Update made to Markov Chains: Basic Concepts lecture

lectures/markov_chains_I.md

@@ -4,7 +4,7 @@ jupytext:
     extension: .md
     format_name: myst
     format_version: 0.13
-    jupytext_version: 1.14.4
+    jupytext_version: 1.16.1
 kernelspec:
   display_name: Python 3 (ipykernel)
   language: python
@@ -76,7 +76,7 @@ nonnegative $n$-vector $p$ that sums to one.
 For example, $p = (0.2, 0.2, 0.6)$ is a probability mass function over $3$ outcomes.
 
 A **stochastic matrix** (or **Markov matrix**)  is an $n \times n$ square matrix $P$
-such that each row of $P$ is a probability mass function over $n$ outcomes.
+such that each row of $P$ is a probability mass function.
 
 In other words,
 
@@ -96,7 +96,7 @@ Before defining a Markov chain rigorously, we'll  give some examples.
 
 
 (mc_eg2)=
-#### Example 1
+#### Example 1: Economic states
 
 From  US unemployment data, Hamilton {cite}`Hamilton2005` estimated the following dynamics.
 
@@ -172,7 +172,7 @@ In particular, $P(i,j)$ is the
 
 
 (mc_eg1)=
-#### Example 2
+#### Example 2: Unemployment
 
 Consider a worker who, at any given time $t$, is either unemployed (state 0)
 or employed (state 1).
@@ -220,7 +220,7 @@ Then we can address a range of questions, such as
 We'll cover some of these applications below.
 
 (mc_eg3)=
-#### Example 3
+#### Example 3: Political transition dynamics
 
 Imam and Temple {cite}`imampolitical` categorize political institutions into
 three types: democracy $\text{(D)}$, autocracy $\text{(A)}$, and an intermediate
@@ -231,17 +231,17 @@ Each institution can have two potential development regimes: collapse $\text{(C)
 Imam and Temple {cite}`imampolitical` estimate the following transition
 probabilities:
 
-
 $$
-P :=
-\begin{bmatrix}
-0.86 & 0.11 & 0.03 & 0.00 & 0.00 & 0.00 \\
-0.52 & 0.33 & 0.13 & 0.02 & 0.00 & 0.00 \\
-0.12 & 0.03 & 0.70 & 0.11 & 0.03 & 0.01 \\
-0.13 & 0.02 & 0.35 & 0.36 & 0.10 & 0.04 \\
-0.00 & 0.00 & 0.09 & 0.11 & 0.55 & 0.25 \\
-0.00 & 0.00 & 0.09 & 0.15 & 0.26 & 0.50
-\end{bmatrix}
+\begin{array}{c|cccccc}
+ & \text{DG} & \text{DC} & \text{NG} & \text{NC} & \text{AG} & \text{AC} \\
+\hline
+\text{DG} & 0.86 & 0.11 & 0.03 & 0.00 & 0.00 & 0.00 \\
+\text{DC} & 0.52 & 0.33 & 0.13 & 0.02 & 0.00 & 0.00 \\
+\text{NG} & 0.12 & 0.03 & 0.70 & 0.11 & 0.03 & 0.01 \\
+\text{NC} & 0.13 & 0.02 & 0.35 & 0.36 & 0.10 & 0.04 \\
+\text{AG} & 0.00 & 0.00 & 0.09 & 0.11 & 0.55 & 0.25 \\
+\text{AC} & 0.00 & 0.00 & 0.09 & 0.15 & 0.26 & 0.50 \\
+\end{array}
 $$
 
 ```{code-cell} ipython3
@@ -285,6 +285,20 @@ plt.colorbar(pc, ax=ax)
 plt.show()
 ```
 
+The probabilities can be represented in matrix form as follows
+
+$$
+P :=
+\begin{bmatrix}
+0.86 & 0.11 & 0.03 & 0.00 & 0.00 & 0.00 \\
+0.52 & 0.33 & 0.13 & 0.02 & 0.00 & 0.00 \\
+0.12 & 0.03 & 0.70 & 0.11 & 0.03 & 0.01 \\
+0.13 & 0.02 & 0.35 & 0.36 & 0.10 & 0.04 \\
+0.00 & 0.00 & 0.09 & 0.11 & 0.55 & 0.25 \\
+0.00 & 0.00 & 0.09 & 0.15 & 0.26 & 0.50
+\end{bmatrix}
+$$
+
 Looking at the data, we see that democracies tend to have longer-lasting growth
 regimes compared to autocracies (as indicated by the lower probability of
 transitioning from growth to growth in autocracies).
@@ -308,7 +322,7 @@ A **distribution** $\psi$ on $S$ is a probability mass function of length $n$, w
 A **Markov chain** $\{X_t\}$ on $S$ is a sequence of random variables taking values in $S$
 that have the **Markov property**.
 
-This means that, for any date $t$ and any state $y \in S$,
+This means that, for any time $t$ and any state $y \in S$,
 
 ```{math}
 :label: fin_markov_mp
@@ -331,7 +345,7 @@ P(x, y) := \mathbb P \{ X_{t+1} = y \,|\, X_t = x \}
 By construction,
 
 * $P(x, y)$ is the probability of going from $x$ to $y$ in one unit of time (one step)
-* $P(x, \cdot)$ is the conditional distribution of $X_{t+1}$ given $X_t = x$
+* $P(x, \cdot)$ is the conditional distribution(probability mass function) of $X_{t+1}$ given $X_t = x$
 
 We can view $P$ as a stochastic matrix where
 
@@ -437,7 +451,7 @@ Here's a short time series.
 mc_sample_path(P, ψ_0=(1.0, 0.0), ts_length=10)
 ```
 
-It can be shown that for a long series drawn from `P`, the fraction of the
+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.
 
 (We will explain why {ref}`later <ergodicity>`.)
@@ -610,7 +624,7 @@ $$
 ```{index} single: Markov Chains; Future Probabilities
 ```
 
-Recall the stochastic matrix $P$ for recession and growth {ref}`considered above <mc_eg2>`.
+Recall the stochastic matrix $P$ for recession and growth considered in {ref}`Example 1: Economic states <mc_eg2>`.
 
 Suppose that the current state is unknown --- perhaps statistics are available only at the *end* of the current month.
 
@@ -632,7 +646,11 @@ The distributions we have been studying can be viewed either
 1. as probabilities or
 1. as cross-sectional frequencies that the law of large numbers leads us to anticipate for large samples.
 
-To illustrate, recall our model of employment/unemployment dynamics for a given worker {ref}`discussed above <mc_eg1>`.
+ ```{note}
+A cross-sectional frequency measures how a particular variable (e.g., employment status) is distributed across a population at a specific time, providing information of the proportions of individuals in each possible state of that variable.
+```
+
+To illustrate, recall our model of employment/unemployment dynamics for a given worker discussed in {ref}`Example 2: Unemployment <mc_eg1>`.
 
 Consider a large population of workers, each of whose lifetime experience is
 described by the specified dynamics, with each worker's outcomes being
@@ -726,7 +744,7 @@ We will come back to this when we introduce irreducibility in the {doc}`next lec
 
 ### Example
 
-Recall our model of the employment/unemployment dynamics of a particular worker {ref}`discussed above <mc_eg1>`.
+Recall our model of the employment/unemployment dynamics of a particular worker discussed in {ref}`Example 2: Unemployment <mc_eg1>`.
 
 If $\alpha \in (0,1)$ and $\beta \in (0,1)$, then the transition matrix is everywhere positive.
 
@@ -840,7 +858,7 @@ HTML(anim.to_jshtml())
 
 Here
 
-* $P$ is the stochastic matrix for recession and growth {ref}`considered above <mc_eg2>`.
+* $P$ is the stochastic matrix for recession and growth considered in {ref}`Example 1: Economic states <mc_eg2>`.
 * The highest red dot is an arbitrarily chosen initial marginal probability distribution  $\psi_0$, represented as a vector in $\mathbb R^3$.
 * The other red dots are the marginal distributions $\psi_0 P^t$ for $t = 1, 2, \ldots$.
 * The black dot is $\psi^*$.