Interaction Order Estimation in Tensor Curie–Weiss Models
In this paper, we consider the problem of estimating the interaction parameter <i>p</i> of a <i>p</i>-spin Curie–Weiss model at inverse temperature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics&...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-02-01
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| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/27/3/245 |
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| Summary: | In this paper, we consider the problem of estimating the interaction parameter <i>p</i> of a <i>p</i>-spin Curie–Weiss model at inverse temperature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, given a single observation from this model. We show, by a contiguity argument, that joint estimation of the parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> and <i>p</i> is impossible, which implies that the estimation of <i>p</i> is impossible if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is unknown. These impossibility results are also extended to the more general <i>p</i>-spin Erdös–Rényi Ising model. The situation is more delicate when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is known. In this case, we show that there exists an increasing threshold function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>β</mi><mo>*</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, such that for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, consistent estimation of <i>p</i> is impossible when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>β</mi><mo>*</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>></mo><mi>β</mi></mrow></semantics></math></inline-formula>, and for <i>almost all</i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, consistent estimation of <i>p</i> is possible for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>β</mi><mo>*</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo><</mo><mi>β</mi></mrow></semantics></math></inline-formula>. |
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| ISSN: | 1099-4300 |