Can random mutations cause an increase in information in the genome?

"Increase the information content" seems to me to be a dubious requirement.On mutations, I found a lot of interesting information in Some mutations sound too good to be true. I suggest you take a look.Apparently some mutations result in the duplication of a segment of DNA. This increases the total size of the DNA, and ought to count as in increase in the information content.Let's look at some of your comments on information. That's fine with me. The implication is that DNA contains no information, and no information is needed for biological organisms to develop from DNA. But this view of "information" makes moot your concern about "increase in information in the genome." Now you appear to be contradicting the previous definition (the one that required information to be the product of a mental process). So which is it to be - the previous definition or this one? And if you want it to be this one, how will you define "ordered complexity"?

That's what I expected you would say. And then if that extra copy is changed to be different information, you will say that was lost information. Nevertheless, with duplication followed by modification you get new information.You are using a vague definition of "information", and then expressing opinion to assert that there was nothing new.If you look at scientific laws, they are closely connected to empirical procedures, often with measurements. And they are defined precisely. You would need a definition of "information" such that we can tell from the definition how much information there is in a genome. And then you would have to show how that definition is relevant to biology and the evolution argument.My prediction: if you ever come up with a suitable definition, it will turn out that biological systems are indeed creating new information by means of mutation. I have no problems with theistic evolution, if that is what you are proposing. But those are poor analogies to evolution. Sure. The parent organism designed it as a message to the cellular machinery on what proteins to construct. So we see it designed by a biological system as a message to another biological system.No need for you to reply to this. I can see that the thread has kept you pretty busy.

Speaking as a mathematician, I have to disagree."Random" need not imply that all outcomes are equi-probable.

I'm not a biologist, so I am probably the wrong person to ask. As I understand it, this simply means that there is no observed direction to the mutations.

I wouldn't put it that way. The mutation rate is small enough, that it doesn't compare well to multiple rolls of dice.If you could replay a small segment of life, then I would expect observable differences in the individual creatures that are there as a result of mutations.The overall outcome of evolution might be similar. But that's because the direction is determined by averages over large populations, so that natural selection is a major factor in how a species evolves.

You are misusing "random".Suppose I toss a coin. The result is random, with equal probabilities for heads or tails.Now suppose that I toss a coin 100 times, and count how many times the coin showed heads. It is predictable that the answer won't be too far from 50. But it is still random, even though predictable. The outcomes are no longer equi-probable. Instead, they follow the binomial probability distribution.You are confusing "random" with "equi-probable". They are not the same thing at all.You might find it interesting to google "monte carlo methods".