[markov_chains_I] Editorial Suggestions

[HumphreyYang]

Comments by @longye-tian.

Content

• [x] The first exercise -> The last exercise.
• [x] Remove unused variables and shorten the visualization code for the network graph.
• [x] unbold future in Defiinig Markov Chains section.
• [x] study a Markov chains -> study Markov chains.
• [x] Use tuples whenever possible.
• [x] Give some comments on the code in the qe library.
• [x] Add a link for marginal probability $\psi_t(x)$.
• [x] Unify postmultiply.
• [x] Add a sentence clarifying $P^m(x,y)$.
• [x] Law of Large Numbers -> law of large numbers.
• [x] Check the capitalization of subtitles.
• [x] for all $t$ -> for all $t \ge 0$.
• [x] $\psi_t(0)$ is employment at [time $t$].
• [x] Delete the word 'Theorem' in the Theorem box.
• [x] Update the solution for exercise 1.

  Code

• [x] Update visualization code for the section on the Hamilton matrix.
• [x] Animate convergence path?

[mbek0605]

Comments by @Jiarui-ZH

Content

• [x] In “A stochastic matrix (or Markov matrix) is an 𝑛×𝑛 square matrix 𝑃 such that each row of 𝑃 is a probability mass function over 𝑛 outcomes.” delete “over 𝑛 outcomes”.
• [x] Change titles “32.2.2.1. Example 1” and “32.2.2.2. Example 2” into titles that are intuitive of the respective examples.
• [x] Express the matrix below as a table?
• [x] For Example 3 (the example for democracy (D), autocracy (A), and an intermediate state called anocracy (N)), implement the following order: table; figure; matrix. [TABLE] [FIGURE] [MATRIX]
• [x] Change date to time in section of “32.2.3. Defining Markov chains”
• [x] Add “(probability mass function)” such that “𝑃(𝑥,⋅) is the conditional distribution (probability mass function) of 𝑋𝑡+1 given.”
• [x] In “It can be shown that for a long series drawn from P, the fraction of the sample that takes value 0 will be about 0.25.”, change to “It can be proven that for a long series drawn from P, the fraction of the sample that takes value 0 will be about 0.25.”
• [x] In the section “32.4. Distributions over time”, have subsections dividing into two parts: “One unit of time” and “m steps in the future”
• [x] In 32.4.1, move the function above (after “m-th power of 𝑃”). After the function, add text “For one way to see it, consider again (32.6), but now with a 𝜓𝑡 that puts all probability on state 𝑥.
• [x] Then 𝜓𝑡 is a vector with 1 in position 𝑥 and zero elsewhere.
• [x] Inserting this into (32.6), we see that, conditional on 𝑋𝑡=𝑥, the distribution of 𝑋𝑡+𝑚 is the 𝑥-th row of 𝑃𝑚.”
• [x] In 32.4.2, instead of the link “considered above”, include the section number (32.4.1) with a link.
• [ ] In 32.4.3, in (2), put a note to better explain “cross-sectional frequencies”. Maybe jump to the example rather than include the text after the two-numbered list?
• [x] Search for “considered above” and replace it with section number and make sure the hyperlink works. (In pdf version, the “considered above” link does not work.)
• [ ] Simplify the figures below:

[mmcky]

thanks @Jiarui-ZH. Would you be able to champion this issue to close the last couple of items?