Errata in Figure 10.2

[erdosxx]

In the 10.1.2, with minimizing KL(q||p), we can found that the unique solution
q(z) = q_1^{}(z_1) * q_2^{}(z_2) = N(z_1| \mu_1, \Lambda_11^{-1}) * N(z_2| \mu_2, \Lambda_22^{-1})
as author showed in Exercise 10.2.
We also have same solution as above by minimizing KL(p||q) in (10.17) and (2.98).
So, the solution q(z) is ,in general, not the spherical Gaussian and can not be looked similar
to Figure 10.2.
The correct contour plot of the solution looks like in the following figure in (a) with purpled colored lines.
https://drive.google.com/file/d/1cSnMA-_hheAmnCBLKrp551BcszI15vpq/view?usp=sharing

I think that Figure 10.2 is originated from Mackay, 2003(https://www.inference.org.uk/itprnn/book.pdf)
at page 436 and the related problem is exercise 33.5 (p.434) and in this problem, the solution is
restricted to spherical Gaussian.

If my understanding is correct, Figure 10.2 is not relevant to the contexts in 10.1.2.

Please check my opinion and let me know your comments.

Thank you.

[yousuketakada]

First, please note that Figure 10.2 compares variational Bayes vs. expectation propagation (moment matching) or minimizing KL(q||p) vs. KL(p||q), and that the solutions have different properties as shown in the figure (see also Figure 10.3).
For either case, we do not need any assumption other than that q(Z) factorizes as (10.5) to obtain the spherical distributions as described in the text because the true distribution p(Z) has an elliptical distribution rotated 45 degrees so that the diagonal elements Lambda_11 and Lambda_22 of the precision matrix (and thus those of the covariance matrix) are equal.
I would suggest you run both the algorithms for some specific example by yourself to understand why.

[erdosxx]

I can understand your point and you are right. Thank you for your good comments. :-)

[erdosxx]

Hi Yousuke
I appreciate your good comments and excellent errata.
For this problem, I want to check one more thing with you.

As you commented, I can understand \Lambda_{11} = \Lambda_{22} and the solution has
spherical Gaussian.
However, I think that as author showed, both solutions based on (10.9) and KL(p||q) are same as following.
https://drive.google.com/file/d/1OKM4T6SL1-3zPfqnXcy3TSEXbrup4T6r/view?usp=sharing
So, I think that only one figure in Figure 10.2 is relevant to this solution and cannot have
two different cases.
Is my understanding correct?

Thank you.