"So, what are you working on?"

I have been asked about my PhD research on multiple occasions. The following description keeps people from yawning and/or dozing off while I ramble about what I work on these days.

Imagine that you are running a 100 meter sprint. Once in a while, you are nudged by a mischievous member of the audience causing you to speed up or slow down instantaneously (depending on whether the mischievous individual pushed you from behind or the front, respectively). You continue to run at a constant speed between nudges.

Now, also imagine that there are sensors and timers at every meter mark along the track that log the time you cross each sensor. But these timers are inaccurate; so they record some number that may be slightly bigger or smaller than the actual time you crossed the sensor.

Given these 100 timer readings, can you find the locations where you were nudged by the frolicsome bystander?

Prior, posterior, likelihood, MAP, ML

Let be a parameter and be the observation or data.

The posterior density is the probability density of given the observation, that is, .

The a priori (or prior) density is the probability density of before any observations are made, that is, .

The likelihood density is the probability of observed data, given the model parameters, that is, .

Using Bayes' rule, it is easy to see that

or, posterior prior likelihood.

When learning an unknown parameter from data, two commonly used estimation methods are as follows.

1. Maximum a posteriori (MAP) estimate

2. Maximum likelihood (ML) estimate

Notice that under a uniform prior, MAP and ML are identical.