Posts tagged ‘prior’

#### The chance that A has nukes is p%

I watched a movie in which one of the characters said, “country A has nukes with 80% chance” (perhaps, not 80% but it was a high percentage). One of the statements in that episode is that people will not eat lettuce only if the 1% chance of e coli is reported, even lower. Therefore, with such a high percentage of having nukes, it is right to send troops to A. This episode immediately brought me a thought about astronomers’ null hypothesis probability and their ways of concluding chi-square goodness of fit tests, likelihood ratio tests, or F-tests.

First of all, I’d like to ask how you would like to estimate the chance of having nukes in a country? What this 80% implies here? But, before getting to the question, I’d like to discuss computing the chance of e coli infection, first. Continue reading ‘The chance that A has nukes is p%’ »

#### gamma function (Equation of the Week)

The gamma function [not the Gamma -- note upper-case G -- which is related to the factorial] is one of those insanely useful functions that after one finds out about it, one wonders “why haven’t we been using this all the time?” It is defined only on the positive non-negative real line, is a highly flexible function that can emulate almost any kind of skewness in a distribution, and is a perfect complement to the Poisson likelihood. In fact, it is the conjugate prior to the Poisson likelihood, and is therefore a natural choice for a prior in all cases that start off with counts. Continue reading ‘gamma function (Equation of the Week)’ »

#### [ArXiv] A fast Bayesian object detection

This is a quite long paper that I separated from [Arvix] 4th week, Feb. 2008:
[astro-ph:0802.3916] P. Carvalho, G. Rocha, & M.P.Hobso
A fast Bayesian approach to discrete object detection in astronomical datasets – PowellSnakes I
As the title suggests, it describes Bayesian source detection and provides me a chance to learn the foundation of source detection in astronomy. Continue reading ‘[ArXiv] A fast Bayesian object detection’ »

#### [ArXiv] 4th week, Nov. 2007

A piece of thought during my stay in Korea: As not many statisticians are interested in modern astronomy while they look for data driven problems, not many astronomers are learning up to date statistics while they borrow statistics in their data analysis. The frequency is quite low in astronomers citing statistical journals as little as statisticians introducing astronomical data driven problems. I wonder how other fields lowered such barriers decades ago.

No matter what, there are preprints from this week that may help to shrink the chasm. Continue reading ‘[ArXiv] 4th week, Nov. 2007’ »

A great advantage of Bayesian analysis, they say, is the ability to propagate the posterior. That is, if we derive a posterior probability distribution function for a parameter using one dataset, we can apply that as the prior when a new dataset comes along, and thereby improve our estimates of the parameter and shrink the error bars.

But how exactly does it work? I asked this of Tom Loredo in the context of some strange behavior of sequential applications of BEHR that Ian Evans had noticed (specifically that sequential applications of BEHR, using as prior the posterior from the preceding dataset, seemed to be dependent on the order in which the datasets were considered (which, as it happens, arose from approximating the posterior distribution before passing it on as the prior distribution to the next stage — a feature that now has been corrected)), and this is what he said:

Yes, this is a simple theorem. Suppose you have two data sets, D1 and D2, hypotheses H, and background info (model, etc.) I. Considering D2 to be the new piece of info, Bayes’s theorem is:

[1]

```p(H|D1,D2) = p(H|D1) p(D2|H, D1)            ||  I
-------------------
p(D2|D1)```

where the “|| I” on the right is the “Skilling conditional” indicating that all the probabilities share an “I” on the right of the conditioning solidus (in fact, they also share a D1).

We can instead consider D1 to be the new piece of info; BT then reads:

[2]

```p(H|D1,D2) = p(H|D2) p(D1|H, D2)            ||  I
-------------------
p(D1|D2)```

Now go back to [1], and use BT on the p(H|D1) factor:

```p(H|D1,D2) = p(H) p(D1|H) p(D2|H, D1)            ||  I
------------------------
p(D1) p(D2|D1)

= p(H, D1, D2)
------------      (by the product rule)
p(D1,D2)```

Do the same to [2]: use BT on the p(H|D2) factor:

```p(H|D1,D2) = p(H) p(D2|H) p(D1|H, D2)            ||  I
------------------------
p(D2) p(D1|D2)

= p(H, D1, D2)
------------      (by the product rule)
p(D1,D2)```

So the results from the two orderings are the same. In fact, in the Cox-Jaynes approach, the “axioms” of probability aren’t axioms, but get derived from desiderata that guarantee this kind of internal consistency of one’s calculations. So this is a very fundamental symmetry.

Note that you have to worry about possible dependence between the data (i.e., p(D2|H, D1) appears in [1], not just p(D2|H)). In practice, separate data are often independent (conditional on H), so p(D2|H, D1) = p(D2|H) (i.e., if you consider H as specified, then D1 tells you nothing about D2 that you don’t already know from H). This is the case, e.g., for basic iid normal data, or Poisson counts. But even in these cases dependences might arise, e.g., if there are nuisance parameters that are common for the two data sets (if you try to combine the info by multiplying *marginalized* posteriors, you may get into trouble; you may need to marginalize *after* multiplying if nuisance parameters are shared, or account for dependence some other way).

what if you had 3, 4, .. N observations? Does the order in which you apply BT affect the results?

No, as long as you use BT correctly and don’t ignore any dependences that might arise.

if not, is there a prescription on what is the Right Thing [TM] to do?

Always obey the laws of probability theory! 9-)

#### [ArXiv] Bayesian Star Formation Study, July 13, 2007

From arxiv/astro-ph:0707.2064v1
Star Formation via the Little Guy: A Bayesian Study of Ultracool Dwarf Imaging Surveys for Companions by P. R. Allen.

I rather skip all technical details on ultracool dwarfs and binary stars, reviews on star formation studies, like initial mass function (IMF), astronomical survey studies, which Allen gave a fair explanation in arxiv/astro-ph:0707.2064v1 but want to emphasize that based on simple Bayes’ rule and careful set-ups for likelihoods and priors according to data (ultracool dwarfs), quite informative conclusions were drawn:
Continue reading ‘[ArXiv] Bayesian Star Formation Study, July 13, 2007’ »