σα̂ σβ̂

m = \sqrt{2E_{1}E_{2}(1 − cos θ_{12})}

P (n|s, b) = \frac{(s + b)^{n}}{n!} e^{−(s+b)}

Q = −2 ln \frac{L_{s+b}}{L_{b}} = 2s − 2 ∑_{i=1}^{n} ln [1 + \frac{s}{b} \frac{f(x_{i}|s)}{f(x_{i}|b)}]

λ(μ) = \frac{L(μ)}{L(μ̂)}

μ̂ = argmax_{μ} L(μ)

p = 1 − Φ (\sqrt{q_{0}}) Z = \sqrt{q_{0}} = μ̂/σμ̂

L(μ) = \frac{(μs + b)^{n}}{n!} e^{−(μs+b)} ∏_{i=1}^{n} f(x_{i}|μ)

f(x|μ) = \frac{μs}{μs + b}f(x|s) + \frac{b}{μs + b}f(x|b)

x_{1}, …, x_{n} n ∼ Poisson(μs + b)

y(x) = \frac{p(x|s)}{p(x|b)}

\frac{p(x|s)}{p(x|b)} > c

t_{p}(x) = \frac{L_{p}(x|s)}{L_{p}(x|b)} = \frac{L(x|ˆν̂(s), s)}{L(x|ˆν̂(b), b)}

t_{m}(x) = \frac{L_{m}(x|s)}{L_{m}(x|b)} = \frac{∫ L(x|ν, s)π_{ν}(ν) dν}{∫ L(x|ν, b)π_{ν}(ν) dν}

P (θ < θ_{up}(x)|θ) = ? P (θ < θ_{up}|x) = ∫_{−∞}^{θ_{up}} p(θ|x) dθ = 95%

π_{ν}(ν) = π(ν|y) ∝ L(y|ν)π_{0}(ν) p(θ, ν|x) ∝ L(x|θ, ν)π_{θ}(θ)π_{ν}(ν)

p(θ, ν|x, y) ∝ L(x|θ, ν)L(y|ν)π_{θ}(θ)π_{0}(ν)

p(θ|x) = ∫ p(θ, ν|x) dν ∝ ∫ L(x|θ, ν)π_{ν}(ν)π_{θ}(θ) dν = L_{m}(x|θ)π_{θ}(θ)

L_{m}(x|θ) = ∫ L(x|θ, ν) π_{ν}(ν) dν

p(θ|x) = ∫ p(θ, ν|x) dν p(θ, ν|x) ∝ L(x|θ, ν)π(θ, ν)

π_{θ}(θ)

π_{ν}(ν)

π(θ, ν) = π_{θ}(θ)π_{ν}(ν)

π(θ, ν) p_{θ} = ∫_{t_{θ,obs}}^{∞} f(t_{θ}|θ, ν) dt_{θ}

t_{θ} = −2 ln \frac{L(θ, ˆ ν̂(θ))}{L(θ̂, ν̂)}

L_{p}(θ) = L(θ, ˆ ν̂(θ))

ˆν̂(θ) = argmax_{ν} L(θ, ν)

L(θ) L(x|θ)

L_{p}(μ) = L(μ, ˆ θ̂(μ)) L_{m}(μ) = ∫ L(μ, θ)π_{θ}(θ) dθ

L(L_{1}, L_{2}, T, τ̃_{0}, α̃_{1}, α̃_{2}|λ, τ, τ_{0}, α_{1}, α_{2}) =

\frac{1}{\sqrt{2π}σ_{T}} e^{−(T −τ)^{2}/2σ_{T}^{2}} ∏_{i=1}^{2} \frac{1}{\sqrt{2π}σ_{i}} e^{−(L_{i}−λ+α_{i}(τ−τ_{0}))^{2}/2σ_{i}^{2}}

×\frac{1}{\sqrt{2π}στ̃_{0}} e^{−(τ̃_{0}−τ_{0})^{2}/2σ2 ; τ̃_{0}}

∏_{i=1}^{2} \frac{1}{\sqrt{2π}σα̃_{i}} e^{−(α̃_{i}−α_{i})^{2}/2σ2 ; α̃_{i}}

cov[y_{1}, y_{2}] = α_{1}α_{2}σ_{T}^{2} cov[y_{i}, T] = α_{i}σ_{T}^{2} y_{i} = L_{i} + α_{i}(T − τ_{0})

L(T, L_{1}, L_{2}|λ, τ) = \frac{1}{\sqrt{2π}σ_{T}} e^{−(T −τ)^{2}/2σ_{T}^{2}} ∏_{i=1}^{2} \frac{1}{\sqrt{2π}σ_{i}} e^{−(L_{i}−λ+α_{i}(τ−τ_{0}))^{2}/2σ_{i}^{2}}

L_{i} ∼ Gauss(λ − α_{i}(τ − τ_{0}), σ_{i})

T ∼ Gauss(τ, σ_{T})

E[L_{i}] = λ − α_{i}(T − τ_{0}), i = 1, 2

cov[θ_{i}, θ_{j}] = ∫ θ_{i}θ_{j}p(θ|x) dθ − ∫ θ_{i}p(θ|x) dθ ∫ θ_{j}p(θ|x) dθ

ρ_{xy} = \frac{cov[x, y]}{σ_{x}σ_{y}}

cov[x, y] = E[xy] − E[x]E[y]

t_{θ} = −2 ln \frac{L(θ)}{L(θ̂)}

V̂_{ij}^{−1} ≈ −\frac{∂^{2} ln L}{∂θ_{i} ∂θ_{j}} |_{θ=θ̂}

b = E[θ̂] − θ

V_{ij} = cov[θ̂_{i}, θ̂_{j}]

θ̂ = argmax_{θ} L(x|θ)

p(θ|x) = \frac{L(x|θ)π(θ)}{∫ L(x|θ)π(θ) dθ}

Z = \sqrt{2 ((s + b) ln (1 + \frac{s}{b}) − s)}

t_{float} = −2 ln \frac{L(0)}{L(μ̂, m̂)} ≡ −2 ln λ(0)

χ^{2}(μ) + τ ∑_{i} [(μ_{i+1} − μ_{i}) − (μ_{i} − μ_{i−1})^{2}]

χ^{2} = ∑_{i=1}^{N} \frac{(ν_{0i} − n_{i})^{2}}{ν_{0i}}

ν_{0i} = ∑_{j=1}^{M} R_{ij}μ_{0j} + β_{i}

Φ(μ) = α (ln L(μ) + ln L_{θ}(θ)) + S(μ)

C_{i} = 0.1

n_{i} = 100

β_{i} = 0

μ̂_{i} = C_{i}n_{i} = 10

σμ̂_{i} = C_{i}\sqrt{n_{i}} = 1.0

E[μ̂] = R^{−1}(E[n] − β) = μ

U_{ij} = cov[μ̂_{i}, μ̂_{j}] = ∑_{k,l=1}^{N}(R^{−1})_{ik}(R^{−1})_{jl} cov[n_{k}, n_{l}]

= ∑_{k=1}^{N}(R^{−1})_{ik}(R^{−1})_{jk} ν_{k}

\frac{∂^{2} ln L}{∂μ^{2}} = −\frac{n}{μ^{2}}

ln L(μ) = ∑_{i=1}^{N}(n_{i} ln ν_{i} − ν_{i})

ν̂ = n

μ̂ = R^{−1}(n − β)

P (n_{i}; ν_{i}) = \frac{ν_{i}^{n_{i}}}{n_{i}!} e^{−ν_{i}}

μ = R^{−1}(ν − β)

μ = (μ_{1}, …, μ_{M}), μ_{tot} = ∑_{j=1}^{M} μ_{j}

p = (p_{1}, …, p_{M}) = μ/μ_{tot}

ν = (ν_{1}, …, ν_{N})

ε_{j} = ∑_{i=1}^{N} R_{ij}

β = (β_{1}, …, β_{N})

E[n] = ν = Rμ + β ∑_{i=1}^{N} R_{ij} = ∑_{i=1}^{N} P (observed in bin i | true value in bin j)

= P (observed anywhere | true value in bin j) = ε_{j}

ν_{i} = ∑_{j=1}^{M} R_{ij}μ_{j} + β_{i}

n = (n_{1}, …, n_{N})

ν_{i} = E[n_{i}] = ∑_{j=1}^{M} R_{ij}μ_{j} , i = 1, …, N

R_{ij} = P (observed in bin i | true in bin j) f_{meas}(x) = ∫ R(x|y)f_{true}(y) dy

p_{j} = ∫_{bin j} f(y) dy
μ_{j} = μ_{tot}p_{j}

π(b) = \frac{1}{\sqrt{2π}σ_{b}} e^{−(b−b_{meas})^{2}/2σ_{b}^{2}}

p_{float} ≈ p_{fix} + ⟨N(c)⟩ c = t_{fix} = Z_{fix}^{2} Z_{fix} = Φ^{−1}(1 − p_{fix})

N = \frac{⟨N(c)⟩}{1 − F_{χ_{2}^{2}}(c)}

⟨N(c)⟩ ≈ ⟨N(c_{0})⟩e^{−(c−c_{0})/2} F_{trials} ≡ \frac{p_{float}}{p_{fix}} ≈ \sqrt{\frac{π}{2}} \frac{⟨N(t_{float,obs})⟩}{1 − F_{χ_{2}^{2}}(t_{float,obs})}Z_{fix}

F_{trials} ≡ \frac{p_{float}}{p_{fix}} ≈ \sqrt{\frac{π}{2}} N Z_{fix}

P (t_{float} > t_{float,obs}|H_{0}) ≤ 1 − F_{χ_{1}^{2}}(t_{float,obs}) + ⟨N(t_{float,obs})⟩

−2 ln λ(μ) = −2 ln(L(μ)/L(μ̂)) < 1 i.e., ln L(μ) > ln L(μ̂) − \frac{1}{2}

\frac{P (x|H_{1})}{P (x|H_{0})} ≥ k

x ∼ Gauss(μ, σ)

α = 1 − Φ (\frac{x_{c} − μ_{0}}{σ} )

power = 1 − β = P (x > x_{c}|μ) = 1 − Φ (\frac{μ_{0} − μ}{σ} + Φ^{−1}(1 − α))

x_{c} = μ_{0} + σΦ^{−1}(1 − α)

L(x, y|θ, ν) = L(x|θ, ν)L(y|ν)

π(ν|y) ∝ L(y|ν)π_{0}(ν)

L_{m}(x|θ) = ∫ L(x|θ, ν)π(ν) dν

ˆν̂(θ)

p_{θ} = ∫_{q_{θ,obs}}^{∞} f(q_{θ}|θ, ν) dq_{θ}

L(x|θ) → L(x|θ, ν)

L(x|θ) = θx

L(x|θ) = θx + αx^{2} + βx^{3} + ⋯

p_{θ} = ∫_{q_{θ,obs}}^{∞} f(q_{θ}|θ) dq_{θ}

p(s|n) = ∫ p(s, b|n) db

p(x|H_{0}), p(x|H_{1})

s_{up} = \frac{1}{2}F_{χ^{2}}^{−1}(1 − α; 2(n + 1)) − b

α

s_{up} = − ln α ≈ 3.00

P (D) = 0.001
P (no D) = 0.999

P (+|D) = 0.98
P (−|D) = 0.02

P (+|no D) = 0.03
P (−|no D) = 0.97

p(D|+) = \frac{P (+|D)P (D)}{P (+|D)P (D) + P (+|no D)P (no D)}

= \frac{0.98 × 0.001}{0.98 × 0.001 + 0.03 × 0.999}
= 0.032