Wait, did you interpret my comment as supporting the "irreducible complexity" argument? My whole point was that it is a bad argument. I was criticizing the Hangman analogy because it seems to invite the same sort of mistake that the "irreducible complexity" people make.

I have come to consider this isomorphism between Bayesian inference and natural selection or Darwinian processes in general as a deep insight into the workings of nature.

You might like this. ("In fact, I realized, Bayes's rule just is the discrete-time replicator equation, with different hypotheses being so many different replicators, and the fitness function being the conditional likelihood. ")

Wait, did you interpret my comment as supporting the "irreducible complexity" argument? My whole point was that it is a bad argument. I was criticizing the Hangman analogy because it seems to invite the same sort of mistake that the "irreducible complexity" people make.

What a wonderful post!

I find it intellectually exhilarating as I have not been introduced to Solomonoff before and his work may be very informative for my studies. I have come at inference from quite a different direction and I am hopeful that an appreciation of Solomonoff will broaden my scope.

One thing that puzzles me is the assertion that:

Therefore an algorithm that is one bit longer is half as likely to be the true algorithm. Notice that this intuitively fits Occam's razor; a hypothesis that is 8 bits long is much more likely than a hypothesis that is 34 bits long. Why bother with extra bits? We’d need evidence to show that they were necessary.

First my understanding of the principle of maximum entropy suggests that prior probabilities are constrained only by evidence and not by the length of the hypothesis test algorithm. In fact Jaynes argues that 'Ockham's' razor is already built into Bayesian inference.

Second given that the probability is reduced by half with every bit of added algorithm length wouldn't that imply that algorithms' having 1 bit were the most likely and have a probability of 1/2. In fact I doubt if any algorithm at all is describable with 1 bit. Some comments as well as the body of the article suggest that the real accomplishment of Solomonoff's approach is to provide the set of all possible algorithms/hypothesis and that the probabilities assigned to each are not part of a probability distribution but rather are for the purposes of ranking. Why do they need to be ranked? Why not assign them all probability 1/N where N = 2^(n+1) - 2, the number of algorithms having length up to and including length n.

Clearly I am missing something important.

Could it be that ranking them by length is for the purpose of determining the sequence in which the possible hypothesis should be evaluated? When ranking hypothesis by length and then evaluating them against the evidence in sequence from shorter to longer our search will stop at the shortest possible algorithm, which by Occam's razor is the preferred algorithm.

Excellent post.

I have pondered the same sort of questions. Here is an excerpt from my 2009 book.

My father is 88 years old and a devout Christian. Before he became afflicted with Alzheimer’s he expected to have an afterlife where he would be reunited with his deceased daughter and other departed loved ones. He doesn’t talk of this now and would not be able to comprehend the question if asked. He is now almost totally unaware of who he is or what his life was. I sometimes tell him the story of his life, details of what he did in his working life, stories of his friends, the adventures he undertook. Sometimes these accounts stir distant memories. I have recently come to understand that there is more of ‘him’ alive in me then there is in him. When he dies and were he to enter the afterlife in his present state and be reunited with my sister he would not recognize or remember her. Would he be restored to some state earlier in his life? Would he be the same person at all?

I originally wrote this to illustrate problems with the religious idea of resurrection. I now believe that this problem of identity is common to all complex evolving systems including 'ourselves'. For example species evolve over their lifetime and although we intuitively know that we are identifying something distinct when we name a species such as homo-sapiens the exact nature of the distinction is slippery. The debate in biology over the definition of species has been long, heated and unresolved. Some definition referring to species are attempts along the line of interbreeding populations that do not overlap with other populations. However this is a leaky definition. For example it has recently been found that modern human populations contain some Neanderthal DNA. Our 'species' interbred in the past, should we still be considered separate species?

I only have a layperson's knowledge of evolutionary biology, so my criticisms might miss some important subtlety, but it seems to that your analogy is significantly misleading in a couple of ways. It does convey the idea that random guesses with incremental feedback is a better search strategy than if the feedback were holistic (e.g. if you were guessing whole words and the only feedback were whether the guess is correct or not). In so far as someone's worry about natural selection is that they're mistaking it for the latter sort of search, the analogy may be helpful. But if you want to convey something more specific about how natural selection works, then I'm afraid the analogy isn't all that great.

One drawback of the analogy is in the nature of the environmental feedback. In Hangman, a letter gets fixed if (and only if) it is part of the correct answer. In genuine natural selection, though, a mutation doesn't get fixed because it is part of a complex set of mutations that collectively confer some phenotypic benefit. The environment isn't forward-looking like that; it doesn't say "This mutation is part of what is needed for optimality, so I'm going to hold onto it for that reason." Each individual mutation, in order to get fixed in the population, must confer some immediate reproductive benefit. Merely being one element of some complex group of mutations that is collectively beneficial is insufficient. The hangman analogy doesn't capture this aspect of natural selection.

This actually leads the analogy to kind of play into the hands of "irreducible complexity" critiques of natural selection. The proponents of such critiques presume that the individual parts of some complex adaptation only benefit the organism to the extent that they are part of that complex adaptation, and hence one cannot explain their selection without supposing that there is some forward-looking element to selection which holds onto those individual changes just because they will eventually contribute to a complex adaptation. This forward-looking aspect is then offered as evidence of intelligent design.

Another big drawback is that the analogy doesn't capture the competitive nature of natural selection. Natural selection occurs in populations, and requires both variation in traits among individuals in the population and competition for resources among those individuals. The Hangman analogy suggests that the environment already has a fixed template for the ideal phenotype and that it punishes organisms (or genes) individually for failing to approach this ideal and rewards them for getting closer to the ideal. If you have a population, and things worked in the Hangman way, there would be no correlation between rewards and punishments. But that's not how natural selection works. Genes are rewarded for contributing to their vehicles (organisms) being more reproductively successful than other organisms in the population. A reward just consists in reproducing more than your competitors, and a punishment just consists in reproducing less, so rewards and punishments are correlated. One allele can't get rewarded without another one getting punished.

I was also inspired by one of Dawkins' books suggesting something similar. It was some years ago but I believe Dawkins suggested writing a type of computer script which would mimic natural selection. I wrote a script and was quite surprised at the power it demonstrated.

As I remember the general idea is that you can type in any string of characters you like and then click the 'evolve' button. The computer program then:

1) generates and displays a string of random characters of the same length as the entered string.

2) compares the new string with the displayed string and retains all characters that are the same and in the same position.

3) generates random characters in the string where they did not match in 2 and displays the full string.

4) If the string in 3 matches the string entered by the computer the program stops otherwise it goes to step 2.

The rapidity with which this program converges on the one entered it quite surprising.

This simulation is somewhat different from natural selection especially in that the selection rules are hard coded but I think it does demonstrate the power of random changes to converge when there is strong selection pressure.

A fascinating aid in demonstrating natural selection was built by Darwin's cousin Francis Galton in 1877. A illustration and description can be found here. The amazing thing about this device is that, as described in the article, it has been re-discovered and re-purpose to illustrate the process of Bayesian inference.

I have come to consider this isomorphism between Bayesian inference and natural selection or Darwinian processes in general as a deep insight into the workings of nature. I view natural selection as a method of physically performing Bayesian inference, specifically as a method for inferring means for reproductive success. My paper on this subject may be found here

No, Newton's theory of gravitation does not provide knowledge. Belief in it is no longer justified; it contradicts the evidence now available.

However, prior to relativity, the existing evidence justified belief in Newton's theory. Whether or not it justified 100% confidence is irrelevant; if we require 100% justified confidence to consider something knowledge, no one knows or can know a single thing.

So, using the definition you gave, physicists everywhere (except one patent office in Switzerland) knew Newton's theory to be true, because the belief "Newton's theory is accurate" was justified. However, we now know it to be false.

Currently, we have a different theory of gravity. Belief in it is justified by the evidence. By your standard, we know it to be true. That's patently ridiculous, however, since physicists still seek to expand or disprove it.

You tried to define knowledge as simply 'justified belief'. The example scientific theory was believed to be true, and that belief was justified by the evidence then available. But, as we now know, that belief was false. By your definition, however, they can still be said to have 'known' the theory was true.

That is the problem with the definition not including the 'true' caveat.

The "Socrates is mortal" one is a good example because nowadays its conclusion has a probability of less than one.