In my last post I presented a simple model of meta-theoretic induction. Let’s instantiante it with concrete data and run through it. Say we have

E_{1} E_{2} E_{3} |
Observations made for different domains 1-3 |

S_{1} S_{2} S_{3} |
Simple theories for domains 1-3 |

C_{1} C_{2} C_{3} |
Complex theories for domains 1-3 |

S |
Meta-theory favoring simple theories |

C |
Meta-theory favoring complex theories |

That is, we have three domains of observation with corresponding theories. We also have two meta-theories that will produce priors on theories. The meta-theories themselves will be supported by theories’ sucesses or failures. Successes of simple theories support S, successes of complex theories support C. Now define the content of the theories through their likelihoods

E_{n} |
P(E_{n}|S_{n}) |
P(E_{n}|C_{n}) |
---|---|---|

E_{1} |
3/4 | 1/4 |

E_{2} |
3/4 | 1/4 |

E_{3} |
3/4 | 3/4 |

Given that E_{1}, E_{2} and E_{3} are evidence, this presents a scenario where theories S_{1} and S_{2} were successful, whereas theories C_{1} and C_{2} were not. S_{3} and C_{3} represent theories that are equally well supported by previous evidence (E_{3}) but with different future predictions. This is the crux of the example, where the simplicity bias enters into the picture. Our meta-theories are defined by

*P(S _{n}|S) = 3/4, *

*P(S*

_{n}|C) = 1/4*P(C _{n}|C) = 3/4, *

*P(C*

_{n}|S) = 1/4Meta-theory S favors simple theories, whereas meta-theory C favors complex theories. Finally, our priors are neutral

*P(S _{n}) = P(C_{n}) = 1/2*

*P(S) = P(C) = 1/2*

We want to process evidence E_{1 }E_{2}, and see what happens at the critical point, where S_{3} and C_{3} make the same predictions. The sequence is as follows

- Update meta theories S and C with E
_{1}and E_{2} - Produce a prior on S
_{3}and C_{3}with the updated C and S - Update S
_{3}and C_{3}with E_{3}

The last step produces probabilities for S_{3} and C_{3}; these theories make identical predictions but *will have different priors granted by S and C*. This will formalize the statement

Simpler theories are more likely to be true because they have been so in the past

### The model as a bayesian network

Instead of doing all the above by hand (using equations **3**,**4**,**5**,**6**), it’s easier to construct the corresponding bayesian network and let some general algorithm do the work. Formulating the model this way makes it much easier to understand, in fact it seems almost trivial. Additionally, our assumptions of conditional independence (**1** and **2**) map directly into the bayesian network formalism of nodes and edges, quite convenient!

Node M represents the meta-theory, with possible values *S* and *C, *the H nodes represent theories, with possible values S_{n} and C_{n}. Note the lack of edges between H_{n} and E_{x} formalizing (**1**), and the lack of edges between M and E_{n} formalizing (**2**) (these were our assumptions of conditional independence).

I constructed this network using the SamIam tool developed at UCLA. With this tool we can construct the network and then monitor probabilities as we input data into the model, using the tool’s* Query Mode*. So let’s do that, fixing the actual outcome of the evidence nodes E1, E2 and E3 (click to enlarge)

Theories S_{1} and S_{2} make correct predictions and are thus favoured by the data over C_{1} and C_{2}. This in turn favours the meta-theory S, which is assigned a probability of 73% over meta-theory C, with 26%. Now, theories S_{3} and C_{3} make the *same* predictions about E_{3}, but because of our meta-theory being better supported, they are assigned different probabilities. Again, recall our starting point

Simpler theories are more likely to be true because they have been so in the past

We can finally state this technically, as seen here

The simple theory S_{3} is favored at 61% over C_{3} with 38%, even though they make the same predictions. In fact, we can see how this works if we look at what happens with and without meta-theoretic induction

where as expected the mirrors of S_{3} and C_{3} would be granted the same probabilities. So everything seems to work, our meta-theory discriminates different theories and is itself justified via experience, as was the objective

Occam seems like an unjustified and arbitrary principle, in effect, an unsupported bias. Surely, there should be some way to anchor this widely applicable principle on something other than arbitrary choice. We need a way to represent a meta-theory such that it favours some theories over others

andsuch that it can bejustified through observations.

**But**, what happens when we add a meta-theory like *Occam(t)* into the picture? What happens when we apply the same argument at the meta-level that prompted the meta-theoretic justitification of simplicity we’ve developed? We define a meta-theory *S-until-T* with

*P(S _{1}|S-until-T) = *

*P(S*

_{2}|S-until-T) = 3/4*P(S _{3}|S-until-T) = 1/4*

which yields this network

Now both S and S-until-T accrue the same probability through evidence and therefore produce the same prior on S_{3} and C_{3}, 50%. It seems we can’t escape our original problem.

Because both

OccamandOccam(t)are supported by the same amount of evidence, equal priors will be assigned to S_{3}and C_{3}. The only way out of this is forOccamandOccam(t)to have different priorsthemselves.But this leaves us back where we started!We are just recasting the original problem at the meta level, we end up begging the question[1] or in an infinite regress.

In conclusion, we have succeeded in formalizing meta-theoretic induction in a bayesian setting, and have verified that it works as intended. However, it ultimately does not solve the problem of justificating simplicity. The simplicity principle remains a prior belief independent of experience.

(The two networks used in this post are metainduction1.net and metainduction2.net, you need the SamIam tool to open these files)

[1] Simplicity is justified if we previously assume simplicity

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