Question

Conditional on \(M=m, Y_{1}, \ldots, Y_{n}\) is a random sample from the \(N\left(m, \sigma^{2}\right)\) distribution. Find the unconditional joint distribution of \(Y_{1}, \ldots, Y_{n}\) when \(M\) has the \(N\left(\mu, \tau^{2}\right)\) distribution. Use induction to show that the covariance matrix \(\Omega\) has determinant \(\sigma^{2 n-2}\left(\sigma^{2}+n \tau^{2}\right)\), and show that \(\Omega^{-1}\) has diagonal elements \(\left\\{\sigma^{2}+(n-1) \tau^{2}\right) /\left\\{\sigma^{2}\left(\sigma^{2}+n \tau^{2}\right)\right\\}\) and offdiagonal elements \(-\tau^{2} /\left\\{\sigma^{2}\left(\sigma^{2}+n \tau^{2}\right)\right\\}\)

Short answer

The unconditional joint distribution is multivariate normal with covariance matrix \( \Omega = \sigma^2 I_n + \tau^2 \mathbf{1}\mathbf{1}^T \). The determinant of \( \Omega \) is \( \sigma^{2n-2}( \sigma^2 + n \tau^2) \), and \( \Omega^{-1} \) has described diagonal and off-diagonal elements.

Step-by-step solution

Understand the Conditional Distribution

Given that conditional on \( M = m \), the variables \( Y_1, Y_2, \ldots, Y_n \) are independent and identically distributed as \( N(m, \sigma^2) \), this implies that individually, \( Y_i | M = m \sim N(m, \sigma^2) \) for each \( i \).

Define the Joint Distribution Conditioning on M

Conditional on \( M = m \), the joint distribution of \( Y_1, Y_2, \ldots, Y_n \) is a multivariate normal distribution. This can be expressed as \( (Y_1, Y_2, \ldots, Y_n | M = m) \sim N(m \mathbf{1}, \sigma^2 I_n) \) where \( \mathbf{1} \) is a vector of ones and \( I_n \) is the \( n \times n \) identity matrix.

Apply the Law of Total Probability

To find the unconditional distribution, integrate out \( M \) using the law of total probability: \[ P(Y_1, \ldots, Y_n) = \int P(Y_1, \ldots, Y_n | M = m) P(M = m) \, dm \]Since \( M \sim N(\mu, \tau^2) \), plug in the gaussian PDFs to get the joint distribution.

Find the Covariance Matrix Ω

The covariance matrix \( \Omega \) of the unconditional joint distribution can be derived by considering both variances from the \( M \) and \( Y_i | M \) separately. This results in: \[ \Omega = \sigma^2 I_n + \tau^2 \mathbf{1} \mathbf{1}^T \] where \( I_n \) is the identity matrix and \( \mathbf{1} \mathbf{1}^T \) is a matrix of all 1s.

Calculate the Determinant of Ou00eb

The determinant of the covariance matrix \( \Omega \) can be calculated by treating it as a sum of an identity matrix and a rank-1 update. Using the matrix determinant lemma:\[ \det(\Omega) = (\sigma^2)^{n-1}(\sigma^2 + n\tau^2) \]By seeing it as an application of the identity plus outer product matrix.

Deriving the Inverse of Ω

Using the Woodbury matrix identity, we derive the inverse of \( \Omega \). The key points are:- Diagonal Elements: \( \frac{\sigma^2 + (n-1) \tau^2}{\sigma^2(\sigma^2 + n\tau^2)} \)- Off-diagonal Elements: \(-\frac{\tau^2}{\sigma^2(\sigma^2 + n\tau^2)} \)This form ensures the product \( \Omega \Omega^{-1} \) is the identity.

Prove by Induction

To confirm the determinant calculation, apply induction on \( n \). Base case \( n=1 \) is trivially \( \sigma^2 + \tau^2 \). For \( n \to n+1 \), assume the formula holds and show that adding an additional variable maintains the determinant structure due to the matrix operations involved.

Key concepts

Covariance Matrix

One of the fundamental concepts in understanding the multivariate normal distribution is the covariance matrix, denoted as \( \Omega \) in our case. A covariance matrix not only represents the variance of multiple variables but also the covariance between each pair. For a set of variables \( Y_1, Y_2, \ldots, Y_n \), the covariance matrix provides a comprehensive look at how each pair of these variables co-vary.
Understanding \( \Omega \) involves considering both the individual variance \( \sigma^2 \) from the distribution of each variable \( Y_i \) given \( M \), and the variance \( \tau^2 \) from the distribution of \( M \) itself. When combined using matrix operations, this results in a structure:\[ \Omega = \sigma^2 I_n + \tau^2 \mathbf{1} \mathbf{1}^T \]
Here, \( \sigma^2 I_n \) represents the diagonal matrix with \( \sigma^2 \) on its diagonal, indicating each variable \( Y_i \)'s variance independently, while \( \tau^2 \mathbf{1} \mathbf{1}^T \) captures the shared variance across the variables due to \( M \).
This matrix is crucial for wrapping up how multivariate distributions differ from simply stacking single-variable distributions together.

Determinant

In understanding the multivariate normal distribution, the determinant of the covariance matrix \( \Omega \) provides key insights into the dataset's spread and volume. The determinant essentially tells us about the 'size' of the distribution formed by \( Y_1, Y_2, \ldots, Y_n \).
With the matrix \( \Omega = \sigma^2 I_n + \tau^2 \mathbf{1} \mathbf{1}^T \), its determinant is calculated using a special lemma for matrix determinants, known as the matrix determinant lemma. This yields:
\[ \det(\Omega) = (\sigma^2)^{n-1}(\sigma^2 + n\tau^2) \]
This result demonstrates the cumulative effect of the variances and covariances on the overall structure of the data's distribution, allowing us to analyze the balance between individual components' variability and their collective consistency. This is foundational in quantifying the uncertainty a dataset contains.

Law of Total Probability

The law of total probability is a cornerstone concept when working with conditional probabilities in statistics. It allows us to integrate out a variable and find an unconditional distribution based on conditional structures.
In our exercise, we use the law of total probability to find the unconditional joint distribution of \( Y_1, \ldots, Y_n \). This is accomplished by considering all possible values of \( M \), weighting each with its probability, and integrating across these weighted possibilities. Mathematically, this is expressed as:
\[ P(Y_1, \ldots, Y_n) = \int P(Y_1, \ldots, Y_n | M = m) P(M = m) \, dm \]
Filling in with Gaussian functions, we achieve a seamless blend of conditional and unconditional scenarios, especially when \( M \) has a normal distribution. This process is crucial in statistical methodology as it establishes a global understanding of how data behaves beyond the conditions previously set.

Induction in Statistics

Induction is a powerful technique in mathematics and statistics for proving statements about an infinite sequence of cases. In this context, it is used to verify the formula for the determinant of the covariance matrix \( \Omega \).
To apply induction, we first verify the base case when \( n=1 \) to see that the expression trivially holds as \( \sigma^2 + \tau^2 \). Next, in the induction step, we assume our formula to be true for some \( n \) and demonstrate it holds for \( n+1 \).
This step involves matrix operations, showing that extending \( \) by one also extends the matrix dimensions and hence elucidates how the determinant structure remains constant under incremental increase. Each step relies on understanding how covariance matrices scale with the addition of data points, giving us confidence in the construction of complex statistical models.