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If both probability measures have smooth densities, the total variation can be bounded by Wasserstein (even by a weaker metric Levy-Prokhorov), see this paper.

More precisely, see Lemma 5.1 and proof of Theorem 2.1. The basic idea is given below.

If both $p$ and $q$ are smooth densities, $d_V(p, p_\gamma)$ and $d_V(q, q_\gamma)$ are sufficiently small for every small gamma, where $d_V$ is the total variation and $p_\gamma$ is a convolution of p and the uniform density as defined in the paper.

Note that $d_V(p, q) < d_V(p, p_\gamma) + d_V(p_\gamma, q_\gamma) + d_v(q_\gamma, q)$ by the triangle inequality.

Lemma 5.1 guarantees that if $p$ and $q$ are close in Levy-Prokhorov metric, then $d_V(p_\gamma, q_\gamma)$ is also small provided that $\gamma$ is much bigger than the Levy-Prochorov distance.

If both probability measures have smooth densities, the total variation can be bounded by Wasserstein (even by a weaker metric Levy-Prokhorov), see the 2017 paper A novel approach to Bayesian consistency, by myself and Stephen G. Walker.

More precisely, see Lemma 5.1 and proof of Theorem 2.1. The basic idea is given below.

If both $p$ and $q$ are smooth densities, $d_V(p, p_\gamma)$ and $d_V(q, q_\gamma)$ are sufficiently small for every small gamma, where $d_V$ is the total variation and $p_\gamma$ is a convolution of p and the uniform density as defined in the paper.

Note that $d_V(p, q) < d_V(p, p_\gamma) + d_V(p_\gamma, q_\gamma) + d_v(q_\gamma, q)$ by the triangle inequality.

Lemma 5.1 guarantees that if $p$ and $q$ are close in Levy-Prokhorov metric, then $d_V(p_\gamma, q_\gamma)$ is also small provided that $\gamma$ is much bigger than the Levy-Prochorov distance.

If both probability measures have smooth densities, the total variation can be bounded by Wasserstein (even by a weaker metric Levy-Prokhorov), see this paper.

More precisely, see Lemma 5.1 and proof of Theorem 2.1. The basic idea is given below.

If both p and q are smooth densities, d_V(p, p_gamma) and d_V(q, q_gamma) are sufficiently small for every small gamma, where d_V is the total variation and p_gamma is a convolution of p and the uniform density as defined in the paper.

Note that d_V(p, q) < d_V(p, p_gamma) + d_V(p_gamma, q_gamma) + d_v(q_gamma, q) by the triangle inequality.

Lemma 5.1 guarantees that if p and q are close in Levy-Prokhorov metric, then d_V(p_gamma, q_gamma) is also small provided that gamma is much bigger than the Levy-Prochorov distance.

If both probability measures have smooth densities, the total variation can be bounded by Wasserstein (even by a weaker metric Levy-Prokhorov), see this paper.

More precisely, see Lemma 5.1 and proof of Theorem 2.1. The basic idea is given below.

If both $p$ and $q$ are smooth densities, $d_V(p, p_\gamma)$ and $d_V(q, q_\gamma)$ are sufficiently small for every small gamma, where $d_V$ is the total variation and $p_\gamma$ is a convolution of p and the uniform density as defined in the paper.

Note that $d_V(p, q) < d_V(p, p_\gamma) + d_V(p_\gamma, q_\gamma) + d_v(q_\gamma, q)$ by the triangle inequality.

Lemma 5.1 guarantees that if $p$ and $q$ are close in Levy-Prokhorov metric, then $d_V(p_\gamma, q_\gamma)$ is also small provided that $\gamma$ is much bigger than the Levy-Prochorov distance.

If both probability measures have smooth densities, the total variation can be bounded by Wasserstein (even by a weaker metric Levy-Prokhorov), see this paper.

If both probability measures have smooth densities, the total variation can be bounded by Wasserstein (even by a weaker metric Levy-Prokhorov), see this paper.

More precisely, see Lemma 5.1 and proof of Theorem 2.1. The basic idea is given below.

If both p and q are smooth densities, d_V(p, p_gamma) and d_V(q, q_gamma) are sufficiently small for every small gamma, where d_V is the total variation and p_gamma is a convolution of p and the uniform density as defined in the paper.

Note that d_V(p, q) < d_V(p, p_gamma) + d_V(p_gamma, q_gamma) + d_v(q_gamma, q) by the triangle inequality.

Lemma 5.1 guarantees that if p and q are close in Levy-Prokhorov metric, then d_V(p_gamma, q_gamma) is also small provided that gamma is much bigger than the Levy-Prochorov distance.