[back]

Monotonic Bernoulli Trials
Amrit Pratap, Yaser Abu-Mostafa, Pietro Perona

Abstract. When estimating a number of bernoulli variables which have a certain monotonicity constraint, if the number of samples for each variable is small, then the estimates will not satisfy the monotonicity constraint. Better performance is achieved by endorcing the monotonicity constraint on the estimation procedure.

Motivations and Aims. Psychophysics [1] deals with the functional relationship between physical characteristics and subjective sensation. An important concept in psychophysics is that of threshold, the boundary between the detectable and the undetectable. This boundary is not abrupt and for each level of stimulus x, there’s a probability of detection p(x). The threshold is set at a level of stimulus that produces a certain probability of detection.

We formulate this concept mathematically as follows. Consider a family of Bernoulli random variables V(x) with parameters p(x) which are known to be monotonic in x i.e. p(x) ≤ p(y) for x ≤ y. Suppose for X = {x₁,…,x_M}, we’ve m(xi) samples of V(xi), then we can estimate p(x_i) as the average number of positive responses y_i. If the number of samples is small then these averages will not necessarily satisfy the monotonicity condition. Also of interest is given the trials, we would like to know where to sample from next in order to find an x which achieves a certain value of p(x).

Approaches. Maximum Likelihood Approach: MonFit
MonFit does a maximum likelihood estimation under the constraints that the underlying variables are monotonic. It’s equivalent to finding the closest monotonic series in the mean square sense.

It solves the optimization problem


under the constraints



The MonFit estimator has a closed form solution


where

Bayesian Approach: BayesFit

In Bayesfit, we assume a uniform prior on all monotonic sequences and compute the posterior distribution under the monotonic assumption. We then estimate the parameters as the posterior expected value.

The Posterior distribution of the parameters given the samples is calculated in [2] as

where is the estimate from MonFit.
The BayesFit estimate is given by


Results. Both BayesFit and MonFit are biased estimators. Figures 1 and 2 show the expected bias in the two estimators.

Figure 1. Bias in MonFit.

Figure 2. Bias in BayesFit

We compare the two estimators for a randomly chosen response function and a sin response function. In Figures 3 and 4, we show the two estimates when there are two samples per point.

Figure 3. BayesFit and MonFit estimates for a randomly chosen response function.

Figure 4. BayesFit and MonFit Estimates for Sin response function.

References
[1] Palmer S.E. Vision science: Photons to Phenomenology
[2] Barlow R.E. et al Statistical Inference under Order Restrictions