added motivation for pi over histogram (#147)

Author: cwehmeyer

notebooks/04-msm-analysis.ipynb

@@ -179,11 +179,18 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The attribute `msm.pi` tells us, for each discrete state, the absolute probability of observing said state in global equilibrium. Mathematically speaking, the stationary distribution $\\pi$ is the left eigenvector of the transition matrix $\\mathbf{T}$ to the eigenvalue $1$:\n",
+    "The attribute `msm.pi` tells us, for each discrete state, the absolute probability of observing said state in global equilibrium.\n",
+    "Mathematically speaking, the stationary distribution $\\pi$ is the left eigenvector of the transition matrix $\\mathbf{T}$ to the eigenvalue $1$:\n",
     "\n",
     "$$\\pi^\\top \\mathbf{T} = \\pi^\\top.$$\n",
     "\n",
-    "We can use the stationary distribution to, e.g., visualize the weight of the dicrete states and, thus, to highlight which areas of our feature space are most probable. Here, we show all data points in a two dimensional scatter plot and color/weight them according to their discrete state membership:"
+    "Please note that the $\\pi$ is fundamentaly different from a normalized histogram of states:\n",
+    "for the histogram of states to accurately describe the stationary distribution, the data needs to be sampled from global equilibrium, i.e, the data points need to be statistically independent.\n",
+    "The MSM approach, on the other hand, only requires local equilibrium, i.e., statistical independence of state transitions.\n",
+    "Thus, the MSM approach requires a much weaker and, in practice, much easier to satisfy condition than simply counting state visits.\n",
+    "\n",
+    "We can use the stationary distribution to, e.g., visualize the weight of the dicrete states and, thus, to highlight which areas of our feature space are most probable.\n",
+    "Here, we show all data points in a two dimensional scatter plot and color/weight them according to their discrete state membership:"
    ]
   },
   {