## Bayes and temperature

by GongJian

(sorry, this post is going to get little bit technical… we’ll find a way of signalling when this is going to happen)

Yvi was reading Wilson’s article in SciAm on the RG… and the first page, although apparently naive, contained something shocking. It had three pictures which showed the idea of an ordered phase, critical point and disordered phase in an Ising ferromagnet. You have three (other) pics right here…

The left image shows the aspect of a ferromagnet below the critical temp Tc, the second at Tc and the third above Tc. And the funny thing is that we can recognize, more or less, the temperature from a single snapshot of the system. BUT we’re always told that temp is not a property of a single configuration, it is always a property of the ensemble!! How come?

We have two events which we’ll consider to be random: T is to have a certain temperature, and C is to get a certain configuration. Then, the conditional probability P(C|T) is just the Boltzmann factor, 1/Z exp(-beta E(C)). What about P(T|C)? I mean: what is the probability of the temperature being a given one, when we observe a single configuration? Somehow, this question makes sense. Otherwise, Wilson would not have put the pictures, would he?

We can use Bayes theorem to “swap” the conditional probability:

P(T|C)·P(C)=P(C|T)·P(T)

So, P(T|C) = P(C|T)· P(T)/P(C), the problem, as always in bayesian statistics, is the a priori probability for the temperature… Of course, you can start with any distribution P_0(T), and then iterate that prescription as you’re given more and more configurations: P(T|C_1,C_2…C_n). My hypothesis is that this process will have a single fixed point, independently of the initial distribution, with some mild restrictions on its support. What do you think?

Thanks to noema, flor and migeru. The latter pointed to maximal entropy methods, still I would like to find the relation… Ah, and see this nice post I found.

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### 6 Responses to Bayes and temperature

1. Erynus dice:

Esto me recuerda al experimento que hice en mi blog sobre un “test de Rorschach” basado en una imagen obtenida de “colorear” el primer millon de decimales de Pi. La explicacion la tengo aqui http://erynus.blogspot.com/2007/07/pintando-con-pi.html
y la imagen al final de este post: http://erynus.blogspot.com/2007/07/pintando-con-pi.html .

2. Erynus dice:

Mis disculpas, el segundo link es este: http://erynus.blogspot.com/2007/07/shai-dorsai.html
Sorry.

3. webjinni dice:

Muy bueno, Erynus. Me figuro que se podría hacer algo mejor cambiando de base: quiero decir, la base 10 no tiene nada de especial, sólo que tenemos 10 dedos! 🙂 ¿Quizás en binario? Prometo pensarlo…

4. Erynus dice:

Yo pense hacerlo al principio en 256 colores. Pero la sustitucion es un coñazo. Si algun dia tengo tiempo y ganas, experimento.

5. webjinni dice:

Otro problema para mí es el tamaño de la matriz: ¿por qué NxN, para algún N? Se me ocurre una idea mejor: una disposición de los dígitos así:

1
23
456

Eso no impondría un ancho fijo, cual lecho de Procusto 😉

6. […] I already made an entry of this here, but this one is explained […]