How Kalman Filters Work, Part 1

Let's suppose you've agreed to a rather odd travel program, where you're going to be suddenly transported to a randomly selected country, and your job is to figure out where you end up. So, here you are in some new country, and all countries are equally likely. You make a list of places and probabilities that you're in those places (all equally likely at about 1/200 for 200 countries). You look around and appear to be in a restaurant. Some countries have more restaurants (per capita/per land area) than others, so you decrease the odds that you're in Algeria or Sudan and increase the odds that you're in Singapore or other high-restaurant-density places. That is, you just multiply the probability that you were in a country with the probability of finding oneself in a restaurant in that country, given that one were already in the country, to obtain the new probability. After a few moments, the waitress brings you sushi, so you decrease the odds for Tajikistan and Paraguay and correspondingly increase the odds on Japan, Taiwan, and such places where sushi restaurants are relatively common. You pick up the chopsticks and try the sushi, discovering that it's excellent. Japan is now by far the most likely place, and though it's still possible that you're in the United States, it's not nearly as likely (sadly for the US). Those "probabilities" are getting really hard to read with all those zeros in front. All that matters is the relatively likelihood, so perhaps you scale that last column by the sum of the whole column. Now it's a probability again, and it looks something like this: Now that you're pretty sure it's Japan, you make a new list of places inside Japan to see if you can continue to narrow it down. You write out Fukuoka, Osaka, Nagoya, Hamamatsu, Tokyo, Sendai, Sapporo, etc., all equally likely (and maybe keep Taiwan too, just in case). Now the waitress brings unagi. You can get unagi anywhere, but it's much more common in Hamamatsu, so you increase the odds on Hamamatsu and slightly decrease the odds everywhere else. By continuing in this manner, you may eventually be able to find that you're eating at a delicious restaurant in Hamamatsu Station -- a rather lucky random draw.