Added definitions of unimodal and multimodal.

02-Discrete-Bayes.ipynb

@@ -392,7 +392,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "This distribution is called a **categorical distribution**, which is the term used to describe a discrete distribution describing the probability of observing $n$ outcomes. It is a **multimodal distribution** because we have multiple beliefs about the position of our dog. Of course we are not saying that we think he is simultaneously in three different locations, merely that so far we have narrowed down our knowledge in his position to be one of these three locations. My (Bayesian) belief is that there is a 33.3% chance of being at door 0, 33.3% at door 1, and a 33.3% chance of being at door 8. More specifically, \n",
+    "This distribution is called a **categorical distribution**, which is the term used to describe a discrete distribution describing the probability of observing $n$ outcomes. It is a **multimodal distribution** because we have multiple beliefs about the position of our dog. Of course we are not saying that we think he is simultaneously in three different locations, merely that so far we have narrowed down our knowledge in his position to be one of these three locations. My (Bayesian) belief is that there is a 33.3% chance of being at door 0, 33.3% at door 1, and a 33.3% chance of being at door 8.\n",
+    "\n",
+    "A few words about the *mode* of a distribution. This terms from elementary statistics. Given a set of numbers, such as {1, 2, 2, 2, 3, 3, 4}, the *mode* is the number that occurs most often. For this set the mode is 2. A set can contain more than one mode. The set {1, 2, 2, 2, 3, 3, 4, 4, 4} contains the modes 2 and 4, because both occur three times. We say the former set is *unimodal*, and the latter is *multimodal*.\n",
     "\n",
     "I hand coded the `belief` array in the code above. How would we implement this in code? Well, hallway represents each door as a 1, and wall as 0, so we will multiply the hallway variable by the percentage, like so;"
    ]

README.md

@@ -9,6 +9,16 @@ Introductory text for Kalman and Bayesian filters. All code is written in Python
 
 ![alt tag](https://raw.githubusercontent.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/animations/05_dog_track.gif)
 
+What are Kalman and Bayesian Filters?
+-----
+
+We measure things in the world with sensors. For example, thermometers measure temperature, anemometers measure wind speed, scales measure weights. We measure distances with things like sonar, lasers, and calipers. 
+
+Unfortunately, no sensor is perfect. For example, a thermometer might only be accurate to within 2 degrees. A laser rangefinder might be accurate to a meter. A bathroom scale might be accurate to 1 kilogram, whereas a kitchen scale might be accurate to a gram or less. 
+
+Furthermore, a sensor might not present all of the information we would like. Suppose we want to know the map coordinates of a structure that we are surveying. A laser rangefinder tells us the distance to an object, but it does not tell us where in the world that object is. The measurement is *relative* to the position of the rangefinder. If the rangefinder does not provide an angle (bearing) to the measured object we can only state that the object lies somewhere within a donut shaped circle around the rangefinder. If we do know the location of the rangefinder and the angle then we can compute the coordinates of the structure. But how do we know the rangefinder's position? With sensors. How do we measure the angle? With sensors. And sensors are inaccurate, so both of those measurements have a degree of error in them.
+
+Is there a way to reduce the amount of error in the measurements? You may have some ideas already. What if you were to take ten measurements and average them? For many kinds of sensor errors this does improve our estimate. You might imagine 
 
 Motivation
 -----