S&T2011, Vienna - Austria,June 8-10, 2011

Bayesian inference to study low-levelradioactivity in the environment:

Application to the detection of xenonisotopes of interest for the CTBTO

I. Rivals1 , C. Fabbri1, G. Euvrard1 & X. Blanchard2

1 Équipe de Statistique Appliquée, ESPCI ParisTech, France2 CEA, DAM, DIF, France

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Introduction

• The decision that a given detection level corresponds to the effectivepresence of a radionuclide, or to its absence, is widely made using aclassic hypothesis test (Currie).• Shortcomings of the classic framework / promising perspectivesof Bayesian statistics (part of recent standards [ISO 11929-7, 2005])

⇒ we propose a novel Bayesian approach providing estimates of:

– the probability of zero radioactivity, together with physicallymeaningful point and interval estimates,

– the prior density of the radioactivity, obtained by fitting previouslyrecorded radioactivity data.

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The classic framework [Currie, 1968]True net radioactivity level µ?Observed net count : x = xg – xbwith blank count xb → Poisson(µb) and gross count xg → Poisson(µ + µb)Independence ⇒ E(x) = µ, var(x) = σ2 = µ + 2 µbLarge enough counting time ⇒ ≈ Gaussian CDF:

Test of H0: µ = 0 against H1: µ > 0 with a type I error (false alarm) risk αCurrie’s critical level LC:

• if x > lc, H0 is rejected with error risk α . 1 – α confidence interval:

⇒ the confidence interval may include negative values

• if x ≤ lc, H0 is accepted. Currie’s detection limit LD:

⇒ unknown probability of missing a real event (type II error)

No use of past observed values of x and xb.

µ !x ±"#1(1# $ 2) xb + xg

LC = !"1(1" #) 2xb

LD = 2 !LC + "#1(1# $)( )2

F(x) = ! x " µ

#$%&

'()

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The Bayesian framework (1)True radioactivity µ ≡ random variable⇒ “H0 is true” (µ = 0) and “H1 is true” (µ > 0) form a complete set of events,with a priori and a posteriori (given the observed net count x) probabilities:

Medical diagnosis– H0 ≡ “healthy”, H1 ≡ “sick”, x ≡ result of a medical test (“+” or “–”)– the a priori probability P(H0) and the conditional probabilities f(x|Hk) ≡P(x|Hk) are estimated with the frequencies observed on a largerepresentative sample of patients ⇒ no theoretical problem

Radioactivity detectionReal-valued x depending on Hk, i.e. on µ ⇒ conditional densities f(x|Hk)

– Θ0 = {0} (under H0) and Θ1 = ] 0 ; +∞[ (under H1)– f(x|µ) : the Gaussian density (= ϕ( (x–µ)/σ) )– π(µ|Hk): a priori density of the activity µ under Hk

priorprobabilities

posteriorprobability

f(x |Hk ) = f(x |µ)!(µ |Hk )dµ

µ"#k$

P(Hi | x) =f(x |Hi)P(Hi)f(x |Hk )P(Hk )

k=0,1!

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The Bayesian framework (2)Denominator of the a posteriori probabilities = prior density of the trueradioactivity µ:

If π(µ) can be estimated:• marginal density of the net count x:

• posterior probability of H0 (no radioactivity):

• posterior density µ:

• point estimate of the true radioactivity:

• 1 – γ credibility interval [µ– ; µ+]:

f(µ | x) = f(x |µ)!(µ)

f(x)

f(x) = f(x |µ)!(µ)dµ"

P(H0 | x) =

f(x |µ = 0)P(H0 )f(x)

µ* = E(µ | x) = µf(µ | x)dµ!

!2= f(µ | x)dµ

µ+

+"

# = f(µ | x)dµ$"

µ$

#

!(µ) = !(µ |Hk )P(Hk )

k=0,1"

positive valuesonly

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Bayesian priors (1)The prior density of the true radioactivity µ should be given by:

Improper prior approach [Zähringer & Kirchner, 2008]:

– no Dirac peak at zero ⇒ P(H0|x) = 0 whatever the observed x– not integrable ⇒ the marginal density f(x) cannot be estimated

Implicit prior approach [Vivier et al., 2009]:– P(H0|x) = 1 – Φ(x/σ) ⇒ P(H0|x=0) = 0.5 whatever σ (depends on µb)– no explicit prior ⇒ the marginal density f(x) cannot be estimated

!(µ) = I[0;+"[ (µ)µ

1

Dirac peakat zero

probability of no radioactivity radioactivity density under H1

!(µ) = P(H0 ) "0(µ) + P(H1) !(µ |H1)

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Bayesian priors (2)Proposed prior (proper):

Possible functional forms for π(µ|H1) :

• uniform (d > 0):

• exponential (τ > 0):

• half-Gaussian (λ > 0):

We calculated all the Bayesian estimates for these three priors.

Estimate P(H0) and π(µ|H1), i.e. the parameter d, τ or λ by fitting themarginal density f(x) = ∫ f(x|µ) π(µ) dµ to past records of observed x.

probability of no radioactivity radioactivity density under H1

!(µ) = P(H0 ) "0(µ) + P(H1) !(µ |H1)

Dirac peakat zero

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Empirical fit of the prior parameters• d, τ, and λ are merged into a single parameter, d, by taking τ and λ sothat d is the 95th centile of the prior density:• P(H0) ≡ p0

Expression of the marginal density of x (uniform prior):

In fact, σ2 = µ + 2µb. Numerical simulations show that all the results holdwhen σ2 varies by simply replacing σ2 by x + 2µb.

Fit f(x) to past records of x with maximum likelihood, two options:– as a function of p0, d and µb,– first estimate µb using measured values of xb ⇒ check whether µb ≈ cte then fit f(x) as a function of p0 and d only (chosen option).

f(x) =

p0

!" x

!#$%

&'(+

1) p0

d* x

!#$%

&'() * x ) d

!#$%

&'(

#

$%&

'(

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Experimental results: empirical fit of the priorsSix month of 2009 daily measurements for a dozen of CTBTO stations:– Hawaï station USX79 (SAUNA)– Xe131m data

• Poisson distribution of blank xb with µb ≈ 4

• ML fit of the marginal density f(x) of the observed net counts ⇒ good results with the uniform and exponential priors: there is an a priori probability p0 ≈ 2/3 of a small net radioactivity (d ≈ 5-6).

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Experimental results: Bayesian estimates

For each observed net count x:– estimate of the posterior probability of zero radioactivity P(H0|x)– point estimate µ*, credibility interval [µ+ ; µ–] for the true activity µ

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Experimental results: comparison to theclassic framework

The posterior probability of zero activity P(H0|x) is now shown as a functionof x – LC (LC is Currie’s critical level): x – LC > 0: reject H0, else accept H0

⇒ x – LC > 0 (reject H0) coincides with P(H0|x) < 0.5

In the Bayesian framework, H0 should be rejected when:P(H0|x) < CII/(CI+CII)

where CI and CII are the costs associated to type I (false alarm) and type II(missing a real event) errors respectively ⇒ quantifying these costswould define the decision threshold for CTBTO.

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ConclusionRealizationsNovel approach based on rigorous Bayesian principles (proper priors):– probability estimate of a truly radioactive sample– physically meaningful estimates of its radioactivity– a priori knowledge is taken into account via the fit of the prior to dataobserved in the past

OutlooksExtend the method to:• time-varying blank level• analysis of several isotopes jointly (Xe131m, Xe133m, Xe133, Xe135)• analysis of other fission and activation products (aerosols) of interestfor CTBTO

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Supplementary slide 1: references• L. A. Currie (1968)Limits for qualitative detection and quantitative determination; application toradiochemistryAnal. Chem. 335, 586-593.

• ISO 11929-7 (2005)Determination of the Detection Limit and Decision Threshold for IonizingRadiation Measurements, Part 7: Fundamentals and General Applications.

• M. Zähringer & G. Kirchner (2008)Nuclide ratios and source identification from high-resolution gamma-ray spectrawith Bayesian decision methods.Nuclear Instruments and Methods in Physics Research A 594, 400-406.

• A. Vivier, Gilbert Le Petit, B. Pigeon & X. Blanchard (2009)Probabilistic assessment for a sample to be radioactive or not: application toradioxenon analysisJournal of Radioanalytical and Nuclear Chemistry 282, 743–748.

• I. Rivals, C. Fabbri, G. Euvrard & X. BlanchardA Bayesian method with empirically fitted priors for the evaluation ofenvironmental radioactivity: application to low-level radioxenon measurementsSubmitted to Nuclear Instruments and Methods in Physics Research A.

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Supplementary slide 2: results for Stockholmstation SEX63

Constant low blank level µb ≈ 6Good fit with the exponential prior: p0 = 0.13 (low a priori probability of noradioactivity), d = 7.1 (small activity when present)⇒ consistent with the station location in northern Europe (vicinity of nuclearpower plants)

⇒ with a threshold of 0.5 for P(H0|x), the proposed Bayesian approachdetects more radioactive events than the classic approach.

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Supplementary slide 3: validity of the Gaussianapproximation even for very small counts

Let X → Poisson(a) and Y → Poisson(b), and Z = X – Y. Then:

Application to:– gross count: Xg → Poisson(µg = 8)– blank count: Xb → Poisson(µb = 5)– net count: X = Xg – Xb (µ = 3)

P(Z = z) = e!(a+b) a

b"#$

%&'

z/2

I|z| 2 ab( ) with Iz (u) = (u / 2)k (u / 2)2j

j!( j+ z)!j=0

+(

)