It has been a while since I mixed mathematics and genealogy. It feels like time to do that again.
Genealogy is all about information. We gather information and use it to figure out information about ancestors and relatives. There is a branch of engineering and mathematics that is actually called information theory. It describes many aspects of how we transmit and store information, even what information is. The basic unit of information is the bit. Bits are the 1s and 0s that computers use, the ons and offs of the transistors inside computer and the trues and falses in a game of twenty questions. One might think that the amount of information in a document or message is just the number of bits it takes to express that message. There are several reasons that isn’t quite right. My favorite, partly because of its name, is the concept of “surprise.”
Surprise!
Surprise sounds like it ought to be a fun concept, and it is actually quite intuitive. Imagine your in a bank. As you wait in line for a teller, you shift your gaze around the room. You notice that there is an alarm bell on the wall. That bell can do two things. It can ring and it can not ring. That sounds nice and binary. The bell can express a single bit of information. As you stand in line, you are not surprised that the alarm is not ringing. You expect it to be silent. From that point of view you might say that it is not conveying any information. It actually is, but only very little. If on the other hand, the bell starts ringing, you (and everyone else in the bank) will be very surprised. It feels like the bell is now conveying a lot of information, but isn’t it still just a one bit message?
The key is that the less likely a message is, the more surprising the message is, and the more information it is conveying. By the formal definition of surprise, a message that has to occur has no surprise and carries no information. A message that is impossible would convey infinite surprise, and infinite information as well. (the next time you are surprised by something, you probably won’t consider that it has a formal definition, but you can always try). It is an interesting concept. We are often surprised by discoveries in genealogy and probably wouldn’t even call them discoveries unless they were at least somewhat surprising. The closer a document is to impossible, the more surprised we are, and it feels right to consider it as carrying more information. The next time you find the second death record for an ancestor recorded years after the first, you are not only entitled to be surprised but to be in awe of the amount of information being conveyed—it is either telling you that your previous information is wrong and giving you the new information, or it is conveying the information that your next step is to get a hold of Stephen King’s phone number.
Ancestral Surprise
Another thing that leads to surprise in genealogy, is going back another generation. It gets harder as we go and feels more surprising every time we manage to do it. I was surprised to learn that there is a basis for this in information theory as well.
Pedigree charts are very binary things. When we number them with ahnentafel numbers, the numbers get larger as we go back to more distant ancestors, and they do so in a special way. Every generation back adds another bit to the length of an ahnentafel number if you express it in binary. You are 1, your father 10, your mother 11, her mother 111 and so on.
The codes computer use to represent typed characters are also very binary. For example, in one coding (ASCII) an A is represented by 0100 0001, a B by 0100 0010, and so on. Often when trying to be efficient, the most probable messages will be given the shortest codes. Morse code represents the most commonly used letters with the shortest codes. E is a single dot in Morse. Q is dash-dash-dot-dash. If you construct a diagram to decide on binary codes in the most efficient way, the most probable codes using the fewest bits, you end up constructing a thing that looks exactly like a pedigree chart. The less likely a coded item is to be used, the more bits in the code for that item. The more bits, the lower the probability, and from above we know that means the greater the surprise. Now, the binary ahnentafel numbers get one bit longer per generation, which means we now have a mathematical proof that discovering great-great-great-great-great-great-grandma is very surprising indeed.