Proofreading changes.

A kind reader suggested these corrections.

02_Discrete_Bayes/discrete_bayes.ipynb

@@ -275,7 +275,7 @@
       "\n",
       "That approach requires a fairly sophisticated understanding of several fields of mathematics, and it still leaves much of the work of understanding and forming an intuitive grasp of the situation in the hands of the reader.\n",
       "\n",
-      "I will use a different way to develop the topic, to which I owe the work of Dieter Fox and Sebastian Thrun a great debt. It depends on building an intuition on how Bayesian statistics work by tracking an object through a hallwya - they use a robot, I use a dog (I like dogs). The first published example of this that I can find weems to be Fox 1999 [1], with a fuller example in Fox [2] 2003. Sebastian Thrun also uses this formulation in his excellent Coursera course Artificial Intelligence for Robotics [3] In fact, if you like watching videos, I highly recommend pausing reading this book in favor of first few lessons of that course, and then come back to this book for a deeper dive into the topic.\n",
+      "I will use a different way to develop the topic, to which I owe the work of Dieter Fox and Sebastian Thrun a great debt. It depends on building an intuition on how Bayesian statistics work by tracking an object through a hallway - they use a robot, I use a dog (I like dogs). The first published example of this that I can find weems to be Fox 1999 [1], with a fuller example in Fox [2] 2003. Sebastian Thrun also uses this formulation in his excellent Coursera course Artificial Intelligence for Robotics [3] In fact, if you like watching videos, I highly recommend pausing reading this book in favor of first few lessons of that course, and then come back to this book for a deeper dive into the topic.\n",
       "\n",
       "Let's now use a simple thought experiment, much like we did with the g-h filter, to see how we might reason about the use of probabilities for filtering and tracking."
      ]
@@ -338,7 +338,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "So I start listening to Simon's transmissions on the network, and the first data I get from the sesnor is \"door\". From this I conclude that he is in front of a door, but which one? I have no idea. I have no reason to believe is in front of the first, second, or third door. But what I can do is assign a probability to each door. All doors are equally likely, so I assign a probability of 1/3 to each door. "
+      "So I start listening to Simon's transmissions on the network, and the first data I get from the sensor is \"door\". From this I conclude that he is in front of a door, but which one? I have no idea. I have no reason to believe is in front of the first, second, or third door. But what I can do is assign a probability to each door. All doors are equally likely, so I assign a probability of 1/3 to each door. "
      ]
     },
     {
@@ -589,13 +589,13 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Recall how quickly we were able to find an exact solution to our dog's position when we incorporated a series of measurements and movement updates. However, that occured in a fictional world of perfect sensors. Might we be able to find an exact solution even in the presense of noisy sensors?\n",
+      "Recall how quickly we were able to find an exact solution to our dog's position when we incorporated a series of measurements and movement updates. However, that occured in a fictional world of perfect sensors. Might we be able to find an exact solution even in the presence of noisy sensors?\n",
       "\n",
       "Unfortunately, the answer is no. Even if the sensor readings perfectly match an extremely complicated hallway map we could not say that we are 100% sure that the dog is in a specific position - there is, after all, the possibility that every sensor reading was wrong! Naturally, in a more typical situation most sensor readings will be correct, and we might be close to 100% sure of our answer, but never 100% sure. This may seem head-spinningly complicated, but lets just go ahead and program the math, which as we have seen is quite simple.\n",
       "\n",
       "First let's deal with the simple case - assume the movement sensor is perfect, and it reports that the dog has moved one space to the right. How would we alter our pos array?\n",
       "\n",
-      "I hope after a moment's thought it is clear that we should just shift all the values one space to the right. If we previously thought there was  a 50% chance of simon being at position 3, then after the move to the right we should believe that there is a 50% chance he is at position 4. So let's implement that. Recall that the hallway is circular, so we will use modulo arithmetic to perform the shift correctly"
+      "I hope after a moment's thought it is clear that we should just shift all the values one space to the right. If we previously thought there was  a 50% chance of Simon being at position 3, then after the move to the right we should believe that there is a 50% chance he is at position 4. So let's implement that. Recall that the hallway is circular, so we will use modulo arithmetic to perform the shift correctly"
      ]
     },
     {
@@ -669,7 +669,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "We want to solve real world problems, and we have already stated that all sensors have noise. Therefore the code above must be wrong. What if the sensor reported that our dog moved one space, but he actually moved two spaces, or zero? Once again this may initially sound like an insummountable problem, but let's just model it in math. Since this is just an example, we will create a pretty simple noise model for the sensor - later in the book we will handle far more sophisticated errors.\n",
+      "We want to solve real world problems, and we have already stated that all sensors have noise. Therefore the code above must be wrong. What if the sensor reported that our dog moved one space, but he actually moved two spaces, or zero? Once again this may initially sound like an insurmountable problem, but let's just model it in math. Since this is just an example, we will create a pretty simple noise model for the sensor - later in the book we will handle far more sophisticated errors.\n",
       "\n",
       "We will say that when the sensor sends a movement update, it is 80% likely to be right, and it is 10% likely to overshoot one position to the right, and 10% likely to undershoot to the left. That is, if we say the movement was 4 (meaning 4 spaces to the right), the dog is 80% likely to have moved 4 spaces to the right, 10% to have moved 3 spaces, and 10% to have moved 5 spaces.\n",
       "\n",
@@ -759,7 +759,7 @@
      "source": [
       "Here the results are more complicated, but you should still be able to work it out in your head. The 0.04 is due to the possibility that the 0.4 belief undershot by 1. The 0.38 is due to the following: the 80% chance that we moved 2 positions $(.4 * .8)$ and the 10% chance that we undershot $(.6*.1)$. Overshooting plays no role here because if we overshot both .4 and .6 would be past this position. **I strongly suggest working some examples until all of this is very clear, as so much of what follows depends on understanding this step.**\n",
       "\n",
-      "If you look at the probabilities after performing the update you probably feel dismay. In the example above we started with probabilitys of .4 and .6 in two fields; after performing the update the probabilities are not only lowered, but they are strewn out across the map."
+      "If you look at the probabilities after performing the update you probably feel dismay. In the example above we started with probabilities of .4 and .6 in two fields; after performing the update the probabilities are not only lowered, but they are strewn out across the map."
      ]
     },
     {
@@ -825,7 +825,7 @@
      "metadata": {},
      "source": [
       "\n",
-      "After 500 iterations we have lost all information, even though we were 100% sure that we started in position 1. Feel free to play with the numbers to see the effect of different number of updates. For example, after 100 updates we have a small amount of information left.\n"
+      "After 500 iterations we have lost all information, even though we were 100% sure that we started in position 1. Feel free to play with the numbers to see the effect of differing number of updates. For example, after 100 updates we have a small amount of information left.\n"
      ]
     },
     {
@@ -848,7 +848,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The problem of loosing information during an update may make it seem as if our system would quickly devolve into no knowledge. However, recall that our process is not an endless series of updates, but of *measure->update->measure->update->measure->update...* The output of the measure step is fed into the update. The update step, with a degraded certainty, is then fed into the measure step. \n",
+      "The problem of losing information during an update may make it seem as if our system would quickly devolve into no knowledge. However, recall that our process is not an endless series of updates, but of *measure->update->measure->update->measure->update...* The output of the measure step is fed into the update. The update step, with a degraded certainty, is then fed into the measure step. \n",
       "\n",
       "Let's think about this intuitively. After the first measure->update round we have degraded the knowledge we gained by the measurement by a small amount. But now we take another measurement. When we try to incorporate that new measurement into our belief, do we become more certain, less certain, or equally certain. Consider a simple case - you are sitting in your office. A co-worker asks another co-worker where you are, and they report \"in his office\". You keep sitting there while they ask and answer \"has he moved\"? \"No\" \"Where is he\" \"In his office\". Eventually you get up and move, and lets say the person didn't see you move. At that time the questions will go \"Has he moved\" \"no\" (but you have!) \"Where is he\" \"In the kitchen\". Wow! At that moment the statement that you haven't moved conflicts strongly with the next measurement that you are in the kitchen. If we were modelling these with probabilities the probability that you are in your office would lowever, and the probability that you are in the kitchen would go up a little bit. But now imagine the subsequent conversation: \"has he moved\" \"no\" \"where is he\" \"in the kitchen\". Pretty quickly the belief that you are in your office would fade away, and the belief that you are in the kitchen would increase to near certainty. The belief that you are in the office will never go to zero, nor will the belief that you are in the kitchen ever go to 1.0 because of the chances of error, but in practice your co-workers would be correct to be quite confident in their system.\n",
       "\n",
@@ -913,7 +913,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "So after the first update we have assigned a high probability to each door position, and a low probability to each wall position. The update step shifted these probabilities to the right, smearing them about a bit. Now lets look at what happens at the next sense."
+      "So after the first update we have assigned a high probability to each door position, and a low probability to each wall position. The update step shifted these probabilities to the right, smearing them about a bit. Now let's look at what happens at the next sense."
      ]
     },
     {
@@ -1192,9 +1192,9 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      " The largest probabilities are in position 0 and position 5. This does not fit our physical intuition at all. A dog cannot be in two places at once (my dog Simon certainly tries - his food bowl and my lap often have equal allure to him). We would have to use heuristics to decide how to interpret this distribution, and there is usually no satisfactory answer. This is not always a weakness - a considerable amount of literature has been written on *Multi-Hypothesis Tracking (MHT)*. We cannot always distill our knowledge to one conclusion, and MHT uses various techniques to maintain multiple story lines at once, using backtracking schemes to go *back in time* to correct hypothesis once more information is known. This will be the subject of later chapters. In other cases we truly have a multimodal situation - we may be optically tracking pedistrians on the street and need to represent all of their positions. \n",
+      " The largest probabilities are in position 0 and position 5. This does not fit our physical intuition at all. A dog cannot be in two places at once (my dog Simon certainly tries - his food bowl and my lap often have equal allure to him). We would have to use heuristics to decide how to interpret this distribution, and there is usually no satisfactory answer. This is not always a weakness - a considerable amount of literature has been written on *Multi-Hypothesis Tracking (MHT)*. We cannot always distill our knowledge to one conclusion, and MHT uses various techniques to maintain multiple story lines at once, using backtracking schemes to go *back in time* to correct hypothesis once more information is known. This will be the subject of later chapters. In other cases we truly have a multimodal situation - we may be optically tracking pedestrians on the street and need to represent all of their positions. \n",
       " \n",
-      "In practice it is the exponential increase in computation time that leads to this filter being the least frequently used of all filters in this book. Many problems are best formulated as discrete or multimodal, but we have other filter choices with better performance. With that said, if I had a small problem that this technique could handle I would choose to use it; it is trivial to implement, debug, and understand, all virtues in my book."
+      "In practice it is the exponential increase in computation time that leads to the discrete Bayes filter being the least frequently used of all filters in this book. Many problems are best formulated as discrete or multimodal, but we have other filter choices with better performance. With that said, if I had a small problem that this technique could handle I would choose to use it; it is trivial to implement, debug, and understand, all virtues in my book."
      ]
     },
     {
@@ -1222,18 +1222,18 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The code is very small, but the result is huge! We will go into the math more later, but we have implemented a form of a Bayesian filter. It is commonly called a Histogram filter. The Kalman filter is also a Bayesian filter, and uses this same logic to produce it's results. The math is a bit more complicated, but not by much. For now, we will just explain that Bayesian statistics compute the liklihood of the present based on the past. If we know there are two doors in a row, and the sensor reported two doors in a row, it is likely that we are positioned near those doors. Bayesian statistics just formalizes that example, and Bayesian filters formalize filtering data based on that math by implementing the sense->update->sense->update process. \n",
+      "The code is very small, but the result is huge! We will go into the math more later, but we have implemented a form of a Bayesian filter. It is commonly called a Histogram filter. The Kalman filter is also a Bayesian filter, and uses this same logic to produce its results. The math is a bit more complicated, but not by much. For now, we will just explain that Bayesian statistics compute the likelihood of the present based on the past. If we know there are two doors in a row, and the sensor reported two doors in a row, it is likely that we are positioned near those doors. Bayesian statistics just formalize that example, and Bayesian filters formalize filtering data based on that math by implementing the sense->update->sense->update process. \n",
       "\n",
-      "We have learned how to start with no information and derive information from noisy sensors. Even though our sensors are very noisey (most sensors are more then 80% accurate, for example) we quickly converge on the most likely position for our dog. We have learned how the update step always degrades our knowledge, but the addition of another measurement, even when it might have noise in it, improves our knowlege, allowing us to converge on the most likely result.\n",
+      "We have learned how to start with no information and derive information from noisy sensors. Even though our sensors are very noisy (most sensors are more than 80% accurate, for example) we quickly converge on the most likely position for our dog. We have learned how the update step always degrades our knowledge, but the addition of another measurement, even when it might have noise in it, improves our knowledge, allowing us to converge on the most likely result.\n",
       "\n",
       "If you followed the math carefully you will realize that all of this math is exact. The bar charts that we are displaying are not an *estimate* or *guess* - they are mathematically exact results that exactly represent our knowledge. The knowledge is probabilistic, to be sure, but it is exact, and correct.\n",
       "\n",
-      "However, we are a long way from tracking an airplane or a car. This code only handles the 1 dimensional case, whereas cars and planes operate in 2 or 3 dimensions. Also, our position vector is *multimodal*. It expresses multiple beliefs at once. Imagine if your GPS told you \"it's 20% likely that you are here, but 10% likely that you are on this other road, and 5% likely that you are at one of 14 other locations. That would not be very useful information. Also, the data is discrete. We split an area into 10 (or whatever) different locations, whereas in most real world applications we want to work with continuous data. We want to be able to represent moving 1 km, 1 meter, 1 mm, or any arbitrary amount, such as 2.347 cm. \n",
+      "However, we are a long way from tracking an airplane or a car. This code only handles the 1 dimensional case, whereas cars and planes operate in 2 or 3 dimensions. Also, our position vector is *multimodal*. It expresses multiple beliefs at once. Imagine if your GPS told you \"it's 20% likely that you are here, but 10% likely that you are on this other road, and 5% likely that you are at one of 14 other locations\". That would not be very useful information. Also, the data is discrete. We split an area into 10 (or whatever) different locations, whereas in most real world applications we want to work with continuous data. We want to be able to represent moving 1 km, 1 meter, 1 mm, or any arbitrary amount, such as 2.347 cm. \n",
       "\n",
-      "Finally, the bar charts may strike you as being a bit less certain than we would want. A 25% certaintly may not give you a lot of confidence in the anwser. Of course, what is important here is the ratio of this probability to the other probabilities in your vector. If the next largest bar is 23% then we are not very knowledgable about our position, whereas if the next largest is 3% we are in fact quite certain. But this is not clear or intuitive. However, there is an extremely important insight that Kalman filters implement that will signficantly improve our accuracy from the same data.\n",
+      "Finally, the bar charts may strike you as being a bit less certain than we would want. A 25% certaintly may not give you a lot of confidence in the answer. Of course, what is important here is the ratio of this probability to the other probabilities in your vector. If the next largest bar is 23% then we are not very knowledgable about our position, whereas if the next largest is 3% we are in fact quite certain. But this is not clear or intuitive. However, there is an extremely important insight that Kalman filters implement that will signficantly improve our accuracy from the same data.\n",
       "\n",
       "\n",
-      "**If you can understand this chapter you will be able to understand and implement Kalman filters** I cannot stress this enough. If anything is murky, go back and reread this chapter and play with the code. The rest of this book will build on the algorithms that we use here. If you don't intuitively understand why this histogram filter works, and can at least work through the math, you will have little success with the rest of the material. However, if you grasp the fundamental insight - multiplying probabilities when we measure, and shifting probabilities when we update leads to a converging solution - then you understand everything important you need to grasp the Kalman filter. "
+      "**If you can understand this chapter you will be able to understand and implement Kalman filters.** I cannot stress this enough. If anything is murky, go back and reread this chapter and play with the code. The rest of this book will build on the algorithms that we use here. If you don't intuitively understand why this histogram filter works, and can't at least work through the math, you will have little success with the rest of the material. However, if you grasp the fundamental insight - multiplying probabilities when we measure, and shifting probabilities when we update, leading to a converging solution - then you understand everything important you need to grasp the Kalman filter. "
      ]
     },
     {