Joint estimation of parameters in Ising model
Joint estimation of parameters in Ising model
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DOI：
10.1214/19-aos1822
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发表时间：
2018-01
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通讯作者：
Promit Ghosal;S. Mukherjee
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文献类型：
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作者：
Promit Ghosal;S. Mukherjee
我们研究了具有非负耦合矩阵$A_n$的Ising模型的逆温度和磁化参数$（\beta，B）$的联合估计，其中A_n$的大小为$n\times n$，给出了Ising模型的一个样本。我们给出了二元伪随机估计的相合率的一般界。利用这一点，我们证明了当$A_n$是有界度图的邻接矩阵时，估计率$n^{-1/2}$总是可能的。如果$A_n$是一个平均度为$+\infty$的图的标度邻接矩阵，情况就稍微微妙一些。在这种情况下，如果图不是正则的（在渐近意义上），以速率$n^{-1/2}$估计仍然是可能的。最后，我们证明了如果图是Erd\“os-Renyi图且参数p>0 free of n，则这两个参数的一致估计是不可能的，从而证实了在度较大的近似正则图上估计是困难的.
We study joint estimation of the inverse temperature and magnetization parameters $(\beta,B)$ of an Ising model with a non-negative coupling matrix $A_n$ of size $n\times n$, given one sample from the Ising model. We give a general bound on the rate of consistency of the bi-variate pseudolikelihood estimator. Using this, we show that estimation at rate $n^{-1/2}$ is always possible if $A_n$ is the adjacency matrix of a bounded degree graph. If $A_n$ is the scaled adjacency matrix of a graph whose average degree goes to $+\infty$, the situation is a bit more delicate. In this case estimation at rate $n^{-1/2}$ is still possible if the graph is not regular (in an asymptotic sense). Finally, we show that consistent estimation of both parameters is impossible if the graph is Erd\"os-Renyi with parameter $p>0$ free of $n$, thus confirming that estimation is harder on approximately regular graphs with large degree.