Pathwise Shrinkage and Probabilistic Calibration of the Multiplicative Half-Cauchy Process

Siddhesh Kulkarni

stat.MEOct 15, 2026

Pathwise Shrinkage and Probabilistic Calibration of the Multiplicative Half-Cauchy Process

Authors: Siddhesh Kulkarni

Organizations: a Bristol Myers Squibb, 5 Giralda Farms, Madison, NJ 07940, USA

Abstract

We characterize the multiplicative half-Cauchy process through its log-scale random-walk representation, establishing shrinkage, expansion, and critical regimes, almost-sure summability, local rebounds, and effective dimension. We also derive a semi-analytic calibration rule linking interpretable prior probability statements to its scale parameter.

Figures & tables

$p$	Calibrated $\zeta$	$\mathbb{E}(N_{0.05})$
0.50	0.7169	20.63
0.75	0.5077	7.60
0.90	0.3694	4.75

Table 1: Effect of the target probability on MHCP calibration and threshold-based effective dimension for

H=10

and

\varepsilon=0.05

Figure 1: MHCP pathwise behavior and calibration. (a) Ten simulated log-scale trajectories in shrinking (

\zeta=0.7

), critical (

\zeta=1

), and expanding (

\zeta=1.3

) regimes, with theoretical median trends; simulation seed 2026. (b) Values of

\zeta

satisfying

P(\eta_{H}\leq 0.05)=p

for

p=0.50

0.75

, and

0.90

. At

H=10

, the corresponding values are

0.7169

0.5077

, and

0.3694

. The dotted line marks

\zeta=1

; the open circle denotes

(H,p)=(10,0.90)

Bayesian Inference

Bayesian inference updates probability distributions over unknowns with evidence to quantify uncertainty and guide prediction and decisions.

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Split conformal prediction provides finite-sample marginal coverage under exchangeability, but this guarantee averages over the random calibration sample. We study instead the law of the calibration-conditional coverage induced by a realized conformal threshold. In the continuous i.i.d. setting this law is exactly

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Abstract

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