# BayesHistogram.jl

Optimal histogram binning based on piecewise constant model.

Paper: Studies in Astronomical Time Series Analysis. VI. Bayesian Block Representations [https://arxiv.org/abs/1207.5578]

## Introduction

Have you ever hated the default histogram binning rules in your favourite analysis and plotting library?

You can try to solve your problem by relying on BayesHistogram.jl! :)

This package provides the function bayesian_blocks, which determines the bin sequence that maximises the probability of observing your data, assuming it can be described by a histogram.

In other words, the package implements a complicated algorithm that returns the optimal histogram, respecting some simple constraints that can be customised.

If you can't take it any more, go directly to the usage examples or, for more details, to the file make_plot.jl or if you want to know all the internal details test/run_tests.jl.

The optimal histogram is determined by four ingredients:

1. The likelihood of a given bin.
2. The a-priori probability of observing that bin.
3. The maximum resolution at which you want to separate the data (default is Inf).
4. The minimum possible number of observations in each bin (default is 1).

How can you influence these factors?

1. It is not modifiable, for a long time now the Likelihood principle and its natural Bayesian extension have been part of the toolbox of every statistician: it is a solid principle that can be reasonably trusted :)

2. It is modifiable, the package implements a wide choice of possibilities, the prior can be chosen among the following alternatives:

• BIC (default): Bayesian information criterion, requires no parameters and is asymptotically consistent.
• AIC: Akaike information criterion: minimizes prediction error, requires no parameters (in some cases adds too many bins, but the problem can be solved using (2) and (3)).
• HQIC: Hannan-Quinn criterion, has intermediate behaviour between BIC and AIC, is close to consistency, tries to minimise prediction error.
• FPR(p): Scargle Criterion, a data bin is added if it has a false positive rate lower than p.
• Geometric(gamma): varying the parameter gamma changes the average number of bins to be observed.
• Pearson(p): this is useful when you want bins containing about N*p observations, where N is the total number of events.
• NoPrior: for non-Bayesians, always requires the tuning of (3) and (4).
• ?, You can implement a customised prior by following the examples in the src/priors.jl file
3. If you need to alter this parameter, simply add the keyword argument resolution = ? to the bayesian_blocks function, a typical value might be 100.

4. Similar to (3), this can be configured via min_counts = ?.

Thank you for reading this (brief?) introductory section.

## Installation

using Pkg


## Usage examples

for looking at every option available type in the repl ?bayesian_blocks.

Examples:

using Plots, BayesHistogram
X = exp.(randn(5000)./3)
bl = bayesian_blocks(X)
# plot using "to_pdf"
support, density = to_pdf(bl)
plot(support, density)
# or using "edges" parameter and an histogramming procedure
histogram(X, bins=bl.edges, normalize = :pdf)


we can change the prior as follows:

bl = bayesian_blocks(X, prior=AIC(), resolution=40)
plot(to_pdf(bl)...)
bl = bayesian_blocks(X, prior=FPR(0.2), resolution=40)
plot!(to_pdf(bl)...)


we can also plot the errors:

plot(to_pdf(bl)..., color="black")
scatter!(bl.centers, bl.heights, yerr = bl.error_heights, color="black")


we can also estimate averages and their errors:

estimate(bl) do v; v end
# result: (1.0610279949641472, 0.014646660658986687)

# the result can be refined by increasing the number of integration points

estimate(bl,100) do v; v^2 end
# result: (1.2574274634942957, 0.021142852215391358)