The Efficiency of Next-Generation Gibbs-Type Samplers: An Illustration Using a Hierarchical Model in Cosmology

Abstract

Supernovae occur when a star’s life ends in a violent thermonuclear explosion, briefly outshining an entire galaxy before fading from view over a period of weeks or months. Because so-called Type Ia supernovae occur only in a particular physical scenario, their explosions have similar intrinsic brightnesses which allows us to accurately estimate their distances. This in turn allows us to constrain the parameters of cosmological models that characterize the expansion history of the universe. In this paper, we show how a cosmological model can be embedded into a Gaussian hierarchical model and fit using observations of Type Ia supernovae. The overall model is an ideal testing ground of new computational methods. Ancillarity-Sufficiency Interweaving Strategy (ASIS) and Partially Collapsed Gibbs (PCG) are effective tools to improve the convergence of Gibbs samplers. Besides using either of them alone, we can combine PCG and/or ASIS along with Metropolis-Hastings algorithm to simplify implementation and further improve convergence. We use four samplers to draw from the posterior distribution of the cosmological hierarchical model, and confirm the efficiency of both PCG and ASIS. Furthermore, we find that we can gain more efficiency by combining two or more strategies into one sampler.

Appendix

This appendix details the sampling steps of the MH within Gibbs, MH within PCG, ASIS and PCG within ASIS samplers, i.e., Samplers 1, 2, 3 and 4, for the cosmological hierarchical model. The posterior distribution of (ξ, X, Ω _m , Ω _Λ , α, β, R _c ², R _x ², σ _res ²) is

$$ \displaystyle\begin{array}{rcl} & & p(\xi,X,\varOmega _{m},\varOmega _{\varLambda },\alpha,\beta,R_{c}^{2},R_{ x}^{2},\sigma _{\mathrm{ res}}^{2}\vert Y ) \\ & & \quad \phantom{ \propto }\propto \vert \varSigma _{C}\vert ^{-1/2}\vert \varSigma _{ P}\vert ^{-1/2}\vert \varSigma _{ 0}\vert ^{-1/2}\mathrm{exp}\bigg\{ -\frac{1} {2}\left [(Y - AX - L)^{T}\varSigma _{ C}^{-1}(Y - AX - L)\right. \\ & & \quad \phantom{\propto } +\left.(X - J\xi )^{T}\varSigma _{ P}^{-1}(X - J\xi ) + (\xi -\xi _{ m})^{T}\varSigma _{ 0}^{-1}(\xi -\xi _{ m})\right ]\bigg\} \frac{1} {R_{c}^{2}} \frac{1} {R_{x}^{2}} \frac{1} {\sigma _{\mathrm{res}}^{2}}, {}\end{array}$$

(13)

where ξ _m  = (0, 0, M _m ) and $\varSigma _{0} =\mathrm{ Diag}(\sigma _{c_{0}}^{2},\sigma _{x_{0}}^{2},\sigma _{M_{0}}^{2})$; Σ _C , Σ _P , J, A and L are defined in Sect. 3.2. Setting $\tilde{X} = AX + L$, the joint posterior distribution of $(\xi,\tilde{X},\varOmega _{m},\varOmega _{\varLambda },\alpha,\beta,R_{c}^{2},R_{x}^{2},\sigma _{\mathrm{res}}^{2})$ is

$$\displaystyle\begin{array}{rcl} & & p(\xi,\tilde{X},\varOmega _{m},\varOmega _{\varLambda },\alpha,\beta,R_{c}^{2},R_{ x}^{2},\sigma _{\mathrm{ res}}^{2}\vert Y ) \\ & & \quad \propto \vert \varSigma _{C}\vert ^{-1/2}\vert \varSigma _{ P}\vert ^{-1/2}\vert \varSigma _{ 0}\vert ^{-1/2}\mathrm{exp}\bigg\{ -\frac{1} {2}\Big[(Y -\tilde{ X})^{T}\varSigma _{ C}^{-1}(Y -\tilde{ X}) \\ & & \quad +\left.\left.(A^{-1}\tilde{X} - A^{-1}L - J\xi )^{T}\varSigma _{ P}^{-1}(A^{-1}\tilde{X} - A^{-1}L - J\xi )\right.\right. \\ & & \quad +(\xi -\xi _{m})^{T}\varSigma _{ 0}^{-1}(\xi -\xi _{ m})\Big]\bigg\} \frac{1} {R_{c}^{2}} \frac{1} {R_{x}^{2}} \frac{1} {\sigma _{\mathrm{res}}^{2}}. {}\end{array}$$

(14)

Furthermore, integrating out (ξ, X), the marginal distribution of (Ω _m , Ω _Λ , α, β, R _c ², R _x ², σ _res ²) is

$$\displaystyle\begin{array}{rcl} p(\varOmega _{m},& & \varOmega _{\varLambda },\alpha,\beta,R_{c}^{2},R_{ x}^{2},\sigma _{\mathrm{ res}}^{2}\vert Y ) \\ & & \phantom{\propto }\propto \vert \varSigma _{C}\vert ^{-1/2}\vert \varSigma _{ P}\vert ^{-1/2}\vert \varSigma _{ A}\vert ^{1/2}\vert K\vert ^{1/2}\vert \varSigma _{ 0}\vert ^{-1/2} \frac{1} {R_{c}^{2}} \frac{1} {R_{x}^{2}} \frac{1} {\sigma _{\mathrm{res}}^{2}} \\ & & \phantom{\propto }\times \mathrm{exp}\left \{-\frac{1} {2}\left [(Y - L)^{T}\varSigma _{ C}^{-1}(Y - L) -\varDelta ^{T}\varSigma _{ A}\varDelta - k_{0}^{T}K^{-1}k_{ 0} +\xi _{ m}^{T}\varSigma _{ 0}^{-1}\xi _{ m}\right ]\right \},{}\end{array}$$

(15)

where Σ _A ⁻¹ = A ^T Σ _C ⁻¹ A +Σ _P ⁻¹, K ⁻¹ = −J ^T Σ _P ⁻¹ Σ _A Σ _P ⁻¹ J + J ^T Σ _P ⁻¹ J +Σ ₀ ⁻¹, Δ = A ^T Σ _C ⁻¹(Y − L), and k ₀ = K(J ^T Σ _P ⁻¹ Σ _A Δ +Σ ₀ ⁻¹ ξ _m ).

When MH updates are required in any of the samplers, we use truncated normal distributions as the proposal distributions. These distributions are centered at the current draws with variance-covariance matrices adjusted to obtain an acceptance rate of around 25%. The truncation enforces prior constraints and in all cases the MH updates are bivariate.

When generating parameters from a truncated distribution, we repeat drawing from the corresponding unconstrained distribution until the truncation condition is satisfied. In the cosmological example, rejection sampling is not computationally demanding, since the ranges of the prior distributions are fairly large.

The steps of Sampler 1 are

• 1. Sample (ξ, X), which consists of two sub-steps:

  • Sample ξ from N(k ₀, K);

  • Sample X from N(μ _A , Σ _A ), where μ _A  = Σ _A (Δ +Σ _P ⁻¹ J ξ).

• 2. Use MH to sample (Ω _m , Ω _Λ ) from p(Ω _m , Ω _Λ  | Y, ξ, X, α, β, R _c ², R _x ², σ _res ²), which is proportional to p(ξ, X, Ω _m , Ω _Λ , α, β, R _c ², R _x ², σ _res ² | Y ), under the constraint (Ω _m , Ω _Λ ) ∈ [0, 1] × [0, 2].

• 3. Sample (α, β) from N(μ _B , Σ _B ) with constraint (α, β) ∈ [0, 1] × [0, 4], where

  $$\displaystyle{ \varSigma _{B}^{-1} = \left [\begin{array}{cc} \sum \limits _{i=1}^{n} \frac{x_{i}^{2}} {\hat{\sigma }_{m_{Bi}}^{2}} & \sum \limits _{i=1}^{n}\frac{-x_{i}c_{i}} {\hat{\sigma }_{m_{Bi}}^{2}} \\ \sum \limits _{i=1}^{n}\frac{-x_{i}c_{i}} {\hat{\sigma }_{m_{Bi}}^{2}} & \sum \limits _{i=1}^{n} \frac{c_{i}^{2}} {\hat{\sigma }_{m_{Bi}}^{2}}\end{array} \right ]\mbox{ and }\mu _{B} =\varSigma _{B}\left [\begin{array}{c} \sum _{i=1}^{n}\frac{x_{i}(M_{i}-\hat{m}_{Bi}+\mu _{i})} {\hat{\sigma }_{m_{Bi}}^{2}} \\ \sum _{i=1}^{n}\frac{-c_{i}(M_{i}-\hat{m}_{Bi}+\mu _{i})} {\hat{\sigma }_{m_{Bi}}^{2}}\end{array} \right ]. }$$

  (16)

• 4. Sample R _c ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(c_{ i}-c_{0})^{2}} {2} \right ]$ with log(R _c ) ∈ [−5, 2].

• 5. Sample R _x ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(x_{ i}-x_{0})^{2}} {2} \right ]$ with log(R _x ) ∈ [−5, 2].

• 6. Sample σ _res ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(M_{ i}-M_{0})^{2}} {2} \right ]$ with log(σ _res) ∈ [−5, 2].

The steps of Sampler 2 are

• 1. Use MH to sample (Ω _m , Ω _Λ ) from p(Ω _m , Ω _Λ  | Y, α, β, R _c ², R _x ², σ _res ²), which is proportional to p(Ω _m , Ω _Λ , α, β, R _c ², R _x ², σ _res ² | Y ), with (Ω _m , Ω _Λ ) ∈ [0, 1] × [0, 2].

• 2. Use MH to sample (α, β) from p(α, β | Y, Ω _m , Ω _Λ , R _c ², R _x ², σ _res ²), which is proportional to p(Ω _m , Ω _Λ , α, β, R _c ², R _x ², σ _res ² | Y ), with (α, β) ∈ [0, 1] × [0, 4].

• 3. Sample (ξ, X), which consists of two sub-steps:

  • Sample ξ from N(k ₀, K);

  • Sample X from N(μ _A , Σ _A ), where μ _A  = Σ _A (Δ +Σ _P ⁻¹ J ξ).

• 4. Sample R _c ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(c_{ i}-c_{0})^{2}} {2} \right ]$ with log(R _c ) ∈ [−5, 2].

• 5. Sample R _x ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(x_{ i}-x_{0})^{2}} {2} \right ]$ with log(R _x ) ∈ [−5, 2].

• 6. Sample σ _res ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(M_{ i}-M_{0})^{2}} {2} \right ]$ with log(σ _res) ∈ [−5, 2].

The steps of Sampler 3 are

• 1. Sample (ξ, X ^⋆), which consists of two sub-steps:

  • Sample ξ from N(k ₀, K);

  • Sample X ^⋆ from N(μ _A , Σ _A ), where μ _A  = Σ _A (Δ +Σ _P ⁻¹ J ξ).

• 2. Use MH to sample (Ω _m ^⋆, Ω _Λ ^⋆) from p(Ω _m , Ω _Λ  | Y, ξ, X ^⋆, α, β, R _c ², R _x ², σ _res ²), which is proportional to p(ξ, X ^⋆, Ω _m ^⋆, Ω _Λ ^⋆, α, β, R _c ², R _x ², σ _res ² | Y ), under the constraint (Ω _m ^⋆, Ω _Λ ^⋆) ∈ [0, 1] × [0, 2];

  Use (Ω _m ^⋆, Ω _Λ ^⋆) to construct L ^⋆.

• 3. Sample (α ^⋆, β ^⋆) from N(μ _B , Σ _B ) with constraint (α ^⋆, β ^⋆) ∈ [0, 1] × [0, 4];

  Use (α ^⋆, β ^⋆) to construct A ^⋆. Then set $\tilde{X} = A^{\star }X^{\star } + L^{\star }$.

• 4. Use MH to sample (Ω _m , Ω _Λ ) from $p(\varOmega _{m},\varOmega _{\varLambda }\vert Y,\xi,\tilde{X},\alpha ^{\star },\beta ^{\star },R_{c}^{2},R_{x}^{2},\sigma _{\mathrm{res}}^{2})$, which is proportional to $p(\xi,\tilde{X},\varOmega _{m},\varOmega _{\varLambda },\alpha ^{\star },\beta ^{\star },R_{c}^{2},R_{x}^{2},\sigma _{\mathrm{res}}^{2}\vert Y )$, under the constraint (Ω _m , Ω _Λ ) ∈ [0, 1] × [0, 2];

  Use (Ω _m , Ω _Λ ) to construct L.

• 5. Sample (α, β) from N(μ _D , Σ _D ) with constraint (α, β) ∈ [0, 1] × [0, 4], where

  $$\displaystyle{ \varSigma _{D}^{-1} = \left [\begin{array}{cc} \sum \limits _{i=1}^{n} \frac{\tilde{x}_{i}^{2}} {\sigma _{\mathrm{res}}^{2}} & \sum \limits _{i=1}^{n}\frac{-\tilde{x}_{i}\tilde{c}_{i}} {\sigma _{\mathrm{res}}^{2}} \\ \sum \limits _{i=1}^{n}\frac{-\tilde{x}_{i}\tilde{c}_{i}} {\sigma _{\mathrm{res}}^{2}} & \sum \limits _{i=1}^{n} \frac{\tilde{c}_{i}^{2}} {\sigma _{\mathrm{res}}^{2}}\end{array} \right ]\mbox{ and }\mu _{D} =\varSigma _{D}\left [\begin{array}{c} \sum _{i=1}^{n}\frac{\tilde{x}_{i}(M_{0}-\tilde{M}_{i})} {\sigma _{\mathrm{res}}^{2}} \\ \sum _{i=1}^{n}\frac{-\tilde{c}_{i}(M_{0}-\tilde{M}_{i})} {\sigma _{\mathrm{res}}^{2}}\end{array} \right ], }$$

  (17)

  where $\tilde{c}_{i}$, $\tilde{x}_{i}$ and $\tilde{M}_{i}$ are the (3i − 2)^th, (3i − 1)^th and (3i)^th components of $(\tilde{X} - L)$;

  Use (α, β) to construct A. Then set $X = A^{-1}(\tilde{X} - L)$.

• 6. Sample R _c ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(c_{ i}-c_{0})^{2}} {2} \right ]$ with log(R _c ) ∈ [−5, 2].

• 7. Sample R _x ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(x_{ i}-x_{0})^{2}} {2} \right ]$ with log(R _x ) ∈ [−5, 2].

• 8. Sample σ _res ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(M_{ i}-M_{0})^{2}} {2} \right ]$ with log(σ _res) ∈ [−5, 2].

The steps of Sampler 4 are

• 1. Use MH to sample (Ω _m , Ω _Λ ) from p(Ω _m , Ω _Λ  | Y, α, β, R _c ², R _x ², σ _res ²), which is proportional to p(Ω _m , Ω _Λ , α, β, R _c ², R _x ², σ _res ² | Y ) with (Ω _m , Ω _Λ ) ∈ [0, 1] × [0, 2]; Use (Ω _m , Ω _Λ ) to construct L.

• 2. Sample (ξ, X ^⋆), which consists of two sub-steps:

  • Sample ξ from N(k ₀, K);

  • Sample X ^⋆ from N(μ _A , Σ _A ), where μ _A  = Σ _A (Δ +Σ _P ⁻¹ J ξ).

• 3. Sample (α ^⋆, β ^⋆) from N(μ _B , Σ _B ) with constraint (α ^⋆, β ^⋆) ∈ [0, 1] × [0, 4];

  Use (α ^⋆, β ^⋆) to construct A ^⋆. Then set $\tilde{X} = A^{\star }X^{\star } + L$.

• 4. Sample (α, β) from N(μ _D , Σ _D ) with constraint (α, β) ∈ [0, 1] × [0, 4];

  Use (α, β) to construct A. Then set $X = A^{-1}(\tilde{X} - L)$.

• 5. Sample R _c ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(c_{ i}-c_{0})^{2}} {2} \right ]$ with log(R _c ) ∈ [−5, 2].

• 6. Sample R _x ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(x_{ i}-x_{0})^{2}} {2} \right ]$ with log(R _x ) ∈ [−5, 2].

• 7. Sample σ _res ² from $\mbox{ Inv-Gamma}\left [\frac{n} {2}, \frac{\sum _{i=1}^{n}(M_{ i}-M_{0})^{2}} {2} \right ]$ with log(σ _res) ∈ [−5, 2].