$\bigg{|}\sum^{n_{k_{l}}}_{i=1}\log\frac{\hat{f}_{n_{k_{l}},k}(Z_{n_{k_{l}},i}^{*})(Z^{*}_{n_{k_{l}},i}-Z^{*}_{n_{k_{l}},i-1})}{\hat{F}_{n_{k_{l}},k}(Z_{n_{k_{l}},i}^{*})-\hat{F}_{n_{k_{l}},k}(Z_{n_{k_{l}},i-1}^{*})}\bigg{|}=\bigg{|}\sum^{n_{k_{l}}}_{i=1}\log\frac{\hat{f}_{n_{k_{l}},k}(Z^{*}_{n_{k_{l}},i})}{\hat{f}_{n_{k_{l}},k}(\widetilde{Z}^{*}_{n_{k_{l}},i})}\bigg{|}.$

By the monotonicity of $\hat{f}_{n_{k_{l}},k}$,

$\displaystyle A_{n}$ $\displaystyle:=$
$\displaystyle\bigg{|}\sum^{n_{k_{l}}}_{i=1}\log\frac{\hat{f}_{n_{k_{l}},k}(Z^{*}_{n_{k_{l}},i})}{\hat{f}_{n_{k_{l}},k}(\widetilde{Z}^{*}_{n_{k_{l}},i})}\bigg{|}\leq\bigg{|}\sum^{n_{k_{l}}}_{i=1}\\{\log\hat{f}_{n_{k_{l}},k}(Z^{*}_{n_{k_{l}},i})-\log\hat{f}_{n_{k_{l}},k}(Z^{*}_{n_{k_{l}},i-1})\\}\bigg{|}$
$\displaystyle=$
$\displaystyle\log\hat{f}_{n_{k_{l}},k}(0)-\log\hat{f}_{n_{k_{l}},k}(Z^{*}_{n_{k_{l}},n_{k_{l}}}).$

We now apply Lemma 9.6(a) with $f_{n_{k_{l}}}=\hat{f}_{n_{k_{l}},k}$. Note
that $\int^{\infty}_{0}\hat{f}^{1-\alpha}_{n_{k_{l}},k}(x)dx\leq
k\hat{f}_{n_{k_{l}}}^{1-\alpha}(0)Z_{n_{k_{l}}}\leq
kM^{1-\alpha}_{n_{k_{l}}}\tilde{b}_{n_{k_{l}}}=kn^{m_{0}(1-\alpha)+m_{0}}\leq
n^{m_{0}(1-\alpha)+m_{0}+1}$ for some $\alpha\in(0,1)$ and for all large $n$.
Therefore, $A_{n}=o_{\mathbb{P}^{*}_{\omega}}(\log n)$ and condition (ii) is
satisfied.

∎

######  Theorem 9.11 (Bootstrap Consistency for Completely Monotone
Densities).

Suppose $f_{0}$ is a density that may not be bounded from above or have a
bounded support. Without loss of generality, assume that $\tau_{f_{0}}=0$.
Suppose that $Z_{1}^{-1}=o_{p}(n^{m_{0}})$ and $Z_{n}=o_{p}(n^{m_{0}})$ for
some $m_{0}>0$. Regardless of whether $f_{0}$ is completely monotone or not,

$\sup_{x\in\mathbb{R}}|\mathbb{P}^{*}(\sqrt{n}(\Lambda^{*}_{n}-\gamma)\leq
x)-\mathbb{P}(N(0,\pi^{2}/6-1)\leq
x)|\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}0,$ (44)

where $\tilde{f}_{n}$ in the bootstrap procedure is the completely monotone
MLE $\hat{f}_{n,\infty}$.

###### Proof of Theorem 9.11.

Since $Z^{-1}_{1}=o_{p}(n^{m_{0}})$ and $Z_{n}=o_{p}(n^{m_{0}})$, for any
subsequence $\\{n_{k}\\}$ of $\\{n\\}$, there exists a further subsequence
$\\{n_{k_{l}}\\}$ and $\Omega_{0}$ such that $\mathbb{P}(\Omega_{0})=1$ and
for all $\omega\in\Omega_{0}$,
$Z^{-1}_{1}(\omega)/n^{m_{0}}_{k_{l}}\rightarrow 0$ and
$Z_{n_{k_{l}}}(\omega)/n^{m_{0}}_{k_{l}}\rightarrow 0$. Now, note that the
distribution function of the completely monotone MLE
$\hat{F}_{n,\infty}(t)=1-\sum^{r_{n}}_{i=1}\hat{p}_{i}e^{-\hat{\lambda}_{i}t}$.
Thus,

$1-e^{-\frac{t}{Z_{n}}}\leq
1-\max_{i=1,\ldots,r_{n}}e^{-\hat{\lambda}_{i}t}\leq\hat{F}_{n}(t)\leq
1-\min_{i=1,\ldots,r_{n}}e^{-\hat{\lambda}_{i}t}\leq 1-e^{-\frac{t}{Z_{1}}}.$

Hence, we have

$\displaystyle\mathbb{P}^{*}_{\omega}(\\{Z^{*}_{n_{k_{l}},1}\\}^{-1}>t)$
$\displaystyle=$ $\displaystyle
1-\mathbb{P}^{*}_{\omega}(\\{Z^{*}_{n_{k_{l}}1}\\}^{-1}\leq t)$
$\displaystyle=$ $\displaystyle
1-\\{\mathbb{P}^{*}_{\omega}(\\{X^{*}_{n_{k_{l}},1}\\}^{-1}\leq
t)\\}^{n_{k_{l}}}$ $\displaystyle=$ $\displaystyle
1-\left\\{\mathbb{P}^{*}_{\omega}\left(X^{*}_{n_{k_{l}},1}>\frac{1}{t}\right)\right\\}^{n_{k_{l}}}$
$\displaystyle=$ $\displaystyle
1-\\{1-\hat{F}_{n_{k_{l}}}(1/t;\omega)\\}^{n_{k_{l}}}$ $\displaystyle\leq$
$\displaystyle 1-e^{-\frac{n_{k_{l}}}{Z_{1}(\omega)t}}$

and

$\mathbb{P}^{*}_{\omega}(Z^{*}_{n_{k_{l}},n_{k_{l}}}>t)=1-\\{\mathbb{P}^{*}_{\omega}(X^{*}_{n_{k_{l}},1}\leq
t)\\}^{n_{k_{l}}}\leq
1-\left(1-e^{-\frac{t}{Z_{n_{k_{l}}}(\omega)}}\right)^{n_{k_{l}}}.$

Therefore, as $l$ goes to infinity,

$\mathbb{P}^{*}_{\omega}(\\{Z^{*}_{n_{k_{l}},1}\\}^{-1}>n^{m_{0}+1}_{k_{l}})\leq
1-e^{-\frac{Z^{-1}_{1}(\omega)}{n^{m_{0}}_{k_{l}}}}\rightarrow 0$

and

$\mathbb{P}^{*}_{\omega}(Z^{*}_{n_{k_{l}},n_{k_{l}}}>n^{m_{0}+1}_{k_{l}})\leq
1-\left(1-\exp\left\\{-\frac{n^{m_{0}}_{k_{l}}}{Z_{n_{k_{l}}}(\omega)}n_{k_{l}}\right\\}\right)^{n_{k_{l}}}\rightarrow
1-e^{0}=0.$

We now apply Lemma 9.8 with
$f_{n_{k_{l}}}=\hat{f}_{n_{k_{l}},\infty}(\cdot;\omega)$,
$\tilde{c}_{1n_{k_{l}}}=n^{m_{0}+1}_{k_{l}}$ and
$\tilde{c}_{2n_{k_{l}}}=n^{\frac{m_{0}+1}{1+\beta}}_{k_{l}}$. Therefore,
condition (i) in Theorem 4.2 is satisfied.

To verify condition (ii) in Theorem 4.2, similar to the $k$-monotone case, it
suffices to show that

$\log\hat{f}_{n_{k_{l}},\infty}(0)-\log\hat{f}_{n_{k_{l}},\infty}(Z^{*}_{n_{k_{l}},n_{k_{l}}})=o_{\mathbb{P}^{*}_{\omega}}(n^{1/2}_{k_{l}})$
(45)

The term $\log\hat{f}_{n_{k_{l}},\infty}(0)$ is bounded above by $\log
Z^{-1}_{n_{k_{l}}}(\omega)\leq\log n^{m_{0}}_{k_{l}}$ for all large $l$. To
apply Lemma 9.6, note that for all large $l$,
$\hat{f}_{n_{k_{l}},\infty}(x)\leq 1/x^{2}$ for all $x\geq
Z^{1+\beta}_{n_{k_{l}}}(\omega)$ (see the proof of Theorem 3.3). Hence, for
all large $l$,

$\displaystyle\int^{\infty}_{0}\hat{f}^{1-\alpha}_{n_{k_{l}},\infty}(x;\omega)dx$
$\displaystyle=$
$\displaystyle\int^{Z^{1+\beta}_{n_{k_{l}}}(\omega)}_{0}\hat{f}^{1-\alpha}_{n_{k_{l}},\infty}(x;\omega)dx+\int^{\infty}_{Z^{1+\beta}(\omega)_{n_{k_{l}}}}\hat{f}^{1-\alpha}_{n_{k_{l}},\infty}(x;\omega)dx$
$\displaystyle\leq$ $\displaystyle
Z^{-(1-\alpha)}_{1}(\omega)Z^{1+\beta}_{n_{k_{l}}}(\omega)+\int^{\infty}_{Z^{1+\beta}_{n_{k_{l}}}(\omega)}\frac{1}{x^{2}}dx$
$\displaystyle\leq$ $\displaystyle
n_{k_{l}}^{m_{0}(1-\alpha)+m_{0}(1+\beta)}+\frac{1}{Z^{1+\beta}_{n_{k_{l}}}(\omega)}$
$\displaystyle\leq$ $\displaystyle
n_{k_{l}}^{m_{0}(1-\alpha)+m_{0}(1+\beta)+1},$

as $Z^{1+\beta}_{n_{k_{l}}}(\omega)>1$ for all large $l$. Therefore, we can
apply Lemma 9.6(a) and (45) is satisfied. ∎

######  Theorem 9.12 (Bootstrap Consistency for Log-concave Densities).

Regardless of whether $f_{0}$ is log-concave or not, suppose that
$\int_{\mathbb{R}}|x|f_{0}(x)dx<\infty$, We have

$\sup_{x\in\mathbb{R}}|\mathbb{P}^{*}(\sqrt{n}(\Lambda^{*}_{n}-\gamma)\leq
x)-\mathbb{P}(N(0,\pi^{2}/6-1)\leq
x)|\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}0,$ (46)

where $\tilde{f}_{n}$ in the bootstrap procedure is the log-concave MLE
$\hat{f}_{n}$.

###### Proof of Theorem 9.12.

In view of Theorem 4.2, it suffices to verify conditions (i) and (ii) in
Theorem 4.2. First, note that the support of $\hat{f}_{n,lc}$ is
$[Z_{1},Z_{n}]$. We now claim that $Z_{n}-Z_{1}=o_{p}(n^{2})$. Indeed, note
that $|Z_{n}-Z_{1}|\leq|Z_{n}|+|Z_{1}|\leq 2\max_{i=1,\ldots,n}|X_{i}|$ and
for any $\delta>0$,

$\mathbb{P}\left(\max_{i=1,\ldots,n}|X_{i}|>\delta
n^{2}\right)=1-\\{1-\mathbb{P}(|X_{1}|>\delta n^{2})\\}^{n}\leq
1-\left\\{1-\frac{\mathbb{E}(|X_{1}|)}{\delta n^{2}}\right\\}^{n}\rightarrow
0,$

as $n\rightarrow\infty$, where the inequality follows from Markov’s
inequality. Hence, $Z_{n}-Z_{1}=o_{p}(n^{2})$, which then implies that for any
subsequence $\\{n_{k}\\}$ of $\\{n\\}$, there exists a further subsequence
$\\{n_{k_{l}}\\}$ and $\Omega_{0}$ such that $\mathbb{P}(\Omega_{0})=1$ and
for any $\omega\in\Omega_{0}$,
$\\{Z_{n_{k_{l}}}(\omega)-Z_{1}(\omega)\\}/n^{2}_{k_{l}}=o(n^{2}_{k_{l}})$.
Now, fix a $\omega\in\Omega_{0}$. Because
$\int_{\mathbb{R}}|x|f_{0}(x)dx<\infty$, Lemma 3 and Theorem 4 in [9] imply
that $\hat{f}_{n,lc}(\cdot;\omega)$ satisfies all the conditions in Lemma 9.9
for all large enough $n$ (where without loss of generality, we assume that the
sets with probability one from Lemma 3 and Theorem 4 in [9] contain
$\Omega_{0}$). Hence, Lemma 9.9(c) implies that condition (i) is satisfied.

In the following, the dependence on $\omega$ is not always written explicitly
for the sake of notational simplicity. To verify condition (ii), by Taylor’s
theorem, for some $\widetilde{Z}^{*}_{n_{k_{l}},i}$ between
$Z^{*}_{n_{k_{l}},i}$ and $Z^{*}_{n_{k_{l}},i-1}$,

$\displaystyle\left|\sum^{n_{k_{l}}}_{i=1}\log\frac{\hat{f}_{n_{k_{l}},lc}(Z_{n_{k_{l}},i}^{*})(Z^{*}_{n_{k_{l}},i}-Z^{*}_{n_{k_{l}},i-1})}{\hat{F}_{n_{k_{l}},lc}(Z_{n_{k_{l}},i}^{*})-\hat{F}_{n_{k_{l}},lc}(Z_{n_{k_{l}},i-1}^{*})}\right|$
(47) $\displaystyle\stackrel{{\scriptstyle d}}{{=}}$
$\displaystyle\left|\log\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},1})+\log(Z^{*}_{n_{k_{l}},1}-Z_{n_{k_{l}},1})-\log(U_{(n_{k_{l}},1)})+\sum^{n_{k_{l}}}_{i=2}\log\frac{\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},i})}{\hat{f}_{n_{k_{l}},lc}(\widetilde{Z}^{*}_{n_{k_{l}},i})}\right|$
$\displaystyle\leq$
$\displaystyle|\log\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},1})|+|\log(Z^{*}_{n_{k_{l}},1}-Z_{n_{k_{l}},1})|+|\log(U_{(n_{k_{l}},1)})|+\left|\sum^{n_{k_{l}}}_{i=2}\log\frac{\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},i})}{\hat{f}_{n_{k_{l}},lc}(\widetilde{Z}^{*}_{n_{k_{l}},i})}\right|,$

where $U_{(n_{k_{l}},1)}$ is the minimum order statistic of a random sample of
standard uniform random variables of size $n_{k_{l}}$. Since
$\hat{f}_{n_{k_{l}},lc}$ is log-concave, it is unimodal. Let $s_{n_{k_{l}}}$
denote the mode of $\hat{f}_{n_{k_{l}},lc}$. Then, we have

$\displaystyle\left|\sum^{n_{k_{l}}}_{i=2}\log\frac{\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},i})}{\hat{f}_{n_{k_{l}},lc}(\tilde{Z}^{*}_{n_{k_{l}},i})}\right|$
$\displaystyle=$ $\displaystyle\Bigg{|}\sum_{i:Z^{*}_{n_{k_{l}},i}\leq
s_{n_{k_{l}}}}\log\frac{\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},i})}{\hat{f}_{n_{k_{l}},lc}(\tilde{Z}^{*}_{n_{k_{l}},i})}+\sum_{i:\tilde{Z}^{*}_{n_{k_{l}},i}\leq
s_{n_{k_{l}}}\leq
Z^{*}_{n_{k_{l}},i}}\log\frac{\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},i})}{\hat{f}_{n_{k_{l}},lc}(\tilde{Z}^{*}_{n_{k_{l}},i})}$
$\displaystyle\quad+\sum_{i:s_{n_{k_{l}}}\leq\tilde{Z}^{*}_{n_{k_{l}},i}}\log\frac{\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},i})}{\hat{f}_{n_{k_{l}},lc}(\tilde{Z}^{*}_{n_{k_{l}},i})}\Bigg{|}$
$\displaystyle\leq$ $\displaystyle
2\\{\log\hat{f}_{n_{k_{l}},lc}(s_{n_{k_{l}}})-\log\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},1})\\}+2\\{\log\hat{f}_{n_{k_{l}},lc}(s_{n_{k_{l}}})-\log\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},n_{k_{l}}})\\}$
$\displaystyle\leq$ $\displaystyle
4\log\hat{f}_{n_{k_{l}},lc}(s_{n_{k_{l}}})-4\log\min\\{\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},1}),\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},n_{k_{l}}})\\}.$

Note that for all large $n$, by Lemma 9.9(a), $\hat{f}_{n_{k_{l}},lc}\leq C$
for some constant $C$. We now verify the condition in Lemma 9.6(a):

$\int^{\infty}_{-\infty}\hat{f}_{n_{k_{l}},lc}(x)dx\leq
C(\sigma_{\hat{f}_{n_{k_{l}}}}-\tau_{\hat{f}_{n_{k_{l}}}})=C(Z_{n_{k_{l}}}(\omega)-Z_{1}(\omega))=o(n^{2}_{k_{l}}).$

Applying Lemma 9.6(a), we have

$\left|\sum^{n_{k_{l}}}_{i=2}\log\frac{\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},i})}{\hat{f}_{n_{k_{l}},lc}(\tilde{Z}^{*}_{n_{k_{l}},i})}\right|=O_{\mathbb{P}^{*}_{\omega}}(\log
n_{k_{l}}).$

Similarly,
$|\log\hat{f}_{n_{k_{l}},lc}(Z^{*}_{n_{k_{l}},1})|=O_{\mathbb{P}^{*}_{\omega}}(\log
n_{k_{l}})$. As

$\mathbb{E}^{*}_{\omega}\\{(X_{n_{k_{l}},1}-\tau_{\hat{f}_{n_{k_{l}}}})^{-\alpha}\\}\leq
C\int^{Z_{n_{k_{l}}}(\omega)}_{Z_{1}(\omega)}(x-\tau_{\hat{f}_{n_{k_{l}}}})^{-\alpha}dx=\frac{C}{1-\alpha}(Z_{n_{k_{l}}}(\omega)-Z_{1}(\omega))^{1-\alpha}=o(n^{2(1-\alpha)}_{k_{l}}),$

$|\log(Z^{*}_{n_{k_{l}},1}-Z_{n_{k_{l}},1})|=O_{\mathbb{P}^{*}_{\omega}}(\log
n_{k_{l}})$ by Lemma 9.6(b). Finally, simply note that $|\log
U_{(n_{k_{l}},1)}|=O_{{\mathbb{P}}^{*}(\omega)}(\log n_{k_{l}})$. Therefore,
condition (ii) is satisfied.

∎

##  10 Details of Simulation Results

Notation | Density
---|---
Exp($\lambda$) | $\lambda e^{-\lambda x}I(x>0)$
Beta($\alpha$, $\beta$) | $\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}I(0<x<1)$
Unif($a,b$) | $\frac{1}{b-a}I(a<x<b)$
Gamma($\alpha$, $\beta$) | $\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}I(x>0)$
Laplace($\lambda$) | $\frac{1}{2\lambda}e^{-|x|/\lambda}I(x\in\mathbb{R})$
Normal($\mu,\sigma$) | $\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}I(x\in\mathbb{R})$
Lognormal($\mu$, $\sigma$) | $\frac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\frac{(\ln x-\mu)^{2}}{2\sigma^{2}}\right)I(x\in\mathbb{R})$
Pareto($\alpha$) | $\alpha x^{-\alpha-1}I(x>1)$
t($\nu$) | $\frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\Gamma\left(\frac{\nu}{2}\right)}\left(1+\frac{x^{2}}{\nu}\right)^{-\frac{\nu+1}{2}}I(x\in\mathbb{R})$
MixExp$(K,p,\lambda)$ | $\sum^{K}_{k=1}p_{k}\lambda_{k}e^{-\lambda_{k}x}$
HalfNormal($\sigma$) | $\frac{2}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x}{\sigma}\right)^{2}}I(x>0)$
Halft($\nu$) | $\frac{2\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\Gamma\left(\frac{\nu}{2}\right)}\left(1+\frac{x^{2}}{\nu}\right)^{-\frac{\nu+1}{2}}I(x>0)$
Table 2: Notations of the distributions and their densities. Using Asymptotic
Null Distribution
---
|  | $n=100$ | $n=250$ | $n=500$ | $n=1000$ | $n=2000$
$H_{0}$ | Exp$(1)$ | 0.0035 | 0.0042 | 0.0048 | 0.0053 | 0.0065
| Exp$(2)$ | 0.0038 | 0.0041 | 0.0044 | 0.0053 | 0.0062
| Beta$(1,1.5)$ | 0.0120 | 0.0138 | 0.0148 | 0.0164 | 0.0182
| Beta$(1,2)$ | 0.0087 | 0.0095 | 0.0105 | 0.0119 | 0.0140
| Beta$(1,3)$ | 0.0058 | 0.0072 | 0.0076 | 0.0089 | 0.0109
| Unif$(0,1)$ | 0.0285 | 0.0337 | 0.0354 | 0.0384 | 0.0396
$H_{1}$ | Beta$(2,1)$ | 0.6596 | 0.9587 | 0.9995 | 1.0000 | 1.0000
| Gamma$(4,5)$ | 0.2280 | 0.5949 | 0.9130 | 0.9981 | 1.0000
Using bootstrap procedure with samples drawn from MLE
|  | $n=100$ | $n=250$ | $n=500$ | $n=1000$ | $n=2000$
$H_{0}$ | Exp$(1)$ | 0.0378 | 0.0412 | 0.0371 | 0.0397 | 0.0424
| Exp$(2)$ | 0.0428 | 0.0404 | 0.0387 | 0.0428 | 0.0427
| Beta$(1,1.5)$ | 0.0512 | 0.0513 | 0.0512 | 0.0471 | 0.0448
| Beta$(1,2)$ | 0.0474 | 0.0484 | 0.0456 | 0.0479 | 0.0429
| Beta$(1,3)$ | 0.0450 | 0.0479 | 0.0481 | 0.0492 | 0.0459
| Unif$(0,1)$ | 0.0677 | 0.0646 | 0.0654 | 0.0591 | 0.0576
$H_{1}$ | Beta$(2,1)$ | 0.7590 | 0.9752 | 0.9994 | 1.0000 | 1.0000
| Gamma$(4,5)$ | 0.4388 | 0.7915 | 0.9719 | 0.9996 | 1.0000
Table 3: Rejection proportions of the NPLRT for the monotone case Using
Asymptotic Null Distribution
---
|  | $n=100$ | $n=250$ | $n=500$ | $n=1000$ | $n=2000$
H0 | Exp(1) | 0.0477 | 0.0457 | 0.0426 | 0.0436 | 0.0425
| Exp(2) | 0.0437 | 0.0434 | 0.0395 | 0.0419 | 0.0401
| Beta(1, 2) | 0.0655 | 0.0608 | 0.0560 | 0.0564 | 0.0552
| Beta(1, 3) | 0.0536 | 0.0528 | 0.0502 | 0.0474 | 0.0472
H1 | Beta(1, 1.5) | 0.1127 | 0.1285 | 0.1635 | 0.2176 | 0.3303
| Beta(2, 1) | 0.9998 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Gamma(4, 5) | 0.7480 | 0.9821 | 1.0000 | 1.0000 | 1.0000
| Unif(0, 1) | 0.4754 | 0.7762 | 0.9595 | 0.9989 | 1.0000
Using bootstrap procedure with samples drawn from MLE
$H_{0}$ | Exp$(1)$ | 0.0480 | 0.0490 | 0.0550 | 0.0500 | 0.0560
| Exp$(2)$ | 0.0490 | 0.0460 | 0.0510 | 0.0470 | 0.0500
| Beta$(1,2)$ | 0.0430 | 0.0470 | 0.0600 | 0.0600 | 0.0500
| Beta$(1,3)$ | 0.0500 | 0.0530 | 0.0560 | 0.0560 | 0.0610
$H_{1}$ | Beta$(1,1.5)$ | 0.0780 | 0.1170 | 0.1540 | 0.2280 | 0.3290
| Beta$(2,1)$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Gamma$(4,5)$ | 0.7290 | 0.9830 | 0.9990 | 1.0000 | 1.0000
| Unif$(0,1)$ | 0.4200 | 0.7480 | 0.9350 | 0.9990 | 1.0000
Table 4: Rejection proportions of the NPLRT for the $k$-monotone case when
$k=2$ Using Asymptotic Null Distribution
---
|  | $n=100$ | $n=250$ | $n=500$ | $n=1000$ | $n=2000$
$H_{0}$ | Exp$(1)$ | 0.0592 | 0.0567 | 0.0535 | 0.0558 | 0.0498
| Exp$(2)$ | 0.0568 | 0.0548 | 0.0527 | 0.0537 | 0.0483
| Beta$(1,3)$ | 0.0737 | 0.0710 | 0.0670 | 0.0583 | 0.0577
$H_{1}$ | Beta$(1,2)$ | 0.1103 | 0.1260 | 0.1615 | 0.2123 | 0.3003
| Beta$(1,1.5)$ | 0.2322 | 0.3687 | 0.5459 | 0.7833 | 0.9637
| Beta$(2,1)$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Gamma$(4,5)$ | 0.8931 | 0.9979 | 1.0000 | 1.0000 | 1.0000
| Unif$(0,1)$ | 0.7291 | 0.9706 | 0.9997 | 1.0000 | 1.0000
Using bootstrap procedure with samples drawn from MLE
$H_{0}$ | Exp$(1)$ | 0.0480 | 0.0490 | 0.0560 | 0.0540 | 0.0420
| Exp$(2)$ | 0.0500 | 0.0570 | 0.0480 | 0.0570 | 0.0580
| Beta$(1,3)$ | 0.0620 | 0.0590 | 0.0510 | 0.0510 | 0.0520
$H_{1}$ | Beta$(1,2)$ | 0.0850 | 0.1220 | 0.1470 | 0.1650 | 0.2630
| Beta1$(1,1.5)$ | 0.1840 | 0.3300 | 0.5200 | 0.7500 | 0.9490
| Beta$(2,1)$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Gamma$(4,5)$ | 0.8610 | 0.9950 | 1.0000 | 1.0000 | 1.0000
| Unif$(0,1)$ | 0.6700 | 0.9630 | 1.0000 | 1.0000 | 1.0000
Table 5: Rejection proportions of the NPLRT for the $k$-monotone case when
$k=3$ Using Asymptotic Null Distribution
---
|  | $n=100$ | $n=250$ | $n=500$ | $n=1000$ | $n=2000$
$H_{0}$ | Exp(1) | 0.089 | 0.084 | 0.069 | 0.059 | 0.059
| MixExp$(2,p_{1},\lambda_{1})$ | 0.069 | 0.082 | 0.070 | 0.068 | 0.058
| MixExp$(3,p_{2},\lambda_{2})$ | 0.076 | 0.063 | 0.086 | 0.056 | 0.059
$H_{1}$ | Gamma(2,1) | 0.477 | 0.756 | 0.945 | 0.999 | 1.000
| Unif$(0,1)$ | 0.976 | 1.000 | 1.000 | 1.000 | 1.000
| HalfNormal$(1)$ | 0.235 | 0.316 | 0.479 | 0.653 | 0.882
| Halft$(20)$ | 0.204 | 0.251 | 0.335 | 0.494 | 0.690
| Halft$(5)$ | 0.132 | 0.119 | 0.136 | 0.174 | 0.222
Using bootstrap procedure with samples drawn from MLE
$H_{0}$ | Exp(1) | 0.071 | 0.040 | 0.051 | 0.050 | 0.049
| MixExp$(2,p_{1},\lambda_{1})$ | 0.049 | 0.043 | 0.055 | 0.053 | 0.052
| MixExp$(3,p_{2},\lambda_{2})$ | 0.042 | 0.055 | 0.057 | 0.050 | 0.047
$H_{1}$ | Gamma(2,1) | 0.352 | 0.686 | 0.920 | 0.994 | 1.000
| Unif$(0,1)$ | 0.949 | 1.000 | 1.000 | 1.000 | 1.000
| HalfNormal$(1)$ | 0.151 | 0.237 | 0.382 | 0.590 | 0.846
| Halft$(20)$ | 0.093 | 0.130 | 0.162 | 0.277 | 0.426
| Halft$(5)$ | 0.085 | 0.091 | 0.091 | 0.167 | 0.196
Table 6: Rejection proportions of the NPLRT for the completely monotone case,
where
$p_{1}=(0.3,0.7),\lambda_{1}=(1,5),p_{2}=(0.3,0.3,0.4),\lambda_{2}=(1,5,10)$.
Using Asymptotic Null Distribution
---
|  | $n=100$ | $n=250$ | $n=500$ | $n=1000$ | $n=2000$
$H_{0}$ | Exp$(1)$ | 0.0621 | 0.0608 | 0.0593 | 0.0562 | 0.0558
| Laplace$(1)$ | 0.0595 | 0.0623 | 0.0640 | 0.0622 | 0.0593
| Normal$(0,1)$ | 0.0311 | 0.0364 | 0.0391 | 0.0413 | 0.0417
| Normal$(0,2)$ | 0.0314 | 0.0375 | 0.0383 | 0.0401 | 0.0433
| Unif$(0,1)$ | 0.0323 | 0.0362 | 0.0394 | 0.0413 | 0.0439
$H_{1}$ | Pareto$(1)$ | 0.9865 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Pareto$(2)$ | 0.6023 | 0.8496 | 0.9697 | 0.9990 | 1.0000
| Lognormal$(0,1)$ | 0.1270 | 0.1937 | 0.2870 | 0.4382 | 0.6619
| Lognormal$(0,2)$ | 0.9988 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Gamma$(0.5,1)$ | 0.8630 | 0.9853 | 0.9998 | 1.0000 | 1.0000
| t$(2)$ | 0.2893 | 0.4418 | 0.6049 | 0.7972 | 0.9488
Using bootstrap procedure with samples drawn from MLE
$H_{0}$ | Exp$(1)$ | 0.0851 | 0.0720 | 0.0603 | 0.0634 | 0.0653
| Laplace$(1)$ | 0.0712 | 0.0657 | 0.0631 | 0.0614 | 0.0513
| Normal$(0,1)$ | 0.0529 | 0.0526 | 0.0513 | 0.0507 | 0.0538
| Normal$(0,2)$ | 0.0551 | 0.0538 | 0.0512 | 0.0499 | 0.0511
| Unif$(0,1)$ | 0.0733 | 0.0658 | 0.0594 | 0.0571 | 0.0560
$H_{1}$ | Pareto$(1)$ | 0.9853 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Pareto$(2)$ | 0.6086 | 0.8405 | 0.9675 | 0.9987 | 1.0000
| Lognormal$(0,1)$ | 0.1865 | 0.2466 | 0.3364 | 0.4904 | 0.6959
| Lognormal$(0,2)$ | 0.9988 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Gamma$(0.5,1)$ | 0.8555 | 0.9859 | 0.9997 | 1.0000 | 1.0000
| t$(2)$ | 0.2869 | 0.4253 | 0.5798 | 0.7772 | 0.9430
Table 7: Rejection proportions of the NPLRT for the log-concave case |  | $n=100$ | $n=250$ | $n=500$ | $n=1000$ | $n=2000$
---|---|---|---|---|---|---
$H_{0}$ | Exp$(1)$ | 0.1885 | 0.1645 | 0.1410 | 0.1219 | 0.1145
| Laplace$(1)$ | 0.3239 | 0.2908 | 0.2501 | 0.2258 | 0.2052
| Normal$(0,1)$ | 0.0130 | 0.0119 | 0.0084 | 0.0069 | 0.0054
| Normal$(0,2)$ | 0.0122 | 0.0121 | 0.0080 | 0.0051 | 0.0055
| Unif$(0,1)$ | 0.0759 | 0.0730 | 0.0763 | 0.0759 | 0.0751
$H_{1}$ | Pareto$(1)$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Pareto$(2)$ | 0.9912 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Lognormal$(0,1)$ | 0.8324 | 0.9820 | 0.9990 | 1.0000 | 1.0000
| Lognormal$(0,1)$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| Gamma$(0.5,1)$ | 0.9929 | 1.0000 | 1.0000 | 1.0000 | 1.0000
| t$(2)$ | 0.8747 | 0.9869 | 0.9992 | 1.0000 | 1.0000
Table 8: Rejection proportions of the trace test in [7] Figure 1: Comparison
with the split LRT in [14]. The beta distribution with $\beta=0.5$ is not log-
concave for any $\alpha$. The beta distribution with $\beta=10$ is log-concave
only when $\alpha\geq 1$. The split LRT performs better when the true density
is far from log-concave (see Figure 2). When the true density is closer to
log-concave (see Figure 3), our proposed LRT has higher powers. Figure 2: The
log-density of Beta($\alpha,0.5$) with different $\alpha$. Figure 3: The log-
density of Beta($\alpha,10$) with different $\alpha$. Figure 4: Comparison
with the split LRT in [14]. The $t$-distribution with any degrees of freedom
is not log-concave. The normal mixture
$\frac{1}{2}\phi(x)+\frac{1}{2}\phi(x-\mu)$ is log-concave only when $\mu\leq
2$. Our proposed LRT has a higher power for all the $t$ distributions and
normal mixtures when $\mu>2$.

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#  M-$\sigma$ RELATIONS ACROSS SPACE AND TIME

David Garofalo  Department of Physics, Kennesaw State University
Damian J. Christian  Department of Physics and Astronomy, California State
University, Northridge
Chase Hames  Department of Physics, Kennesaw State University
Max North  Department of Information Systems, Kennesaw State University
Keegan Thottam1 Samuel Nazaroff1 Alisaie Eckelbarger1

###### Abstract

Feedback from active galactic nuclei (AGN) has long been invoked to explain
the correlation between black hole mass and stellar velocity dispersion
(M-$\sigma$) discovered in low redshift galaxies. We describe the time
evolution of AGN in the M-$\sigma$ plane based on our gap model (Garofalo,
Evans & Sambruna 2010) for black hole accretion and jet formation illustrating
a fundamental difference between jetted and non-jetted AGN. While the latter
tend to evolve diagonally upward with black hole mass increasing along with
stellar dispersion, we show that jetted AGN tend on average to move initially
more upwards because their effect on velocity dispersion is weaker than for
non-jetted AGN. But this initial phase is followed by a shift in the nature of
the feedback, from positive to negative, a transition that is more dramatic on
average in denser cluster environments. The feedback gets its kick from tilted
jets which shut down star formation but increase velocity dispersion values.
As this change in the nature of the feedback takes tens of millions to
hundreds of millions of years, these cluster, merger-triggered jetted AGN,
will evolve more upwards for up to order $10^{8}$ years, followed by an
extremely long phase in which low excitation progressively slows black hole
growth but dramatically affects stellar dispersion. As a result, powerful
jetted AGN evolve for most of their lives almost horizontally on the
M-$\sigma$ plane. The prediction is that strongest AGN feedback on stellar
dispersion is a late universe phenomenon with M87 a good example. We show how
jetted and non-jetted AGN parallel the Sersic and core-Sersic galaxy paths in
the M-$\sigma$ plane found by Sahu et al (2019) and to a prediction that
jetted quasars are not core-Sersic galaxies as found for lower redshift jetted
AGN.

active galactic nuclei — black hole feedback — black hole scaling relations

##  1 Introduction

For a quarter century, evidence has been building that massive black holes
occupy the centers of galaxies and that stellar velocity dispersions in the
bulge are tightly connected with black hole mass via the so-called M-$\sigma$
relation (Magorrian et al 1998; Gebhardt et al 2000; Ferrarese & Merritt 2000;
see Kormendy & Ho 2013 for a review)

with c and $\alpha$ constants with values 3 and 4, respectively (e.g. Zubovas
& King 2019). We plot equation (1) in Figure 1. The nature of the correlation
has been the subject of debate but no clear picture has emerged. The dominant
idea is that AGN feedback is responsible (e.g. Silk & Rees 1998). In Figure 1
we also plot the galaxies of Gultekin et al 2009. Because the majority of AGN
do not produce powerful jets, it is not unreasonable to assume that the
majority of these objects were once radio quiet AGN. As a result, the fit and
these objects give us a sense of how the non-jetted AGN populate the
M-$\sigma$ diagram.

We apply the time evolution in the gap paradigm for black holes (Garofalo,
Evans & Sambruna 2010), to understand the nature of the M-$\sigma$ relation
for different types of AGN. We show that whereas non-jetted and generally more
core-Sersic AGN move both upwards and to the right in the M-$\sigma$ plane,
jetted and generally more Sersic AGN 111Sérsic and core-Sérsic galaxies are
defined by two distinct relations, $M_{BH}\propto\sigma^{5.75+-0.34}$ and
$M_{BH}\propto\sigma^{8.64+-1.10}$, respectively (Sahu et al. 2019)., tend to
move more upwards initially, followed by mostly horizontal motion, which on
average is environment-dependent. We identify characteristic timescales,
characteristic black hole mass growth, and characteristic velocity dispersion,
in order to explain the physical origin of this difference as a key component
of the radio loud/radio quiet dichotomy. This current work focuses on a more
qualitative picture of these parameters as we build on our models for future
more quantitative descriptions. In Section 2 we address the time dependent
feedback on star formation, accretion, and stellar velocities of jetted and
non-jetted AGN and compare to observations. In Section 3 we conclude.

##  2 Paths of AGN in the M-$\sigma$ plane

Two kinds of black hole feedback contribute to enhancing the stellar
dispersion in the gap paradigm (Garofalo, Evans & Sambruna 2010): 1)
radiatively-driven accretion disk winds and 2) tilted jets, with the second
dominant but present only in a subclass of AGN. In this section, we argue that
these two elements drive two different M-$\sigma$ relations and make
predictions for observations. The model describes the origin of both FRI and
FRII jets, with the former less powerful on average that often experience
entrainment due to the surrounding medium while the latter are on average more
powerful and collimated and visible in hotspots (Fanaroff $\&$ Riley 1974).
While FRII jets enhance the star formation rate and may alter the state of
accretion, they do not contribute to stellar dispersion. This is due to the
fact that they are not tilted jets. Because more massive black holes produce
greater AGN feedback, there is a dependence on black hole mass that translates
into a dependence on environment that we now describe.

In Figure 2 we consider a high spinning black hole that is triggered into an
AGN either by secular processes in a spiral galaxy, or by a merger. The cold
gas is funneled into the nucleus and eventually forms a thin, radiatively
efficient disk, that co-rotates with the black hole in the sense that the
angular momentum vectors of the black hole and disk are aligned. This is the
most likely configuration that results from the Bardeen-Petterson effect
(Bardeen & Petterson 1974). Because the innermost stable circular orbit (ISCO)
is closer to the black hole than when the angular momenta are anti-aligned,
the radiative efficiency of the accretion disk is high and this leads to
powerful, radiatively-driven winds, as shown by the arrows in red in Figure 2.
Accretion onto the black hole further spins the black hole up and the ISCO,
therefore, moves further inward toward the black hole. The value of black hole
spin is shown on the right and the rate of star formation is shown on the
left. The feedback on the star formation rate was described in Singh et al
2021. Such systems are subject to jet suppression (Ponti et al 2012; Garofalo,
Evans & Sambruna 2010; Nielsen & Lee 2009), which kicks in around a spin of
0.7, and therefore become radio quiet or jetless AGN. Because the spin is
above 0.7 in Figure 2, such objects are born jetless. Our model has a key
parameter of spin alignment and not just the general agreement that radio loud
(jetted) AGN tend to host SMBH exceeding $10^{8}$ $M_{\odot}$ (Chiaberge &
Marconi 2011). Accretion adds to the mass of the black hole while the
radiative wind contributes to the velocity dispersion (e.g. Silk & Rees 1998).
The cartoons in the column labeled $\sigma$ capture the relative scale of the
effect of the wind on the stellar dispersion via the length of the white
arrows. The arrows in Figure 2 are of intermediate length compared to those in
Figures 3, 4, and 5. Co-rotating systems, again, constitute the majority of
AGN, and Figure 2 objects purport to explain equation 1 and are captured as
the blue line in both Figures 1 and 6. It is important to emphasize that
Figures 2, 3, 4, and 5 are meant to produce a holistic (but not quantitative)
picture of differences in black hole feedback. This does not preclude
quantitative estimates for feedback and timescales for that feedback. While
Figure 1 has low redshift observational data, Figure 6 describes the evolution
in time on the M-$\sigma$ plane for two black holes. The slope of the blue
line captures the increase in black hole mass that is associated with an
increase in the stellar dispersion, through the disk wind, in such systems. If
the black hole spin is lower than in Figure 2, an FRI jet may exist. This is
where the model addresses so-called jetted narrow line Seyfert 1 but does not
add anything deep to the ideas here because jet feedback is not great. In
other words, black holes that are triggered in corotation for any spin value
tend to grow black holes and affect stellar dispersion in a mild way, unlike
for black holes that are triggered in counter-rotation - and especially for
those in richer environments - as we now describe.

In Figure 3 we show the average evolution in time of black holes in isolated
environments such that their black holes are not the most massive but are
triggered by mergers, start with the same spin value as in Figure 2, but
differ in the orientation of the accretion disk, which ends up in a
counterrotating configuration with respect to the black hole. I.e. the angular
momenta of black hole and disk are anti-aligned. These are difficult to form
(which is why Figure 2 captures most post-merger systems in these isolated
field environments as well) because of stability issues (Garofalo, North,
Belga, Waddell 2020; King et al 2005) and are therefore a minority.
Counterrotation is key to the production of FRII jets (Garofalo, Evans &
Sambruna 2010), which results from maximizing both the Blandford-Znajek (BZ)
and Blandford-Payne (BP) effects. Notice that such systems are not hosted in
spiral galaxies because they are the product of mergers. FRII jets increase
the star formation rate as shown in the left column (e.g. Kalfountzou et al
2014; Singh et al 2021). Although the radiative wind from the disk contributes
to the stellar dispersion, it is less effective than in Figure 2 systems
because the radiative efficiency of the disk is lower in counterrotation due
to the larger value of the ISCO compared to corotation (compare lowest panels
of Figure 2 and Figure 3). Notice how the inner edge of the disk evolves from
large value (lowest panel of Figure 3) to low value (highest panel of Figure
3).

As the black hole spin changes in time due to accretion, the radiative wind
increases in tandem with the decreasing distance of the ISCO. But when the
system enters the corotating phase, notice the existence of an FRI jet and an
associated increase in the size of the white arrows for the column labeled
$\sigma$. FRI jets, in fact, have the most dominant effect on stellar
dispersion. This is due to the fact that in the transition through zero black
hole spin, the disk experiences a shift in orientation (Garofalo, Joshi et al
2020). Once the black hole spin increases sufficiently (a = 0.2) such that an
FRI jet is produced, it is also likely to be tilted with respect to the
previous FRII jet, and the interstellar medium is now sprayed head on by the
jet. This is the reason why FRI jets suppress star formation (the Roy
Conjecture – Garofalo, Moravec et al 2022) while having a positive feedback
effect on stellar dispersion.

The possibility of a tilted jet forming in the transition from counter-
rotation to co-rotation is key to understanding the different behavior in the
M-$\sigma$ plane for jetted AGN. But these isolated systems described in
Figure 3 remain radiatively efficient and are therefore subject to the same
jet suppression at high spin of Figure 2, hence the jet disappears. Once the
FRI jet disappears (highest panel of Figure 3), the dominant effect on stellar
dispersion drops and the white arrows in the $\sigma$ column shorten again.
The period during which the FRI jet exists is a few tens of millions of years.
Compared to Figure 2, Figure 3 systems appear to experience lower
contributions to stellar dispersion earlier, and then make up for that later
with the FRI jet. Because of the larger gap region between disk and black hole
in Figure 3 systems, the disk wind is weaker compared to Figure 2 systems
which means that Figure 3 AGN move more upwards in the M-$\sigma$ plane
initially compared to Figure 2 AGN (This is also true of the more dominant
jetted systems we will describe next and is the reason why the red curve (RLQ)
in Figure 6 is initially leftward of the blue curve (RQQ)). The
counterrotating black holes of Figure 3, therefore, must possess more massive
black holes to achieve a given $\sigma$ value. Because of the dominant effect
of FRI jets, such systems will eventually shift to more horizontal motion in
the M-$\sigma$ diagram compared to Figure 2 systems once corotation is
achieved. We now move on to systems whose FRI jets are more dominant.

In Figure 4, we show the average time evolution of black hole systems that
formed via mergers in denser environments that lead to initial conditions that
involve more massive black holes compared to Figure 3 but that also involves
counterrotation (here too, most post-merger systems end up in corotation as in
Figure 2). The same initial conditions of Figure 3 explains the FRII jet and
the high rate of star formation as well as the weaker contribution to stellar
dispersion compared to Figure 2. Crucially, however, the FRII jet feedback is
greater than that of Figure 3 and this leads to the change in accretion state
at late times (see advection dominated accretion flow (ADAF) in the upper
panel – Garofalo, Evans & Sambruna 2010). Because of the absence of a thin
disk at late times, jet suppression fails and the tilted FRI jet lives on as
long as accreting fuel persists. This enables such systems to further enhance
their stellar dispersions as seen by the large white arrows in the $\sigma$
column in the two upper panels of Figure 4. It is important to note that ADAF
accretion feeds black holes at lower rates compared to thin disks (lower than
$10^{-2}$ that of the Eddington rate). Therefore, upward migration slows down
while horizontal motion speeds up on the M-$\sigma$ diagram. Figure 4 systems
evolve in a similar way to Figure 3 systems on the M-$\sigma$ plane except at
late times when they can affect their stellar dispersions in a more dominant
way due to the tilted FRI jet.

In Figure 5, we show the average time evolution of the most massive black hole
systems triggered in the densest environments into counterrotation (Again, the
more likely post-merger state is accretion in corotation as in Figure 2). The
reader can anticipate that feedback will be similar to Figures 3 and 4, only
more dramatic due to the triggering of the most massive black holes. Because
of the dominant effect of FRII jets, such systems evolve rapidly away from
thin disks and into ADAF states, which means that once the spin is large
enough for the FRI jet to form, it will be part of the longest of any AGN
subclass (e.g. M87). Despite a period in which stellar dispersion is not
affected (because the disk is an ADAF and not radiatively efficient), the
contribution to stellar dispersion from FRI jets is most dominant and is
captured by the red path in Figure 6. Such systems move for billions of years
mostly horizontally in the M-$\sigma$ plane.

In order to capture the basic differences between the two kinds of M-$\sigma$
relations that are implied by Figures 2-5, we show, in Figure 6,
characteristic evolution timescales for the objects represented by Figure 2
(blue path) and Figures 4 and 5 (red path). In the lowest panels of Figures 2
and Figures 4 and 5, we are dealing with Eddington-limited accretion onto
rapidly rotating black holes in corotation and counterrotation. For purposes
of direct comparison, we pick a black hole that is triggered with the same
mass ($10^{9}$ solar masses). Because the inner edge of the disk is closer to
the black hole represented in Figure 2, its disk wind is more powerful than
that of Figures 4 and 5, which manifests itself with a larger $\sigma$ value
(the blue curve is slightly rightward of the red one). Counterrotation lasts
$10^{7}$ years for Figure 4 objects and 4 x $10^{8}$ years for Figure 5
objects because the accretion rates are at the Eddington limit for Figure 4
objects and at $10^{-2}$ the Eddington limit within 4 million years for Figure
5 objects. Given this information, one can evaluate the total accreted mass
during this phase. In the transition through zero spin, Figure 4 and 5 systems
experience absence of the Bardeen-Petterson effect and a possible tilt in the
accretion disk, which due to spin-up then leads to a tilted FRI jet and ADAF
accretion. This generates a dramatic shift in the M-$\sigma$ plane, with
slower black hole growth that begins around $10^{-3}$ the Eddington rate and
drops two or more orders of magnitude below that over order a billion years,
but larger velocity dispersions as shown in Figure 6 with the red curve
tilting rightward. These are the types of paths that lead to the low redshift
M-$\sigma$ relation. In short, our model teases out two different behaviors
associated with jetted and non-jetted AGN.

Figure 1: The M-$\sigma$ relation of Gultekin et al (2009). Equation 1 is
over-plotted as the blue-line. $M_{BH}$ is the mass of the black hole divided
by 1 solar mass. Figure 2: Most AGN will be triggered into corotation between
black hole and accretion disk, whether it results from a merger or secular
processes. If such systems are rapidly spinning black holes, they will simply
spin the black hole up further. The radiative wind from the disk will increase
slightly over time and its effect on the stellar dispersion will increase.
Figure 3: For the subclass of AGN that are in isolated environments such that
the merger leads to counter-rotation, the radiative wind from the disk will
contribute relatively less to the stellar velocity dispersion compared to
Figure 2 because counter-rotating disks have lower efficiency and thus weaker
disk winds. A short-lived FRI jet will also increase $\sigma$ for a few tens
of millions of years until jet suppression kicks in, and the system behaves
like the high-spinning black holes of Figure 2 thereafter. HERG = high
excitation radio galaxy. Counter-rotating phase underlined in red. Figure 4:
More massive black holes on average in denser group environments results in
greater AGN feedback on the disk which evolves into an ADAF. Such systems
differ from Figure 3 in that the FRI jet persists which leads to a prolonged
effect on the stellar dispersion and increased horizontal red paths in Figure
6. LERG = low excitation radio galaxy. Counter-rotating phase underlined in
red. Figure 5: Strongest AGN feedback and rapid evolution in ADAF accretion
states which allows the FRI jet to emerge early and have a dominant effect on
stellar dispersion. The feedback on stellar dispersion is weak in the second
panel from bottom, hence no image is shown. Such systems move more
horizontally in Figure 6 than any other AGN subclass.  Figure 6: Theoretical
M-$\sigma$ paths. Blue line: Radio quiet quasars (RQQ). Red path: Radio loud
quasars (RLQ). Both systems begin with a rapidly spinning, billion solar mass
black hole, surrounded by a thin accretion disk in corotation (blue) and
counterrotation (red). The blue system corresponds to Figure 2 while the red
captures average evolution between Figures 4 and 5. Because the ISCO is
smaller for the blue system, the disk wind more strongly affects stellar
dispersion. Feedback on stellar dispersion is reversed when the tilted FRI jet
is generated in the red systems. The change in the black hole engine depends
on the value of the spin which changes due to accretion in a quantifiable way.
Characteristic timescales for those changes are labeled on the upper
horizontal axis.  Figure 7: 3 GHz radio luminosity as a function of stellar
dispersion from Barisic et al (2017) showing absence of correlation between
FRII jet power and stellar velocities.  Figure 8: Virial mass versus stellar
dispersion from van der Wel et al (2021) as evidence for the behavior
predicted in Figure 6. Objects with redshift between 0.6 and 0.8 in red while
objects with redshift between 0.8 and 1 in blue.

The model features described above predict that for large(r) data sets, FRI
LERG will on average tend to have larger $\sigma$ values than FRII HERG. FRII
HERG are predicted to be phases in which the accretion rates are high and the
black hole mass is thus increasing at high rates compared to latter stages.
I.e. they are represented in the initial part of the red curve in Figure 6
with largest slope. Hence, on average, FRII HERG should also have lower black
hole masses than FRI LERG (e.g. Miraghaei & Best 2017).

Let us now describe the effects of AGN feedback on star formation as depicted
in the first column of Figures 2-5. Along with more rightward location on the
M-$\sigma$ diagram, FRI LERG systems should also experience the lowest star
formation rates of any other subclass of AGN (Figure 5 left column of upper
two panels). FRII HERG systems, on the other hand should experience high star
formation rates (left column and lowest panels of Figures 3, 4, and 5). This
has indeed been shown to be the case in the star formation – stellar mass
plane including the intermediate FRII LERG systems (Garofalo & Pouliasis
2022). Because FRII jets do not enhance stellar dispersion while FRI jets do,
jet power as a function of stellar dispersion should not show any correlation
if FRII HERG and FRII LERG are included. We choose only the FRII candidates
from Figure 6 of Barisic et al 2017 where they plot stellar velocity
dispersion and jet powers. For FRII objects and in Figure 7, we show the radio
luminosity (a proxy for jet power) versus stellar dispersion, supporting an
absence of correlation between jet power and stellar dispersion. Because
stellar dispersion increases over time once the FRI jet is triggered, the
prediction is that a correlation should instead exist between the dynamical
FRI jet lifetime and velocity dispersion, unlike for FRII sources.

The FRII phases in our figures are relatively short in the less dense
environments and associated with rapid black hole growth. Because the
accretion rate drops orders of magnitude below the Eddington limit during FRI
ADAF phases (upper panels of Figures 4 and 5), black hole growth slows down
dramatically within 10-100 million years from the triggering of the radio
galaxy. If we could capture the evolution with rapid snapshots in the early
phase, we would expect to see an early large slope followed by a bend toward a
lower slope to capture the dominant effect of tilted FRI jets on stellar
dispersion at late times . But this requires exploring M-$\sigma$ at higher
redshift. The bending in the M-$\sigma$ plane experienced by radio loud
quasars as they evolve into low excitation systems shown in Figure 6 is
difficult to tease out of an observational M-$\sigma$ diagram. It should be
possible to capture this effect with more data and higher redshift. This has
been done in van der Wel et al. (2021), although with virial mass. We take
their virial masses and stellar dispersions from LEGA-C with galaxies at
redshifts $0.6<z<1$ and plot them in Figure 8. We see the bending described in
Figure 6. While caution is in order here due to the fact that virial masses
are often systematically larger than dynamical masses, this should not affect
the bend.

In our final analysis, we note that we are not the first to suggest different
M-$\sigma$ relations. That separate M-$\sigma$ relations for Sersic and core-
Sersic galaxies exist was discovered recently (Sahu, Graham $\&$ Davis 2019).
They make a fit for both populations in the M-$\sigma$ plane and report them
in equations 4 and 6 which we plot in Figure 9. Notice that core-Sersic
galaxies tend to behave in a way that is similar to the RQQ in Figure 6. Core-
Sersic galaxies tend to have dimmer cores compared to Sersic galaxies (Graham
et al 2003), the origin of which is thought to be that such systems are formed
as the result of a dry merger with less gas in the central region and
therefore the merging black holes in the nucleus require stars to solve the
last parsec problem (Milosavjlevic $\&$ Merritt 2001). It has long been
recognized that the last parsec problem is less problematic for
counterrotating disk configurations (Nixon et al 2011). We are on a spectrum
here, where both wet and dry mergers that form co-rotating disk configurations
in their nuclei (again, this is the majority of post-merger configurations for
the nuclear disk), while struggling to merge their black holes, will appeal
less to stars in wet mergers than in dry ones due to the availability of cold
gas in the former. Counterrotating configurations, on the other hand, will not
need as much help to coalesce the black hole binary. This means that all
mergers that lead to co-rotating disk configurations will struggle to merge
their black holes and will have to eject from the nucleus a mass in stars that
can be up to the order of the binary mass (Milosavljevic $\&$ Merrit 2001),
significantly reducing the density in the core. As a result, co-rotating
accretion disks around newly merged black holes will both have lower core
density and thus brightness (core-Sersic profiles) and will tend to be non-
jetted AGN. RQQ are the brightest of non-jetted AGN because of the abundant
amount of cold gas that fuels the AGN, and therefore will have had an
opportunity to coalesce their binary black holes using gas, dust, and stars,
such that their core-Sersic profiles should be different from similar
coalescing black hole binaries that formed in dry mergers in the sense that
less stars are evacuated and the core-Sersic profile less steep. Note that
Figure 6 deals with extremes, i.e. most radio loud (or jetted) and most radio
quiet (or non jetted). Thus, such extremes are not expected to fully explain
Figure 9. It is interesting to note that the red curve of Figure 6 (RLQ) would
bend less away from the blue curve (RQQ) if we incorporated less extreme
objects. Along those same lines, note how Figure 9 curves intersect at lower
black hole mass compared to Figure 6. If we think of extreme jetted AGN or
RLQ, such objects are growing their black holes at the most rapid rate while
affecting their dispersions in the weakest way so it makes sense that
overtaking RQQ will occur at higher mass as seen in Figure 6.

While extreme RQQ and RLQ are prescribed to have larger and smaller M-$\sigma$
slopes, respectively, less extreme AGN will distribute themselves in-between.
Since the majority of post-merger black holes end up in co-rotating disk
configurations, if we consider systems such as those in Figure 2 but with
nascent black hole spin values in the range 0.2 $<$ _a_ $<$ 0.7, there is an
absence of jet suppression and a jetted AGN will form. Given the initial co-
rotating disk configuration, such systems are expected to be core-Sersic
galaxies and therefore the division between Sersic profile and radio loudness
begins to break down. We claim that such jetted AGN are the ones explored in
Capetti & Brienza (2023) and that a mixing, therefore, exists among jetted AGN
such that some have evacuated cores and others do not. The important point to
notice is that the jetted AGN subclass with evacuated cores cannot be the
extreme RLQ that we plotted in Figure 6. The absence of counter-rotation in
this core-Sersic class of jetted AGN ensures absence of low excitation and the
absence of tilted jets, and therefore absence of strong $\sigma$ values,
features that increase the slope of such objects on the M-$\sigma$ plane. In
other words, as we consider less extreme jetted AGN, they will distribute
themselves on the M-$\sigma$ plane with slopes that are in-between the extreme
RLQ and RQQ and we will expect a mixing of core-Sersic and Sersic profiles
with jetted and non-jetted AGN. In addition, they will also necessarily not
experience the relatively low star formation rates associated with tilted jet
suppression of star formation. These are the objects described recently in
Capetti & Brienza (2023) but see also Hamilton (2010).

Figure 9: Fits for the M-$\sigma$ relation for Sersic (red) and core-Sersic
(blue) ellipticals (equations 4 $\&$ 6) from Sahu, Graham & Davis 2019. The
red dot represents M87, an FRI LERG in a cluster environment.

##  3 Conclusions

The strong correlation of central black hole mass and stellar velocity
dispersion (M-$\sigma$ plane) has been the subject of debate, but no clear
picture has emerged. The gap paradigm for black holes has given us a roadmap
for understanding the radio loud/radio quiet dichotomy, with the former the
result of original counterrotation between black hole and accretion disk
(Garofalo, Evans & Sambruna 2010). We have shown here how to understand the
location of different subclasses of AGN in the M-$\sigma$ plane based on a
shift from relatively stronger feedback in the jetted subclass compared to
non-jetted AGN. For two post-merger systems accreting at the Eddington limit,
the system that produces a powerful FRII jet is in counterrotation, which
makes its gap region between disk and black hole larger. This makes for an
initial weaker disk wind and a weaker impact on $\sigma$. FR II sources show
an absence of correlation between jet power and stellar dispersion. Hence,
jetted AGN tend to begin leftward of non-jetted AGN on the M-$\sigma$ plane
and this weaker coupling makes them travel more upwards than horizontally on
the diagram. But once the transition to FRI jets is instantiated, such systems
generate a dominant effect on $\sigma$. But this tends to occur when accretion
rates drop dramatically. Hence, the most powerful jetted AGN live long periods
by moving mostly horizontally in the M-$\sigma$ plane. We have attempted to
separate the paths of objects in the M-$\sigma$ plane that react in different
ways on star formation, stellar dispersion, accretion, and with different
Sersic profiles. Because the FRII to FRI transition in radio galaxies suggests
a bending path for radio loud AGN in the M-$\sigma$ plane as shown in Figure
6, we argued that if we go to higher redshift, we will begin to capture more
such transition objects and this will result in a bend such as seen in Figure
8. However, if we imagine going to even higher redshift, to redshift of 2 or
so, which often corresponds to the peak in estimated merger functions, one
predicts that the high $\sigma$ objects should appear to drop out of the
diagram. This is because such objects have not had time to form (i.e. FRI
LERG). This will be the subject of future work. Since radio loud AGN are about
10-20 % of the total AGN population, it is interesting to explore whether the
bending we are attributing to them in virial mass versus stellar dispersion
(Figure 8) can be achieved by such a small fraction of the population. But if
the clustering of the LEGA-C sample is high, the percentage of radio loud AGN
would be even higher.

We thank the anonymous referee for interesting insights.

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#  Improving Conversational Recommender System by Pretraining Billion-scale
Knowledge Graph

Chi-Man Wong32,Fan Feng3, Wen Zhang4, Chi-Man Vong2, Hui Chen3, Yichi Zhang3,
Peng He3,
Huan Chen1, Kun Zhao3, Huajun Chen45  31Alibaba Group, 2University of Macau,
4Zhejiang University, China

###### Abstract

Conversational Recommender Systems (CRSs) in E-commerce platforms aim to
recommend items to users via multiple conversational interactions. Click-
through rate (CTR) prediction models are commonly used for ranking candidate
items. However, most CRSs are suffer from the problem of data scarcity and
sparseness. To address this issue, we propose a novel knowledge-enhanced deep
cross network (K-DCN), a two-step (pretrain and fine-tune) CTR prediction
model to recommend items. We first construct a billion-scale conversation
knowledge graph (CKG) from  information about users, items and converations,
and then pretrain CKG by introducing knowledge graph embedding method and
graph convolution network to encode semantic and structural information
respectively. To make the CTR prediction model sensible of current state of
users and the relationship between dialogues and items, we introduce user-
state and dialogue-interaction representations based on pre-trained CKG and
propose K-DCN. In K-DCN, we fuse the user-state representation, dialogue-
interaction representation and other normal feature representations via deep
cross network, which will give the rank of candidate items to be recommended.
We experimentally prove that our proposal significantly outperforms baselines
and show it’s real application in Alime.

###### Index Terms:

Conversational Recommender Systems, Knowledge Graph, Entity Representation,
CTR.

##  I Introduction

Conversational Recommender Systems (CRS)[1, 2, 3, 4, 5] are commonly used to
improve customer experience on E-commerce platforms. Many online stores have
customer service chatbots to help users find their ideal items, which
contributes to the revenue greatly. The goal of these chatbots is to identify
users’ intentions by multiple conversational interactions, and then recommend
a list of most suitable items to them. As shown in figure 1, in a certain
dialogue, when a user asks for recommendation, the CRS employs a three-stage
pipeline. First, it makes semantic analysis of the query, then gets a list of
candidate items, and finally ranks candidate items through deep click-through
rate (CTR) models.

Figure 1:  The workflow of K-DCN.

In this paper, we focus on the ranking process of CRS, thus the key task is to
estimate the CTR. CTR prediction models[6, 7, 8] with various deep networks
have been proposed and achieved good performance. However, they mainly rely on
abundant behavioral records of users on items, while these records might be
very sparse because of limited interactions with a few chatbots among millions
of online shops, which would lead to the over-fitting problem.

To solve the data scarcity and sparseness problem, we propose to make user,
item and conversation information into consideration, since certain preference
for items might be expressed in conversation.

To better organize and utilize information, our proposal mainly includes two
parts: 1) construct and pre-train of conversational knowledge graph (CKG); and
2) knowledge-enhanced deep cross network (K-DCN) is employed for fine-tuning.
Firstly, we construct a billion-scale CKG from information about user, item
and conversation, and then is employed to learn a pre-trained model to obtain
better entity representations. There are two common approaches to learn entity
representations in KGs. The one is Graph Neural Network(GNN) based methods[9,
10, 11, 12, 13] like GCN [9], which models structure information of an entity
by aggregating all its neighbors’ information. The other one is knowledge
graph emebdding (KGE) methods like translation-based methods[14, 15, 16, 17,
18, 19, 20, 21, 22], which are good at capture semantic information of
entities. Since both structure and semantic information of entities are
important, we combine GNN-based method and KGE method together for pre-
training CKG via joint learning. In the second part, we propose a novel two-
step ranking model, K-DCN, in which we introduce user-state representations
and dialogue-interaction representations based on pre-trained CKG, to help the
CTR prediction model sensible of the uses’ state and relationships between
dialogues and items. We experiment our proposal on real-life e-commerce CRS.
The results show that our proposal outperforms baselines and converge faster
during training. Finally, we also show the application of our proposal in
Alimi, a chatbot in Taobao platform.

In summary, our contributions are:

  * •

We propose to integrate both user, item and conversation information in the
form of conversation knowledge graph to solve the data scarcity and sparseness
problem of CTR prediction models in CRS.

  * •

We introduce a novel method to organize and utilize various information, via
first get good entity representations from billion-scale pre-trained CKG, and
then fine-tune K-DCN with user-state representations and dialogue-interaction
representation introduced.

  * •

We experimentally show that our proposal outperforms baselines and converge
faster during training on real-life e-commerce datasets, and we also show the
real application of our proposal on Alimi.

##  II Related Work

###  II-A Translation-based KG embedding

Translation-based methods adopt a scoring function $f(h,r,t)$ to measure the
plausibility of a fact (h,r,t) from KG. E.g., in TransE [14] assumes
$f(h,r,t)=\|\mathbf{h}+\mathbf{r}-\mathbf{t}\|$. TransH [15] projects entity
representation onto relation’ hyperplane before calculating score. TransR [16]
and TransD [17] project entities from entity space to relation space via
projection matrix. TransEdge [18] contextualizes relation by edge-centric
embedding. DistMult [19] captures relational semantics by the composition of
relation; RotateE [20] learns various relation patterns by modeling relation
as rotation. However, all of the above methods do not learn the structure
information of KG, which is an important information to provide robustness of
the model.

###  II-B GCN-based KG embedding

GCN-based method represents the embedding of each entity by iteratively
propagating neighbor information. For example, GCN [9] introduces a first-
order approximation of ChebNet to perform graph convolution so that the number
of parameters are restrained and the issue of over-fitting is avoided; AGCN
[10] could learn all graph structure information by introducing a distance
metric learning; DGCN [11]propose a dual graph convolution to encodes both
local and global structure information by normalized adjacency and positive
pointwise mutual information (PPMI) matrix; GAT [12] employed masked self-
attentional layer to learn the importance of neighborhoods’ information by
assigning different weights to them; GeniePath [21] learns the importance of
different sized neighborhoods by using an adaptive path layer, such that both
breadth and depth can be explored for information extraction; Gaan [13]
control each attention head’s importance by sub-convolution network; Although
these methods could encode the neighbor’s information of an entity, the
semantic meaning can not be learned, which would deteriorate the performance.

###  II-C Deep Click-through Rate Prediction model

The development of deep neural networks (DNNs) and embedding techniques
provides a better way to learn feature expression in recommender systems. For
example, Deep & Cross Network(DCN) [6] and DeepFM [7, 23] contain both deep
component and shallow component to capture the feature interaction
automatically. In common Click-through Rate Prediction model, the input data
with sparse and dense features mainly concern about user action and product
attribute, and only use the embedding technique to reduce the dimensionality
of categorical features. But in dialog system, the input must be more specific
about current status. Inspired by the DKN [8], the knowledge-level embeddings
of entities on the dialog system can enrich the use state and dialogue
interaction representation directly.

Figure 2:  The architecture of knowledge-enhanced deep cross network (K-DCN).
Figure 3:  The conversation knowledge graph.

##  III Method

Our method contains two parts: 1) constructing a Conversation Knowledge
Graph(CKG) and pre-training it to encode structure and semantic information;
2) fine-tuning a ranking model DCN based on normal features and features from
pre-trained knowledge graph representations, including user-state
representations and dialogue-interaction representations.

###  III-A Constructing and Pre-training of CKG

####  III-A1 Conversation Knowledge Graph Construction

We make knowledge graph as a way to encode diverse information related to
conversational recommendation system and we construct the CKG from real-world
scenarios of user-chatbot conversation. There are three kinds of information
contained in CKG:

  * •

Information about users: in our platform, each user has a list of tags, such
as gender, shopping history, etc, which could help to catch the interest of
users. Thus we build triples encoding the tag information in CKG in the form
of (user, has, user tag).

  * •

Information about items: our platform contains rich information about items
like category and seller of items, and also properties of items such as color,
brand, etc. For categories and seller information, we build triple in the form
of (item, belong to, category) and (seller, has, item) respectively. For
properties and values, two kinds of triples are built (item, has, value) and
(property, has, value). This item information could contribute the mapping to
users interests greatly.

  * •

Information about conversation: in the chatbot, many sessions are created when
user chat with the bot, the intentions and keywords from the conversation
could help understand the intention of a user more precisely. Thus,we could
build four kinds of triples based on conversation history, (user, created,
session), (session, relate to, seller), (session, has, intention) and
(intention, has, keywords).

The schema of CKG is shown in figure 3. Finally, CKG contains 600 million
entities and 6 billion triples.

####  III-A2 Conversation Knowledge Graph Pre-training

As known that structural and semantic information are both valuable in
knowledge graph, thus our CKG pre-training contains two modules, structural
embedding module and semantic embedding module to capture them respectively.

_Structural Embedding Module:_ Graph neural network(GNN) has been proved to be
useful in encoding structural information[9, 10, 11, 12, 13]. Thus we employ
GCN [9] to learn the structur representation of entities in CKG where a vector
representation $\mathbf{e}\in\mathbb{R}^{d}$ for each entity
$e\in\mathcal{K}$. Gathering all entity embedding together in order, we get
$\mathbf{E}\in\mathbb{R}^{n_{e}\times d}$, where $n_{e}$ is the number of
entities and $d$ is the embedding dimension. A multi-layer GCN is given with a
simple layer-wise propagation rule and the $i$th layer could be represented
as:

$X^{(i+1)}=\sigma(\hat{D}^{-\frac{1}{2}}\hat{A}\hat{D}^{-\frac{1}{2}}X^{(i)}W^{(i)})$
(1)

where $\hat{A}\in\mathbb{R}^{n\times n}$ is adjacency matrix where
$\hat{A}_{ij}=1$ if $e_{i}$ and $e_{j}$ are connected in CKG and
$\hat{A}_{ij}=0$ otherwise. $\hat{D}$ is the diagonal degree matrix of
$\hat{A}$. $n$ is the number of entities in CKG. $\sigma$ is the activation
function and we make it sigmoid function during experiments. $W^{(i)}$ is the
weight matrix in $i$th layer. The output $X^{(i+1)}$ encodes the structure
information from $i$th layer with $X^{(i)}$ as structure information of
previous layers. In the first layer, we make $X^{(1)}=\mathbf{E}$.

_Semantic Embedding Module:_ Many knowledge graph embedding(KGE) methods are
proposed to encode the semantic information implicitly. Considering the
effectiveness and efficiency, we employ TransE in semantic embedding module.
For a triple $(e_{i},r,e_{j})$, the score function of it is defined as:

$f(e_{i},r,e_{j})=||\mathbf{e}_{i}+\mathbf{r}-\mathbf{e}_{j}||$ (2)

where $\mathbf{e}_{i},\mathbf{r}$ and $\mathbf{e}_{j}$ are embeddings of
$e_{i},r$ and $e_{j}$ respectively. With this function, positive triples
should get lower scores and negative ones get higher ones.

_Pretraining:_ we jointly train structure embedding module and semantic
embedding module to learn entity and relation embeddings. During co-training,
we make the output of multi-layer GCN $X^{(i+1)}$ as the entity embedding
matrix for TransE which means $\mathbf{e}_{k}=X_{k}^{(i+1)}(k\in[0,n-1])$. The
training target to minimize is the following margin-based ranking loss:

$L=\sum_{(e_{i},r,e_{j})\in\mathcal{T}}[f(e_{i},r,e_{j})+\gamma-f(e_{i}^{\prime},r^{\prime},e_{j}^{\prime})]_{+}$
(3)

where $(e_{i}^{\prime},r^{\prime},e_{j}^{\prime})$ is the negative sample of
$(e_{i},r,e_{j})$ by randomly replace $e_{i}$ or $e_{j}$ with $e\in CKG$.
Function $[x]_{+}=0$ if $x<0$, otherwise $[x]_{+}=x$. After construction and
pre-training of CKG, the trained entity embeddings will be used to fine-tune
CTR prediction models to improve conversational recommendation system.

###  III-B K-DCN for Fine-tuning

Normally, common sparse and dense features, eg. statistical features, are used
in CTR prediction models, while in conversation recommendation system, user’s
immediate question to chatbot is also important for understanding user’s
intention, thus we consider and construct user-state and dialogue-interaction
representations based on pre-trained CKG as additional features.

####  III-B1 User-state Representation

Users’ state could be captured by their previous behaviors. For one user,
suppose he/she has $k$ behaviors
$\mathcal{B}=\\{b_{i}|b_{i}=\\{e_{1}^{(i)},e_{2}^{(i)},...\\},i\in[1,k]\\}$
where $b_{i}$ is an user-item click sequence. For each $b_{i}$, we first
average the embedding of each item $e\in b_{i}$ from pre-trained CKG and get a
behavior vector $\mathbf{b}_{i}$:

$\mathbf{b}_{i}=\frac{1}{|b_{i}|}\sum_{e_{j}^{(i)}\in
b_{i}}\mathbf{e}_{j}^{(i)}$ (4)

Then we vertically concatenate all behavior vectors as
$\mathbf{B}\in\mathbb{R}^{d\times k}$ as shown in Figure (2) and employ
convolutional neural networks(CNN)[24] on $\mathbf{B}$ to model the local
information of the user-state representations:

$\mathbf{u}=f_{CNN}(\mathbf{B}*\mathbf{w}+b)$ (5)

where $*$ is is the convolution operator, and $b\in\mathbb{R}$ is a bias. We
show the details of this step in right side of Figure (2) marked as user-state
representation layer.

####  III-B2 Dialogue-interaction Representation

To build dialogue interactions, we first extract a set of keywords from users
question $\mathcal{W}_{query}=\\{w_{q}^{(1)},w_{q}^{(2)},...,w_{q}^{(m)}\\}$.
Given a candidate item $c$, we also extract a set of keywords from its title
$\mathcal{W}_{title}=\\{w_{t}^{(1)},w_{t}^{(2)},...,w_{t}^{(n)}\\}$. Then, we
gather all keywords together and get their representations from pre-trained
CKG’s entity embeddings, represented as
$\mathbf{W}=\\{\mathbf{w}^{(1)},\mathbf{w}^{(2)},...,\mathbf{w}^{(m+n)}\\}$.
Then we fed keyword embeddings into a multi-head self attention network [25]
to model the inter-relationship between query and candidate item. Formally:

$a_{ij}=\frac{exp((\mathbf{M}_{a}\mathbf{w}_{i})\circ(\mathbf{M}_{b}\mathbf{w}_{j})))}{\sum_{t=1}^{m+n}exp((\mathbf{M}_{a}\mathbf{w}_{i})\circ(\mathbf{M}_{b}\mathbf{w}_{k}))}$
(6)

where $\circ$ is inner-product operation and
$\mathbf{W}_{a}\in\mathbb{R}^{d\times d}$ and
$\mathbf{W}_{b}\in\mathbb{R}^{d\times d}$ are matrices transforming input
embeddings to a new space. where
$\psi(\mathbf{w}_{i},\mathbf{w}_{j})=innerproduct({M_{a}\mathbf{w}_{i},{M}_{b}\mathbf{w}_{j}})$
is the attention function.${M}_{a},{M}_{b}$ are matrix which

Next, we update the representation through a weighted sum according to
$a_{ij}$:

$\widetilde{\mathbf{w}}^{(i)}=\sum_{j=1}^{m+n}a_{i,j}(\mathbf{W}_{v}\mathbf{e}^{(i)})$
(7)

Finally, we concatenate all updated word embeddings as the dialogue-
interaction representation $\mathbf{d}\in\mathbb{R}^{(m+n)\times d}$:

$\mathbf{d}=[\widetilde{\mathbf{w}}^{(1)},\widetilde{\mathbf{w}}^{(2)},...,\widetilde{\mathbf{w}}^{(m+n)}]$
(8)

####  III-B3 K-DCN Model

Inputs of K-DCN are various feature representations for a candidate item $e$.
Besides aforementioned user-state and dialogue interaction representations
$\mathbf{u}$ and $\mathbf{d}$, other common features are also considered,
including categorical features and statistical features. First, we concatenate
all feature vectors as follows:

$\mathbf{f}=[\mathbf{c}_{e}^{(1)},\mathbf{c}_{e}^{(2)},...,\mathbf{c}_{e}^{(k)},\mathbf{x}_{dense},\mathbf{u},\mathbf{d}]$
(9)

where $\mathbf{c}_{e}^{(1)},\mathbf{c}_{e}^{(2)},...,\mathbf{c}_{e}^{k}$ are
embeddings of categories that target item $e$ belongs to, which will be
randomly initialized during training. $\mathbf{x}_{dense}$ is statistical
feature vector of $e$ where values could be price, sales etc.

With the feature representation $\mathbf{f}$ for $e$, we feed it into the
cross network and deep network respectively which are two common components in
CTR prediction model. The cross network is composed of $n_{c}$ cross layers
and the $i$th layer could be represented as:

$\mathbf{x}^{(i+1)}_{c}=\mathbf{f}(\mathbf{x}^{(i)}_{c})^{T}\mathbf{w}_{i}+\mathbf{b}_{i}+\mathbf{x}^{(i)}_{c}$
(10)

where $\mathbf{x}_{c}^{0}=\mathbf{f}$. The deep network is a $n_{d}$ layers of
fully-connected feed-forward network and the function of the $i$th layer is

$\mathbf{x}^{(i+1)}_{d}=\sigma\left(W_{i}\mathbf{x}^{(i)}_{d}+\mathbf{b}_{l}\right)$
(11)

Then we concatenate the two outputs from the above two networks and feed the
concatenated vector into a standard logits layer. Formally:

$p=\sigma\left(\left[\mathbf{x}_{c}^{(n_{c})},\mathbf{x}_{d}^{(n_{d})}\right]\mathbf{W}_{\text{logits
}}\right)$ (12)

To train K-DCN, we minimize the following log likelihood loss function:

$L_{K-DCN}=-\frac{1}{N}\sum_{i=1}^{N}y_{i}\log\left(p_{i}\right)+\left(1-y_{i}\right)\log\left(1-p_{i}\right)$
(13)

where $N$ is the number of input samples. $y_{i}\in\mathbb{R}^{n_{cand}\times
1}$ is the recommendation label of the $i$th sample and $n_{cand}$ is the
number of candidate items to be recommended.

(a) example A

(b) example B

Figure 4: Two Real-word examples of conversation from Taobao with English
translation on the right.

##  IV Experiment and Discussion

###  IV-A Datasets

K-DCN is pretrained and fine-tuned on a private dataset from scratch. The
dataset for training CTR prediction model is from the conversation of Alime
salebot (As illustrated in Figure 1). As shown in table I,We have sampled 900K
records from 10 categories.

###  IV-B Implementation Details

  * •

CKG. The pre-trained KG model is trained on billion-scale converation
knowledge graph (CKG). CKG has a total number of 500M entities and 6B
triplets. We processd the data into triplets using Alibaba max-reduce
framework (called Maxcompute). We removed the attributes with occurrences that
are less than 5000 in CKG. Such attributues are very sparse that are likely to
deteriorate the model performance. For trainging, we employed the tool Graph-
learn from [26] to perform sampling of 10 neighbors for structure pretrain,
and the batch size is 512. Adam optimizer is employed with initial learning
rate = 0.0001; each training batch size = 1000, and node embedding size = 64.
The model is trained with 50 parameter servers and 200 GPUs for 5 epochs. The
whole training consumed 2 days.

  * •

K-DCN. The K-DCN is implementated on Tensorflow. The hyperparameters are tuned
with grid search. The optimal hyperparameter settings were 2 deep layers of
size 512 and 4 cross layers for the K-DCN model. More specifically, we use 4
kinds of behaviors in user state representation and the size of the
convolutional operator is $N\times itemEmbeddingSize$, where $N$ is 2 and 4.
And the attention head number for text embedding is 4. For different size of
convolutional operator, the outputs are concated as final output. We use mini-
batch stochastic optimization with Adam [27] optimizer and the batch size is
set at 512.

###  IV-C Results and Discussion

We compare our methods with 5 baselines: Deep & Cross Network (DCN) [6] ,
Wide&Deep Network (WDN)[28], Deep Neural Network (DNN) [29], Gradient Boosted
Decision Trees (GBDT)[30], and Logistic Regression (LR) [31] in 10 datasets,
and the results are shown in II. The proposed K-DCN outperforms all the
baselines in all datasets.

Specifically, K-DCN achieves a higher score of 2.6% for the digital
accessories category. The main reason is that the number of training data is
very limited for digital accessories category, which verifies our model which
was pretrained from conversation knowledge can alleviate the issue of data
scarcity. For all datasets, K-DCN has an average improvement of 1.2%, which
verifies that the proposed KG embeddings from conversation knowledge graph can
provide meaningful and useful structure and semantic information to CTR
prediction model. What’s more, as shown in Figure 5, it would also accelerate
the convergence when the pretrained KG embedding were used to initialize the
CTR prediction model. We have also tested our K-DCN with DCN in real
application in Alime (as shown in Figure. 4. From table III, K-DCN has
relative improvement of 4.2% over DCN, which proved the effectiveness of
K-DCN.

TABLE I:  The number of training and testing data for deep CTR prediction model. Category | # Train | # Test
---|---|---
Beauty Skin Care/Body Care/Essential Oil(BS) | 157027 | 16028
Women’s wear(MW) | 139167 | 14205
Sports shoes(SS) | 104182 | 10634
Women’s and Men’s underwear/Home wear(WN) | 98705 | 10075
Sports clothes/Casual wear(SC) | 93180 | 9511
Bed linings (BL) | 87537 | 8935
Cleanser/Sanitary Napkin/Paper/Aromatherapy(CS) | 58410 | 5962
Men’s clothes(MC) | 57263 | 5845
Digital accessories(DA) | 49455 | 5048
Kitchen appliances(KA) | 55069 | 5621
TABLE II:  K-DCN and 5 baselines’ performance in 10 datasets. Category | AUC
---|---
K-DCN | DCN | WDL | DNN | GBDT | LR
BC | 0.617 | 0.608 | 0.6 | 0.598 | 0.584 | 0.58
WM | 0.652 | 0.636 | 0.63 | 0.633 | 0.611 | 0.592
SS | 0.658 | 0.652 | 0.65 | 0.649 | 0.619 | 0.603
WN | 0.663 | 0.65 | 0.647 | 0.646 | 0.616 | 0.594
SC | 0.667 | 0.66 | 0.653 | 0.65 | 0.618 | 0.604
BL | 0.631 | 0.62 | 0.61 | 0.604 | 0.598 | 0.574
CS | 0.625 | 0.624 | 0.622 | 0.621 | 0.61 | 0.602
MC | 0.647 | 0.63 | 0.628 | 0.626 | 0.603 | 0.574
DA | 0.706 | 0.68 | 0.677 | 0.671 | 0.675 | 0.646
KA | 0.654 | 0.64 | 0.642 | 0.64 | 0.638 | 0.622
Average | 0.652 | 0.64 | 0.636 | 0.633 | 0.617 | 0.59
TABLE III:  The online AB Test in Alime. Platform | % Improvement of K-DCN over DCN
---|---
Alime | 4.2%
Figure 5: Convergence speed comparison between K-DCN and DCN.

##  V Conclusion

In this paper, we put forward a two-step CTR prediction model K-DCN. We first
propose a pretrained model KGM based on a billions scale conversation
knowledge graph. Then we propose a knowledge-enhanced deep cross network
(K-DCN) model based on KGM as well as some dense and sparse features.
Experimental results show that our model obtains substantial performance on 10
datasets compared to baselines. K-DCN have been applied to a real
conservational chatbot in one of the largest E-commerce company. In future
work, we plan to design a learning framework which can uniformly exploit
evidence from heterogeneous knowledge sources, such as entity descriptions and
knowledge schema.

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