Incorrect assumptions on conditional distributions

One final concern is that many times even the distribution $ P(y\vert\theta)$ is incorrectly estimated because it is assumed arbitrarily to belong to a family of distributions. For example, it is often very easy to work with Gaussian densities. Therefore, it is tempting to assume that $ p(y\vert\theta)$ is Gaussian. Experiments can be performed to estimate the mean and variance parameters. Even though some best fit will be found, it does not necessarily imply that a Gaussian is a good representation. Conclusions based on this model may be incorrect, especially if the true distribution has a different shape, such as having a larger tail or being multimodal. In many cases, nonparametric methods may be needed to avoid such biases. Such methods do not assume a particular family of distributions. For example, imagine estimating a probability distribution by making a histogram that records the frequency of $ y$ occurrences for a fixed value of $ \theta $. The histogram can then be normalized to contain a representation of the probability distribution without assuming an initial form.

Steven M LaValle 2012-04-20