Bayesian Expectation of the Mean Power of Several Gaussian Data
Author
Giovanni Mana, Carlo Palmisano
Title
Bayesian Expectation of the Mean Power of Several Gaussian Data
Description
Given measurement results affected by additive uncorrelated Gaussian errors, the notebook investigates the Bayesian inferences of the data means, individual means’ squares, and average means’ squares.
Category
Academic Articles & Supplements
Keywords
Bayesian Analysis, Stein paradox
URL
http://www.notebookarchive.org/2024-09-4mgtyvd/
DOI
https://notebookarchive.org/2024-09-4mgtyvd
Date Added
2024-09-10
Date Last Modified
2024-09-10
File Size
1.65 megabytes
Supplements
Rights
CC BY 4.0
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This file contains supplementary data for Mana, G., Palmisano, C. Bayesian inference of the mean power of several Gaussian data. Eur. Phys. J. B 97, 93 (2024). https://www.doi.org/10.1140/epjb/s10051-024-00737-w.
Bayesian Expectation of the Mean Power of Several Gaussian Data
Bayesian Expectation of the Mean Power of Several Gaussian Data
Giovanni Mana and Carlo Palmisano
1,2
3
1) INRIM – Istituto Nazionale di Ricerca Metrologica, Torino, Italy
2) UNITO – Università di Torino, Dipartimento di Fisica, Torino, Italy
3) DMA – Diagnostic Monitoring Applications, Torino, Italy
2) UNITO – Università di Torino, Dipartimento di Fisica, Torino, Italy
3) DMA – Diagnostic Monitoring Applications, Torino, Italy
4.1 Posteriors of the instantaneous signals
4.1 Posteriors of the instantaneous signals
data likelihood, N(xi|μi), and prior distribution, π(μi|a,b) = N(μi|a,b)
data likelihood, N(|), and prior distribution, |a,b) = N(|a,b)
x
i
μ
i
π(
μ
i
μ
i
marginal likelihood Z(xi|a,b)
marginal likelihood |a,b)
Z(
x
i
posterior distribution μi|xi,a,b ∼ N(μi|μi,σμ)
posterior distribution |,a,b ∼ N(|,)
μ
i
x
i
μ
i
μ
i
σ
μ
4.2 hyper-prior
4.2 hyper-prior
marginal likelihood Z(x|a,b)
marginal likelihood Z(x|a,b)
Jeffreys’ hyper-prior
Jeffreys’ hyper-prior
4.3 model probabilities
4.3 model probabilities
model probabilities Q(a,b|x,2sx)
model probabilities Q(a,b|,)
x
2
s
x
parameter values maximising the model probability Q(a,b|x,2sx)
parameter values maximising the model probability Q(a,b|,)
x
2
s
x
4.4 Expectations of the instantaneous signals
4.4 Expectations of the instantaneous signals
4.4.1 m = 1 case
4.4.1 m = 1 case
parameter values maximising the model probability Q(a,b|x,2sx)
parameter values maximising the model probability Q(a,b|,)
x
2
s
x
Figure 1 top, m = 1
Figure 1 top, m = 1
Figure 1 bottom, m = 20, sx = 2
Figure 1 bottom, m = 20, = 2
s
x
4.4.2 m ≥ 2 case, model averaged posterior mean E(μi|xi,x,2sx)
4.4.2 m ≥ 2 case, model averaged posterior mean
E(|,,)
μ
i
x
i
x
2
s
x
4.4.2 m ≥ 2 case, asymptotic expressions
4.4.2 m ≥ 2 case, asymptotic expressions
Figure 2
Figure 2
asymptotic mean and variance of 2sxm∑i12(xi-x)m
asymptotic mean and variance of m
2
s
x
m
∑
i1
2
(-)
x
i
x
4.5 James-Stein estimate
4.5 James-Stein estimate
4.6 Expectations of the instantaneous powers
4.6 Expectations of the instantaneous powers
model averaged posterior mean E(2μi|xi,x,2sx)
model averaged posterior mean E(|,,)
2
μ
i
x
i
x
2
s
x
asymptotic expressions
asymptotic expressions
Figure 3
Figure 3
4.7 Expectation of the mean power
4.7 Expectation of the mean power
expectation of the mean power
expectation of the mean power
variance of the mean power
variance of the mean power
averaged mean
averaged mean
asymptotic expressions
asymptotic expressions
Figure 4
Figure 4
5 Application examples
5 Application examples
Input data
E Tiesinga et al J. Phys. Chem. Ref. Data 50, 033105 (2021), P J Mohr et al 2018 Metrologia 55 125
Input data
E Tiesinga et al J. Phys. Chem. Ref. Data 50, 033105 (2021), P J Mohr et al 2018 Metrologia 55 125
E Tiesinga et al J. Phys. Chem. Ref. Data 50, 033105 (2021), P J Mohr et al 2018 Metrologia 55 125
Clear["Global`*"];Remove["Global`*"];x=1*^3{6.67248,6.67290,6.67398,6.674255,6.67559,6.67422,6.67387,6.67222,6.67425,6.67349,6.67554,6.67191,6.67435,6.674184,6.67484,6.67260};ux=1.*^-2{43,50,70,9.2,27,98,27,87,12,18,16,99,13,7.8,7.7,25};m=Length[x];y=Array[#&,m];σx2=1/Total[1./];=σx2Total[x/];data=x-;data=MapThread[Around,{data,ux}];data=Transpose[{data,y}];F1=ListPlotdata,PlotRange{{-4,4},{0,m+1}},ImageSize400,FrameTrue,AxesFalse,FrameTicks{Automatic,None},FrameLabel"(G - ) ",None,BaseStyle{FontSize12},PlotMarkers{Automatic,6},PlotStyleThickness[0.003],ImageMargins5
2
ux
x
2
ux
x
3
10
G
0
-11
10
-1
kg
3
m
2
s
x=1*^6{6.626070133`,6.626070405`,6.626069934`,6.62607022`,6.62607013`,6.62606994`,6.6260704`,6.62606936`,6.62606891`};ux=1.*^-3{60,77,88,130,160,200,380,380,580};m=Length[x];y=Array[#&,m];σx2=1/Total[1./];=σx2Total[x/];data=x-;data=MapThread[Around,{data,ux}];data=Transpose[{data,y}];F2=ListPlotdata,PlotRange{{-2,1},{0,m+1}},ImageSize400,FrameTrue,AxesFalse,FrameTicks{Automatic,None},FrameLabel"(h - )( J s)",None,BaseStyle{FontSize12},PlotMarkers{Automatic,6},PlotStyleThickness[0.003],ImageMargins5
2
ux
x
2
ux
x
6
10
h
0
-34
10
x=1*^6{1.3806488`,1.38064862`,1.3806487`,1.3806509`,1.3806477`,1.3806502`,1.3806482`,1.3806484`,1.3806497`,1.3806497`,1.3806498`,1.3806516`,1.380643`,1.3806467`};ux=1.*^-2{83,96,140,150,190,250,270,280,370,380,440,530,690,930};m=Length[x];y=Array[#&,m];σx2=1/Total[1./];=σx2Total[x/];data=x-;data=MapThread[Around,{data,ux}];data=Transpose[{data,y}];F3=ListPlotdata,PlotRange{{-15,10},{0,m+1}},ImageSize400,FrameTrue,AxesFalse,FrameTicks{Automatic,None},FrameLabel"(k - )( J/K)",None,BaseStyle{FontSize12},PlotMarkers{Automatic,6},PlotStyleThickness[0.003],ImageMargins5
2
ux
x
2
ux
x
6
10
k
0
-23
10
xyz
xyz
σx2=1/Total[1./];=σx2Total[x/];data=x-;=σx2Total[data/]χ2=Total;L0=Z0=(1+)Print["Z(x|H0) = ",Z0/.{a->0,b->1/Sqrt[3]}];
2
ux
x
2
ux
x
x
2
ux
2
(data-)
x
2
ux
-
2
μ
2
m
(2π)
-
χ2
2
-
2
a
21+
2
b
m
(2π)
2
b
-
χ2
2
6.85269×
-12
10
3.79241×
-7
10
-
2
μ
2
3.79241×
-7
10
-
2
a
21+
2
b
1+
2
b
Z(x|H0) = 3.28432×
-7
10
5.1 H0 hypothesis
5.1 H0 hypothesis
σx2=1/Total[1./];=σx2Total[x/];σx=Sqrt[ux^2+σx2];data=(x-)/σx;=Mean[data];χ2=Total[];Z0=;Print["Z(x|H0) = ",Z0];
2
ux
x
2
ux
x
x
2
(data-)
x
-
m
2
x
2
m
(2π)
-
χ2
2
Z(x|H0) = 4.25568×
-7
10
5.2 H1 hypothesis
5.2 H1 hypothesis
σx2=1/Total[1./];=σx2Total[x/];data=(x-)/ux;(*Mathematica'svarianceestimateis*)vx=Variance[data];=Mean[data];Q=b/.{a->};b0=b/.FindMaximum[{Q,b>0},{b,1}][[2]];Z1=/.{a->,b->b0};Print["Z(x|H1) = ",Z1]Print["{Prob(H0|x), Prob(H1|x)} = ",{Z0,Z1}/(Z0+Z1)]
2
ux
x
2
ux
x
1
m-1
n
∑
i=1
2
-
x
i
μ
m-1
m
x
-
m+mvx
2
(a-)
x
21+
2
b
(3+m)
(1+)
2
b
x
-
m+vx
2
(a-)
x
21+
2
b
m
(1+)
2
b
m
(2π)
x
Z(x|H1) = 2.54689×
-7
10
{Prob(H0|x), Prob(H1|x)} = {0.625599,0.374401}
5.3 Results
5.3 Results
Cite this as: Giovanni Mana, Carlo Palmisano, "Bayesian Expectation of the Mean Power of Several Gaussian Data" from the Notebook Archive (2024), https://notebookarchive.org/2024-09-4mgtyvd
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