Explanatory power is a bit of a loaded term. I believe that we can come to a good understanding of what it is and how it related to machine learning by comparing and contrasting linear regression with neural nets.

The IBM Analytics Education Series has a good introductory analytics presentation that includes brief descriptions of neural nets and linear regression. It's a good video worth your viewing, though it does make one point with which I don't agree.

The speaker says that a challenge with neural nets in business applications is that they are black box, meaning that you can understand the inputs and the outputs but not really how it is deriving the outputs. Later, the speaker says that linear regression is a preferred technique because it has a very strong predictive and explanatory power.

It's not really true that linear regression has more explanatory power than neural nets. Rather, it is easier to understand the problems and the answers that can be solved by linear regression. By comparison, neural nets tend to be used to provide cognitive computing power to harder problems than linear regression can solve.

To put this another way, when you use linear regression, you actually begin by assuming linearity of the relation you want to predict. As the speaker points out, you can also make a non-linear assumption, and you can accommodate this using a data transformation, for example. But the high order bit is that you are asked to *assume *the data relationship, and that assumption is what is giving you *the illusion of explanatory power*. You can explain that the data follows a line, but this is due to *your own assumption*. Note that an important aspect of completing a linear regression model is determining the R^{2} or goodness of fit of the model. This is the part where you make sure that your assumption of linearity is valid. And if the assumption is invalid, then the model has no predictive value, so it does not matter that you can explain how it operates.

Under the interpretation that explanatory power is akin to predictive power, it turns out that neural nets have *greater *predictive power because they can produce results for a wider array of applications than linear regression can. There a neat table that relates the cognitive power of a neural net to the number of hidden layers. From the table, you can see that when a relationship actually is linear, a neural net can solve it without even using any hidden layers of neurons. When one or two hidden layers of neurons is present, neural nets transcend the capabilities of linear regression, in part because they do not require you to make any assumption about what the data relationship actually is.

And that's where the confusions comes in. The linear regression model requires you to assume linearity and so you know at least what geometric shape the relationship looks like. The neural net requires no such assumption, but nor does the trained neural network give you any hint at what the relationship is. The lack of knowing the relationship is confused for having less explanatory power.

But if you look at this a bit more abstractly, the trained linear regression model has the same exact problem of **not providing any additional insight**. A neural net is really just a pile of numbers giving constant weights to the neural connections that can convert inputs to outputs. Similarly, a linear regression model is just a pile of numbers that give constant weights to inputs to be linearly combined into an output. Sure you know the data relationship, but that's because you assumed it. The actual linear regression model gives you no insight into why one dimension has a large slope constant where another has a small slope.

An analogy I like to use is that the value of the neural net is not diminished by our inability to explain how it is that the little gray cells which implement our personal neural nets can produce the cognitive results that they do, and **who among us would prefer to have cognitive powers defined by linear regression instead**?

In terms of explanatory power, our biological neural nets perform an additional key function that we have not hitherto been able to achieve with artificial neural nets. We are able to construct additional information in the output that reveals causal relationships, or insights into the reasons for the phenomena it predicts. Put simply: *we say why something is true*. We **provide a rationale**. This is an aspect of explanatory power that, when achieved, dramatically increases the value and utility of any cognitive analytic. Theorem provers and Prolog programs have been able to do this for the applications to which they apply. In the area of unstructured information processing and data mining, you can see a demo of this concept in Watson Paths.

## Explanatory Power in Machine Learning |