Role of probability
A robot uses sensors to obtain information about it’s state.
In certain cases, such as measuring joint positions and velocities, direct state feedback can be obtained.
However in general, sensor information is noisy and/or only provides partial information about the environment.
In this case, we cannot determine the exact state, but instead try to find what state is “most likely” based off the evidence.
To do this, the conditional probability is used, for state
For a known value of measurement
There are two ways to choose a “best guess”
- Maximise the probability distribution:
- Find the (conditional) expectation over the probability distribution:
Finding the maximum is better in general, since the expectation is only suitable for “unimodal” distributions, where there is a single “clump” of probability density, and the mean characterises the centre of this. For example, if there are two concentrations of probability density, the expectation will give a value between them, which is actually very unlikely.
However, the expectation can be useful, such as with monte-carlo methods (the particle filter), which can better handle the this drawback.
Probabity models
The term “probability model”, will refer to the form of the function
For example, a simple probability model for a gps sensor is:
Interpretation:
- The “true” position of the sensor is
. - The sensor returns a position measurement
. - This measurement follows a gaussian distribution (normal distribution) with mean
and covariance . - Writing
means that the normal distribution is being evaluated at . ie, if and are k-dimensional:
Collections of random variables
Random variable
For a collection of datapoints, defined over times
Like any other random variable, this has a probability distribution
With collections of random variables, the probability models can be much more complex. However, there is usually a way to factorise the probability distribution so it is much more manageable.
For example, in the extreme case that all random variables are independent, the probability distribution becomes:
In general, the factorisation depends on “conditional independence” between variables, which can be visually represented using graphical models.
See: (todo: notes on graphical models)
Log-likelihood
Log-likelihood
This maps from
This is useful because log converts produces into sums:
Additionlly, the logarithmic is monotonic, meaning that:
It also gives a nice result for the gaussian distribution:
When a gaussian log-likelihood appears, this gives a quadratic term to be minimised.
Monte-carlo methods
Monte-carlo methods provide a way to find the expectation of a random variable using random sampling:
In the final line,
The reason this is useful is because:
- For a complex
, it is difficult to produce a sample . - However, the distribution can easily be evaluated at a given value.
- A simpler distribution
can be used to generate samples, and the weighting corrects the resultant mean to correct for the difference between and and how this would affect what samples are more likely.
This can be extended to work with distributions