Pre-read (1):
Pre-read (2):
Pre-read (3):
Ex 1.9.A:
Show that the mode (i.e. the maximum) of the Gaussian distribution (1.46) is given by µ.
Ex 1.9.B:
Similarly, show that the mode of the multivariate Gaussian (1.52) is given by µ.
Solution hint from the book:
Solution by ChatGPT
The mode of a multivariate Gaussian distribution is the point at which the probability density function (PDF) is
maximized. For a multivariate Gaussian distribution with mean vector μ and covariance matrix Σ, the PDF is given
by:
f(x;μ,Σ)=(2π)n/2det(Σ)1/21exp(−21(x−μ)TΣ−1(x−μ))
where:
- x is the vector of random
variables.
- μ is
the mean vector.
- Σ is the
covariance matrix.
- n is the dimensionality of
the multivariate distribution.
- det(Σ) is the determinant of the covariance
matrix.
To find the mode, we need to maximize this PDF with respect to x. Taking the natural logarithm (log)
of the PDF simplifies the maximization problem, as the logarithm is a monotonic function:
lnf(x;μ,Σ)=−2nln(2π)−21ln(det(Σ))−21(x−μ)TΣ−1(x−μ)
To find the mode, we need to find the point x at which the derivative of the log
PDF with respect to x is zero. This can be expressed as:
∂x∂lnf(x;μ,Σ)=0
Solving this equation will give us the mode x. Taking the derivative, we get:
−Σ−1(x−μ)=0
Solving for x, we find:
x=μ
Therefore, the mode of the multivariate Gaussian distribution is given by the mean vector μ.
Solution by Bard
Here's a detailed proof demonstrating that the mode of a multivariate Gaussian
distribution is indeed its mean:
1. Probability Density Function (PDF) of Multivariate Gaussian:
2. Finding the Mode:
- The mode is the point where the PDF has its maximum value.
- To find the mode, we need to find the maximum of f(x) with respect to x.
3. Taking the Gradient:
4. Setting the Gradient to Zero:
5. Solving for x:
6. Mode x = μ:
- This leads to the solution x = μ.
7. Second Derivative Test (Hessian):
- To confirm that this is a maximum, we can examine the Hessian matrix, which is
the matrix of second partial derivatives.
- The Hessian of f(x) is -Σ^(-1), which is negative definite (all eigenvalues are
negative).
- This guarantees that f(x) has a maximum at x = μ.
Therefore, we have proven that the mode of the multivariate Gaussian
distribution is equal to its mean vector μ.