Bayesian Prediction

Background

Make sure you understand the Introduction to Bayesian statistics first. Now we'll derive "Bayes' Rule".

Venn diagram
$P(A|B) = \frac{P(A \& B)}{P(B)}$ , and $P(B|A) = \frac{P(A \& B)}{P(A)}$ , so and therefor $P(A|B) = \frac{P(A)P(B|A)}{P(B)}$
Note that the above mathematics is true when A is a set of worlds or models and hence P(A) is a probability. It also holds when A is defined over a continuous space and so P(A) is a density rather than a probability.
If the set A is parameterised, then we can do this for a whole function at the same time. e.g. Assume A stood for "Location is x", and B stood for "the last observation was O" (see diagram p 199 of Probabilistic Robotics)
- Then for each value of x, P(Location is x | last observation was O) = P(Location is x (regardless of the last observation)) P(last observation was O | Location was x) / P(last observation would have been O)
- Note that P(last observation would have been O) is a constant - ignore it for the moment.
- Note that P(Location is x | last observation was O), P(Location is x (regardless of the last observation)) and P(last observation was O | Location was x) are all functions of x - we could do the calculation for each value of x in parallel.
- If $f_1(x) = C_1e^{(x-\mu_1)^2}$ and $f_2(x) = C_2e^{(x-\mu_2)^2}$ then $f_1(x) \times f_2(x) = C_1e^{(x-\mu_1)^2} \times C_2e^{(x-\mu_2)^2} = C_1C_2e^{(x-\mu_1)^2+(x-\mu_2)^2}$

The goal of Bayesian prediction is to predict the next observation from a sequence of such observations.
We assume a class of models, M, which includes the true model, t.
- Each model will define the probability of each observation at each timestep.
Define a probability distribution over those models; our subjective probability that each model is the true model.
- Initially this could be a uniform distribution, or some sort of complexity prior (See Occam's razor) that makes more complex models less likely
At any point there are two obvious ways to make a prediction about the next observation:
- Choose the most likely model and predict what it predicts, or (this is fast)
- Use a weighted sum of the predictions of all the models. (this is more accurate)
- Note that models might make different predictions at different times.
When we see an observation, we need to then update the probability distribution over the models given that distribution. Bayes' rule is used.
- $P(m|o_t) = \frac{P(m)P(o_t|m)}{P(o_t)}$ , but the observation probability is a constant, so $P(m|o_t) \propto P(m)P(o_t|m)$ , and you can renormalise to return that to a probability distribution.