## 1.Bias and Variance

Understanding how different sources of error lead to bias and variance helps us improve the data fitting process resulting in more accurate models. We define bias and variance in three ways: conceptually, graphically and mathematically.

### 1.1Conceptual Definition

Error due to Bias: The error due to bias is taken as the difference between the expected (or average) prediction of our model and the correct value which we are trying to predict. Of course you only have one model so talking about expected or average prediction values might seem a little strange. However, imagine you could repeat the whole model building process more than once: each time you gather new data and run a new analysis creating a new model. Due to randomness in the underlying data sets, the resulting models will have a range of predictions. Bias measures how far off in general these models' predictions are from the correct value. Error due to Variance: The error due to variance is taken as the variability of a model prediction for a given data point. Again, imagine you can repeat the entire model building process multiple times. The variance is how much the predictions for a given point vary between different realizations of the model.

### 1.2Graphical Definition

We can create a graphical visualization of bias and variance using a bulls-eye diagram. Imagine that the center of the target is a model that perfectly predicts the correct values. As we move away from the bulls-eye, our predictions get worse and worse. Imagine we can repeat our entire model building process to get a number of separate hits on the target. Each hit represents an individual realization of our model, given the chance variability in the training data we gather. Sometimes we will get a good distribution of training data so we predict very well and we are close to the bulls-eye, while sometimes our training data might be full of outliers or non-standard values resulting in poorer predictions. These different realizations result in a scatter of hits on the target.

We can plot four different cases representing combinations of both high and low bias and variance. ### 1.3Mathematical Definition ## 2.An Illustrative Example: Voting Intentions

Let's undertake a simple model building task. We wish to create a model for the percentage of people who will vote for a Republican president in the next election. As models go, this is conceptually trivial and is much simpler than what people commonly envision when they think of "modeling", but it helps us to cleanly illustrate the difference between bias and variance.

A straightforward, if flawed (as we will see below), way to build this model would be to randomly choose 50 numbers from the phone book, call each one and ask the responder who they planned to vote for in the next election. Imagine we got the following results: From the data, we estimate that the probability of voting Republican is 13/(13+16), or 44.8%. We put out our press release that the Democrats are going to win by over 10 points; but, when the election comes around, it turns out they actually lose by 10 points. That certainly reflects poorly on us. Where did we go wrong in our model?

Clearly, there are many issues with the trivial model we built. A list would include that we only sample people from the phone book and so only include people with listed numbers, we did not follow up with non-respondents and they might have different voting patterns from the respondents, we do not try to weight responses by likeliness to vote and we have a very small sample size.

It is tempting to lump all these causes of error into one big box. However, they can actually be separate sources causing bias and those causing variance.

For instance, using a phonebook to select participants in our survey is one of our sources of bias. By only surveying certain classes of people, it skews the results in a way that will be consistent if we repeated the entire model building exercise. Similarly, not following up with respondents is another source of bias, as it consistently changes the mixture of responses we get. On our bulls-eye diagram these move us away from the center of the target, but they would not result in an increased scatter of estimates.

On the other hand, the small sample size is a source of variance. If we increased our sample size, the results would be more consistent each time we repeated the survey and prediction. The results still might be highly inaccurate due to our large sources of bias, but the variance of predictions will be reduced. On the bulls-eye diagram, the low sample size results in a wide scatter of estimates. Increasing the sample size would make the estimates clump closers together, but they still might miss the center of the target.

Again, this voting model is trivial and quite removed from the modeling tasks most often faced in practice. In general the data set used to build the model is provided prior to model construction and the modeler cannot simply say, "Let's increase the sample size to reduce variance." In practice an explicit tradeoff exists between bias and variance where decreasing one increases the other. Minimizing the total error of the model requires a careful balancing of these two forms of error.