What are optimal observer models?

• define: what is optimal anyway?
  • Bayeisan inference
  • Does the brain perform Bayesian inference?
  • The question is: how would a statistician solve the problem?

What kinds of problems do brains solve? What is the 'optimal' solution to these problems? How would brains do this?

Next question: it looks like brains are optimal, but are they really?

Key idea: use sampling approach to inference. Take close look at a Bayesian model. Are they really fully Bayesian? I would argue that they are not, in the sense that they do not make full use of uncertainty.

• How can models of behaviour be translated into common problems of statistical inference?
• What is the connection between models of cognition and comon problems of data analysis?

One of the most common inference problems is:

• assuming data follow a normal distribution \[ y \sim N(\mu, \sigma^2)\]
• explain simple problem of estimating a mean with known variance -> conjugacy
• anything more complex is going to be difficult
• show effect of prior (figures)
• introduce MCMC as solution for more complex problems
• how does this improve on analytically solved Bayesian models? These (e.g. De Vrijer) often do not make full use of uncertainly of parameter estimates, because this becomes very difficult to do…

1. AJ is a statistician: .

He would like to make sense of some measurements taken with a newly acquired accelerometer.

1. FJ is an observer:

He would like to know his tilt angle with respect to gravity.

Using conjugate priors

• imagine we have a sample of observed data
• we want to know the mean and the standard deviation

true_mean <- 10
true_sd <- 2
N <- 50 # sample_size
y <- rnorm(N, true_mean, true_sd)

max_like_est <- mean(y)