Inductive inference is the process of reaching a general conclusion from specific examples.

overview

The general conclusion should apply to unseen examples.

Inductive Learning Hypothesis: any hypothesis found to approximate the target function well over a sufficiently large set of training examples will also approximate the target function well over other unobserved examples.

Example:

Identified relevant attributes: x, y, z

x y z
2 3 5
4 6 10
5 2 7

Model 1:

x + y = z

Prediction: x = 0, z = 0 mapsto y = 0

Model 2:

if x = 2 and z = 5, then y = 3.
if x = 4 and z = 10, then y = 6.
if x = 5 and z = 7, then y = 2.
otherwise y = 1.

Model 2 is likely overfitting.

Good:
Bad:
completely consistent with data.
no justification in the data for the prediction that y = 1 in all other cases.
not in the class of algebraic functions (but nothing was said about class of descriptions).

Inductive bias: explicit or implicit assumption(s) about what kind of model is wanted.

Typical inductive bias:

Example:

Some languages of interest:

Positive and Negative Examples

Positive Examples

x y z
2 3 5
2 5 7
4 6 10


general x, y, z epsilon I
more specific x, y, z epsilon I+
more specific than the first two 1 < x, y, z < 11 ; x, y, z episolon I
even more specific model x + y = z

Negative Examples

x y z Decision
2 3 5 Y
2 5 7 Y
4 6 10 Y
2 2 5 N

Search for Description

Description keeps getting larger or longer.

Finite language - algorithm terminates.

Infinite language - algorithm runs

X = example space/instance space (all possible examples)

D = description space (set of descriptions defined as a language L)

langmap

Success Criterion

Look for a description l epsilon L such that l is consistent with all observed examples.

Example:

L = {x op y = z}, op = {+, -, *, /}

Given a precise specification of language and data, write a program to test descriptions one by one against the examples.

Why is Machine Learning Hard (Slow)?

It is very difficult to specify a small finite language that contains a description of the examples.

e.g., algebraic expressions on 3 variables is an infinite language