 and the adjusted sample variance is S^2 = 9/3. Under the null hypothesis, the test statistic T = Z / (S / √3) follows a t-distribution with 2 degrees of freedom. Since this is a two-sided test (testing for a change in the number of typos), the p-value is calculated as P(|T| ≥ (4/3) / √(9/3)/3) ≈ P(|t(2)| ≥ 0.756) ≈ 0.529.

Advice: When calculating the p-value for a hypothesis test, follow the appropriate formula for the chosen test and the given data. Pay attention to the degrees of freedom and the directionality (one-sided or two-sided) of the test to calculate the correct p-value. Make sure to interpret the p-value correctly in the context of the research question and the chosen significance level.

As always, remember that the advice provided is specific to the given example problem. For a more comprehensive understanding of hypothesis testing and tailored advice for your course ENGG2780 at CUHK, refer to the course materials and resources provided by your instructor.

Based on the given example, here are some advice for hypothesis testing:

1. Understand the problem statement and clearly state the null and alternative hypotheses.

2. Choose the appropriate test based on the type of data and assumptions. For example, use the paired t-test when data is dependent and independent sample t-test when data is independent.

3. Calculate the test statistic and determine the p-value.

4. Choose the significance level (alpha) for the test, which represents the probability of rejecting the null hypothesis when it is true. A common choice is 0.05.

5. Compare the p-value to alpha. If p-value is less than alpha, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

6. Interpret the results in the context of the problem and draw conclusions.

Based on the given example, here are some additional advice for hypothesis testing:

(a) Justification for the choice of test:

The paired t-test is more appropriate in this scenario because the data collected from the typists' work is paired, meaning there is a natural pairing between the morning and afternoon shifts for each typist. The assumption is that the performance of different typists is independent, but we cannot assume that the number of typos is independent of the typist. For instance, if a typist made many errors in the morning, we would expect them to make many errors in the afternoon as well.

Advice: When analyzing paired data, such as the number of typos before and after a shift, the paired t-test should be used. Ensure that you understand the concept of paired data and the assumptions associated with this type of analysis. Justify your choice of test based on the nature of the data and the research question at hand.

(b) Calculation of the p-value:

To calculate the p-value for the chosen paired t-test, compute the differences between the number of afternoon and morning typos for each typist. Then, calculate the sample mean and adjusted sample variance of these differences. In this example, the sample mean is Z = 4/3 and the adjusted sample variance is S^2 = 9/3. Under the null hypothesis, the test statistic T = Z / (S / √3) follows a t-distribution with 2 degrees of freedom. Since this is a two-sided test (testing for a change in the number of typos), the p-value is calculated as P(|T| ≥ (4/3) / √(9/3)/3) ≈ P(|t(2)| ≥ 0.756) ≈ 0.529.

Advice: When calculating the p-value for a hypothesis test, follow the appropriate formula for the chosen test and the given data. Pay attention to the degrees of freedom and the directionality (one-sided or two-sided) of the test to calculate the correct p-value. Make sure to interpret the p-value correctly in the context of the research question and the chosen significance level.

Remember to consult your course materials, instructor, or textbook for further guidance and practice with hypothesis testing.

Yes, that is correct. In Bayesian inference, the prior distribution represents our beliefs or knowledge about the parameter before observing the data, and the posterior distribution represents our updated beliefs after taking the data into account.

In this case, the prior distribution is a Beta distribution with parameters ?? and ??. The Beta distribution is a suitable choice for modeling the prior belief about a probability parameter because it is defined on the interval (0, 1), which is the range of possible values for ??. The Beta distribution has a shape parameter ?? and a scale parameter ??, which control the shape and spread of the distribution.

Given a random sample ??_1, ..., ??_?? from a Bernoulli distribution with parameter ??, the likelihood function is the product of the individual probabilities, which is proportional to ??^(∑_(??=1)^?????_??) (1-??)^(???∑_(??=1)^?????_??).

The posterior distribution is obtained by multiplying the prior distribution and the likelihood function, which results in a Beta distribution with updated parameters. Specifically, the posterior distribution is "Beta"(??+∑_(??=1)^?????_?? ,??+???∑_(??=1)^?????_?? ). The parameter ??+∑_(??=1)^?????_?? represents the updated number of successes, and the parameter ??+???∑_(??=1)^?????_?? represents the updated number of failures.

This result is a fundamental property of the conjugate prior for the Bernoulli likelihood function, where the prior and posterior distributions belong to the same family of distributions (Beta distribution), making the updating process simple and tractable.

The posterior distribution provides a summary of our updated knowledge about the parameter ??, incorporating both prior beliefs and the information from the observed data.

I'm sorry, but that statement is not accurate. In Bayesian inference, when the prior distribution is a Gamma distribution and the likelihood function is a Poisson distribution, the posterior distribution is not a simple Gamma distribution.

Let's clarify the correct posterior distribution for this scenario. Given a random sample ??_1, ..., ??_?? from a Poisson distribution with mean ??, the likelihood function is the product of the individual probabilities, which is proportional to ??^(∑_(??=1)^?????_??) * e^(-n??).

If the prior distribution for ?? is a Gamma distribution with shape parameter ?? and rate parameter ??, denoted as "Gamma"(??, ??), then the posterior distribution is obtained by multiplying the prior distribution and the likelihood function.

The posterior distribution is proportional to ??^(∑_(??=1)^?????_??) * e^(-n??) * ??^(??-1) * e^(-????) = ??^(∑_(??=1)^?????_?? + ??-1) * e^(-(n+??)??).

Hence, the posterior distribution is a Gamma distribution with updated shape parameter ??+∑_(??=1)^?????_?? and rate parameter ??+n. The correct notation for the posterior distribution is "Gamma"(??+∑_(??=1)^?????_??, ??+n).

This result is a consequence of the conjugacy property between the Poisson likelihood and Gamma prior. The posterior distribution belongs to the same family of distributions as the prior distribution (Gamma distribution), facilitating the updating process and making it analytically tractable.

So, in summary, the correct statement is that if the prior distribution is a Gamma distribution "Gamma"(??, ??), and the likelihood function is a Poisson distribution, the posterior distribution is a Gamma distribution "Gamma"(??+∑_(??=1)^?????_??, ??+n).