# Combining probability distributions

This table is yet unfinished.

If I observe the sum of two processes with known distributions, the distribution of the observations is expected to be …

 + Normal Poisson Binomial Uniform Normal Normal See here Poisson Poisson, normal if many summands Binomial Binomial, if common p, Poisson, if many summands, Poisson binomial otherwise, note also the binomial sum variance inequality and this. Uniform Irwin-Hall

Wikipedia has some more. A general discussion of probability distribution convolutions is for example here.

# Feature detectors in animal vision

Image feature detectors are a common concept between mammalian vision and computer vision. When using them, a raster image is not directly processed to identify complex objects (e.g. a flower, or the digit 2). Instead feature detectors map the distribution of simple figures (such as straight edges) within the image. Higher layers of the neural network then use these maps for distinguishing objects.

In the mammalian brain’s visual cortex (which is at the back of the head, at the furthest possible point from the eyes) the image on the retina is recreated as a spatially faithful projection of the excitation pattern on the retina. Overlapping sets of feature detectors use this as input.

From eyeball to visual cortex in humans. Note the Ray-Ban-shaped area at the back of the brain where the retinal excitation pattern is projected to with some distortions. (From Frisby: Seeing: the computational approach to biological vision (2010), p 5)

How we know about retinotopic projection to the visual cortex: an autoradiography of a macaque brain slice shows in dark the neurons that were most active in result of the animal seeing the image on top left. (From Tootell et al., Science (1982) 218, 902-904.)

A feature detector neuron becomes active when its favourite pattern shows up in the projected visual field – or more exactly in the area within the visual field where each detector is looking. A typical class of detectors is specific for edges with a specific angle, where one side is dark, and the other side is light. Other neurons recognise more complex patterns, and some also require motion for activation. These detectors together cover the entire visual field, and their excitation pattern is the input to higher layers of processing. We learned about these neurons first by sticking microelectrodes into the visual cortex and measuring electrical activity. When lucky, the electrode measured the activity of a single neuron; then by showing different visual stimuli the activation pattern of the neuron could be mapped.

A toad’s antiworm detector neuron reacts to a stripe moving across its receptive field. The antiworm may move in any direction, but only crosswise for the neuron to react. The worm detector, for comparison, would react if the stripe moves lengthwise. Toad at the right side with microelectrode, the oscillogram above the screen shows the tapped feature detector neuron’s activity. (Segment from Jörg-Peter Ewert, Gestalt Perception in the Common Toad – 3. Neuroethological Analysis of Prey Recognition.)

# The state of London, October 2017

Brixton, and from there a corridor towards the Thames ending at Vauxhall and Elephant and Castle are winning. The areas around Islington, Hackney and Greenwich are struggling. The Soho and the rest of Westminster keep doing well. This is at least what starting and failing food-related businesses tell about the last six months in London. I felt food is something everyone buys daily, and whether it is cheap or expensive, less or more, is a good indication of socioeconomic developments.

Increase and decrease of food-related businesses in London over the six month up to October 2017. (Map backgrounds are courtesy of Google Maps. Overlays: R, ggmaps.)